Package 'CDM'

Title: Cognitive Diagnosis Modeling
Description: Functions for cognitive diagnosis modeling and multidimensional item response modeling for dichotomous and polytomous item responses. This package enables the estimation of the DINA and DINO model (Junker & Sijtsma, 2001, <doi:10.1177/01466210122032064>), the multiple group (polytomous) GDINA model (de la Torre, 2011, <doi:10.1007/s11336-011-9207-7>), the multiple choice DINA model (de la Torre, 2009, <doi:10.1177/0146621608320523>), the general diagnostic model (GDM; von Davier, 2008, <doi:10.1348/000711007X193957>), the structured latent class model (SLCA; Formann, 1992, <doi:10.1080/01621459.1992.10475229>) and regularized latent class analysis (Chen, Li, Liu, & Ying, 2017, <doi:10.1007/s11336-016-9545-6>). See George, Robitzsch, Kiefer, Gross, and Uenlue (2017) <doi:10.18637/jss.v074.i02> or Robitzsch and George (2019, <doi:10.1007/978-3-030-05584-4_26>) for further details on estimation and the package structure. For tutorials on how to use the CDM package see George and Robitzsch (2015, <doi:10.20982/tqmp.11.3.p189>) as well as Ravand and Robitzsch (2015).
Authors: Alexander Robitzsch [aut, cre], Thomas Kiefer [aut], Ann Cathrice George [aut], Ali Uenlue [aut]
Maintainer: Alexander Robitzsch <[email protected]>
License: GPL (>= 2)
Version: 8.3-1
Built: 2024-03-31 07:00:02 UTC
Source: https://github.com/alexanderrobitzsch/cdm

Help Index


Cognitive Diagnosis Modeling

Description

Functions for cognitive diagnosis modeling and multidimensional item response modeling for dichotomous and polytomous item responses. This package enables the estimation of the DINA and DINO model (Junker & Sijtsma, 2001, <doi:10.1177/01466210122032064>), the multiple group (polytomous) GDINA model (de la Torre, 2011, <doi:10.1007/s11336-011-9207-7>), the multiple choice DINA model (de la Torre, 2009, <doi:10.1177/0146621608320523>), the general diagnostic model (GDM; von Davier, 2008, <doi:10.1348/000711007X193957>), the structured latent class model (SLCA; Formann, 1992, <doi:10.1080/01621459.1992.10475229>) and regularized latent class analysis (Chen, Li, Liu, & Ying, 2017, <doi:10.1007/s11336-016-9545-6>). See George, Robitzsch, Kiefer, Gross, and Uenlue (2017) <doi:10.18637/jss.v074.i02> or Robitzsch and George (2019, <doi:10.1007/978-3-030-05584-4_26>) for further details on estimation and the package structure. For tutorials on how to use the CDM package see George and Robitzsch (2015, <doi:10.20982/tqmp.11.3.p189>) as well as Ravand and Robitzsch (2015).

Details

Cognitive diagnosis models (CDMs) are restricted latent class models. They represent model-based classification approaches, which aim at assigning respondents to different attribute profile groups. The latent classes correspond to the possible attribute profiles, and the conditional item parameters model atypical response behavior in the sense of slipping and guessing errors. The core CDMs in particular differ in the utilized condensation rule, conjunctive / non-compensatory versus disjunctive / compensatory, where in the model structure these two types of response error parameters enter and what restrictions are imposed on them. The confirmatory character of CDMs is apparent in the Q-matrix, which can be seen as an operationalization of the latent concepts of an underlying theory. The Q-matrix allows incorporating qualitative prior knowledge and typically has as its rows the items and as the columns the attributes, with entries 1 or 0, depending on whether an attribute is measured by an item or not, respectively.

CDMs as compared to common psychometric models (e.g., IRT) contain categorical instead of continuous latent variables. The results of analyses using CDMs differ from the results obtained under continuous latent variable models. CDMs estimate in a direct manner the probabilistic attribute profile of a respondent, that is, the multivariate vector of the conditional probabilities for possessing the individual attributes, given her / his response pattern. Based on these probabilities, simplified deterministic attribute profiles can be derived, showing whether an individual attribute is essentially possessed or not by a respondent. As compared to alternative two-step discretization approaches, which estimate continuous scores and discretize the continua based on cut scores, with CDMs the classification error can generally be reduced.

The package CDM implements parameter estimation procedures for the DINA and DINO model (e.g.,de la Torre & Douglas, 2004; Junker & Sijtsma, 2001; Templin & Henson, 2006; the generalized DINA model for dichotomous attributes (GDINA, de la Torre, 2011) and for polytomous attributes (pGDINA, Chen & de la Torre, 2013); the general diagnostic model (GDM, von Davier, 2008) and its extension to the multidimensional latent class IRT model (Bartolucci, 2007), the structure latent class model (Formann, 1992), and tools for analyzing data under the models. These and related concepts are explained in detail in the book about diagnostic measurement and CDMs by Rupp, Templin and Henson (2010), and in such survey articles as DiBello, Roussos and Stout (2007) and Rupp and Templin (2008).

The package CDM is implemented based on the S3 system. It comes with a namespace and consists of several external functions (functions the package exports). The package contains a utility method for the simulation of artificial data based on a CDM model (sim.din). It also contains seven internal functions (functions not exported by the package): this are plot, print, and summary methods for objects of the class din (plot.din, print.din, summary.din), a print method for objects of the class summary.din (print.summary.din), and three functions for checking the input format and computing intermediate information. The features of the package CDM are illustrated with an accompanying real dataset and Q-matrix (fraction.subtraction.data and fraction.subtraction.qmatrix) and artificial examples (Data-sim).

See George et al. (2016) and Robitzsch and George (2019) for an overview and some computational details of the CDM package.

Author(s)

Alexander Robitzsch [aut, cre], Thomas Kiefer [aut], Ann Cathrice George [aut], Ali Uenlue [aut]

Maintainer: Alexander Robitzsch <[email protected]>

References

Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141-157.

Chen, J., & de la Torre, J. (2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37, 419-437.

Chen, Y., Li, X., Liu, J., & Ying, Z. (2017). Regularized latent class analysis with application in cognitive diagnosis. Psychometrika, 82, 660-692.

de la Torre, J., & Douglas, J. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.

de la Torre, J. (2009). A cognitive diagnosis model for cognitively based multiple-choice options. Applied Psychological Measurement, 33, 163-183.

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179–199.

DiBello, L. V., Roussos, L. A., & Stout, W. F. (2007). Review of cognitively diagnostic assessment and a summary of psychometric models. In C. R. Rao and S. Sinharay (Eds.), Handbook of Statistics, Vol. 26 (pp. 979–1030). Amsterdam: Elsevier.

Formann, A. K. (1992). Linear logistic latent class analysis for polytomous data. Journal of the American Statistical Association, 87, 476-486.

George, A. C., & Robitzsch, A. (2015) Cognitive diagnosis models in R: A didactic. The Quantitative Methods for Psychology, 11, 189-205. doi:10.20982/tqmp.11.3.p189

George, A. C., Robitzsch, A., Kiefer, T., Gross, J., & Uenlue, A. (2016). The R package CDM for cognitive diagnosis models. Journal of Statistical Software, 74(2), 1-24.

Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272.

Ravand, H., & Robitzsch, A.(2015). Cognitive diagnostic modeling using R. Practical Assessment, Research & Evaluation, 20(11). Available online: http://pareonline.net/getvn.asp?v=20&n=11

Robitzsch, A., & George, A. C. (2019). The R package CDM. In M. von Davier & Y.-S. Lee (Eds.). Handbook of diagnostic classification models (pp. 549-572). Cham: Springer. doi:10.1007/978-3-030-05584-4_26

Rupp, A. A., & Templin, J. (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement: Interdisciplinary Research and Perspectives, 6, 219–262.

Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic Measurement: Theory, Methods, and Applications. New York: The Guilford Press.

Templin, J., & Henson, R. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287–305.

von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287-307.

See Also

See the GDINA package for comprehensive functions for the GDINA model.

See also the ACTCD and NPCD packages for nonparametric cognitive diagnostic models.

See the dina package for estimating the DINA model with a Gibbs sampler.

Examples

##
##   **********************************
##   ** CDM 2.5-16 (2013-11-29)      **
##   ** Cognitive Diagnostic Models  **
##   **********************************
##

Likelihood Ratio Test for Model Comparisons

Description

This function compares two models estimated with din, gdina or gdm using a likelihood ratio test.

Usage

## S3 method for class 'din'
anova(object,...)

## S3 method for class 'gdina'
anova(object,...)

## S3 method for class 'gdm'
anova(object,...)

## S3 method for class 'mcdina'
anova(object,...)

## S3 method for class 'reglca'
anova(object,...)

## S3 method for class 'slca'
anova(object,...)

Arguments

object

Two objects of class din, gdina, mcdina, slca, gdm, reglca

...

Further arguments to be passed

Note

This function is based on IRT.anova.

See Also

din, gdina, gdm, mcdina, slca

Examples

#############################################################################
# EXAMPLE 1: anova with din objects
#############################################################################

# Model 1
d1 <- CDM::din(sim.dina, q.matr=sim.qmatrix )
# Model 2 with equal guessing and slipping parameters
d2 <- CDM::din(sim.dina, q.matr=sim.qmatrix, guess.equal=TRUE, slip.equal=TRUE)
# model comparison
anova(d1,d2)
  ##     Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2    d2 -2176.482 4352.963     9 4370.963 4406.886 268.2071 16  0
  ##   1    d1 -2042.378 4084.756    25 4134.756 4234.543       NA NA NA

## Not run: 
#############################################################################
# EXAMPLE 2: anova with gdina objects
#############################################################################

# Model 3: GDINA model
d3 <- CDM::gdina( sim.dina, q.matr=sim.qmatrix )

# Model 4: DINA model
d4 <- CDM::gdina( sim.dina, q.matr=sim.qmatrix, rule="DINA")

# model comparison
anova(d3,d4)
  ##     Model   loglike Deviance Npars      AIC      BIC    Chisq df       p
  ##   2    d4 -2042.293 4084.586    25 4134.586 4234.373 31.31995 16 0.01224
  ##   1    d3 -2026.633 4053.267    41 4135.266 4298.917       NA NA      NA

## End(Not run)

Cognitive Diagnostic Indices based on Kullback-Leibler Information

Description

This function computes several cognitive diagnostic indices grounded on the Kullback-Leibler information (Rupp, Henson & Templin, 2009, Ch. 13) at the test, item, attribute and item-attribute level. See Henson and Douglas (2005) and Henson, Roussos, Douglas and He (2008) for more details.

Usage

cdi.kli(object)

## S3 method for class 'cdi.kli'
summary(object, digits=2,  ...)

Arguments

object

Object of class din or gdina. For the summary method, it is the result of cdi.kli.

digits

Number of digits for rounding

...

Further arguments to be passed

Value

A list with following entries

test_disc

Test discrimination which is the sum of all global item discrimination indices

attr_disc

Attribute discriminations

glob_item_disc

Global item discriminations (Cognitive diagnostic index)

attr_item_disc

Attribute-specific item discrimination

KLI

Array with Kullback-Leibler informations of all items (first dimension) and skill classes (in the second and third dimension)

skillclasses

Matrix containing all used skill classes in the model

hdist

Matrix containing Hamming distance between skill classes

pjk

Used probabilities

q.matrix

Used Q-matrix

summary

Data frame with test- and item-specific discrimination statistics

References

Henson, R., DiBello, L., & Stout, B. (2018). A generalized approach to defining item discrimination for DCMs. Measurement: Interdisciplinary Research and Perspectives, 16(1), 18-29. http://dx.doi.org/10.1080/15366367.2018.1436855

Henson, R., & Douglas, J. (2005). Test construction for cognitive diagnosis. Applied Psychological Measurement, 29, 262-277. http://dx.doi.org/10.1177/0146621604272623

Henson, R., Roussos, L., Douglas, J., & He, X. (2008). Cognitive diagnostic attribute-level discrimination indices. Applied Psychological Measurement, 32, 275-288. http://dx.doi.org/10.1177/0146621607302478

Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic Measurement: Theory, Methods, and Applications. New York: The Guilford Press.

See Also

See discrim.index for computing discrimination indices at the probability metric.

See Henson, DiBello and Stout (2018) for an overview of different discrimination indices.

Examples

#############################################################################
# EXAMPLE 1: Examples based on CDM::sim.dina
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

mod <- CDM::din( sim.dina, q.matrix=sim.qmatrix )
summary(mod)
  ##  Item parameters
  ##         item guess  slip   IDI rmsea
  ##  Item1 Item1 0.086 0.210 0.704 0.014
  ##  Item2 Item2 0.109 0.239 0.652 0.034
  ##  Item3 Item3 0.129 0.185 0.686 0.028
  ##  Item4 Item4 0.226 0.218 0.556 0.019
  ##  Item5 Item5 0.059 0.000 0.941 0.002
  ##  Item6 Item6 0.248 0.500 0.252 0.036
  ##  Item7 Item7 0.243 0.489 0.268 0.041
  ##  Item8 Item8 0.278 0.125 0.597 0.109
  ##  Item9 Item9 0.317 0.027 0.656 0.065

cmod <- CDM::cdi.kli( mod )

# attribute discrimination indices
round( cmod$attr_disc, 3 )
  ##      V1     V2     V3
  ##   1.966  2.506 11.169

# look at global item discrimination indices
round( cmod$glob_item_disc, 3 )
  ##  > round( cmod$glob_item_disc, 3 )
  ##  Item1 Item2 Item3 Item4 Item5 Item6 Item7 Item8 Item9
  ##  0.594 0.486 0.533 0.465 5.913 0.093 0.040 0.397 0.656

# correlation of IDI and global item discrimination
stats::cor( cmod$glob_item_disc, mod$IDI )
  ##  [1] 0.6927274

# attribute-specific item indices
round( cmod$attr_item_disc, 3 )
  ##           V1    V2    V3
  ##  Item1 0.648 0.648 0.000
  ##  Item2 0.000 0.530 0.530
  ##  Item3 0.581 0.000 0.581
  ##  Item4 0.697 0.000 0.000
  ##  Item5 0.000 0.000 8.870
  ##  Item6 0.000 0.140 0.000
  ##  Item7 0.040 0.040 0.040
  ##  Item8 0.000 0.433 0.433
  ##  Item9 0.000 0.715 0.715

## Note that attributes with a zero entry for an item
## do not differ from zero for the attribute specific item index

Utility Functions in CDM

Description

Utility functions in CDM.

Usage

## requireNamespace with package message for needed installation
CDM_require_namespace(pkg)
## attach internal function in a package
cdm_attach_internal_function(pack, fun)

## print function in summary
cdm_print_summary_data_frame(obji, from=NULL, to=NULL, digits=3, rownames_null=FALSE)
## print summary call
cdm_print_summary_call(object, call_name="call")
## print computation time
cdm_print_summary_computation_time(object, time_name="time", time_start="s1",
         time_end="s2")

## string vector of matrix entries
cdm_matrixstring( matr, string )

## mvtnorm::rmvnorm with vector conversion for n=1
CDM_rmvnorm(n, mean=NULL, sigma, ...)
## fit univariate and multivariate normal distribution
cdm_fit_normal(x, w)

## fit unidimensional factor analysis by unweighted least squares
cdm_fa1(Sigma, method=1, maxit=50, conv=1E-5)

## another rbind.fill implementation
CDM_rbind_fill( x, y )
## fills a vector row-wise into a matrix
cdm_matrix2( x, nrow )
## fills a vector column-wise into a matrix
cdm_matrix1( x, ncol )

## SCAD thresholding operator
cdm_penalty_threshold_scad(beta, lambda, a=3.7)
## lasso thresholding operator
cdm_penalty_threshold_lasso(val, eta )
## ridge thresholding operator
cdm_penalty_threshold_ridge(beta, lambda)
## elastic net threshold operator
cdm_penalty_threshold_elnet( beta, lambda, alpha )
## SCAD-L2 thresholding operator
cdm_penalty_threshold_scadL2(beta, lambda, alpha, a=3.7)
## truncated L1 penalty thresholding operator
cdm_penalty_threshold_tlp( beta, tau, lambda )
## MCP thresholding operator
cdm_penalty_threshold_mcp(beta, lambda, a=3.7)

## general thresholding operator for regularization
cdm_parameter_regularization(x, regular_type, regular_lam, regular_alpha=NULL,
         regular_tau=NULL )
## values of penalty function
cdm_penalty_values(x, regular_type, regular_lam, regular_tau=NULL,
       regular_alpha=NULL)
## thresholding operators regularization
cdm_parameter_regularization(x, regular_type, regular_lam, regular_alpha=NULL,
       regular_tau=NULL)

## utility functions for P-EM acceleration
cdm_pem_inits(parmlist)
cdm_pem_inits_assign_parmlist(pem_pars, envir)
cdm_pem_acceleration( iter, pem_parameter_index, pem_parameter_sequence, pem_pars,
      PEM_itermax, parmlist, ll_fct, ll_args, deviance.history=NULL )
cdm_pem_acceleration_assign_output_parameters(res_ll_fct, vars, envir, update)

## approximation of absolute value function and its derivative
abs_approx(x, eps=1e-05)
abs_approx_D1(x, eps=1e-05)

## information criteria
cdm_calc_information_criteria(ic)
cdm_print_summary_information_criteria(object, digits_crit=0, digits_penalty=2)

## string pasting
cat_paste(...)

Arguments

pkg

An R package

pack

An R package

fun

An R function

obji

Object

from

Integer

to

Integer

digits

Number of digits used for printing

rownames_null

Logical

call_name

Character

time_name

Character

time_start

Character

time_end

Character

matr

Matrix

string

String

object

Object

n

Integer

mean

Mean vector or matrix if separate means for cases are provided. In this case, n can be missing.

sigma

Covariance matrix

...

More arguments to be passed (or a list of arguments)

x

Matrix or vector

y

Matrix or vector

w

Vector of sampling weights

nrow

Integer

ncol

Integer

Sigma

Covariance matrix

method

Method 1 indicates estimation of different item loadings, method 2 estimation of same item loadings.

maxit

Maximum number of iterations

conv

Convergence criterion

beta

Numeric

lambda

Regularization parameter

alpha

Regularization parameter

a

Parameter

tau

Regularization parameter

val

Numeric

eta

Regularization parameter

regular_type

Type of regularization

regular_lam

Regularization parameter λ\lambda

regular_tau

Regularization parameter τ\tau

regular_alpha

Regularization parameter α\alpha

parmlist

List containing parameters

pem_pars

Vector containing parameter names

envir

Environment

update

Logical

iter

Iteration number

pem_parameter_index

List with parameter indices

pem_parameter_sequence

List with updated parameter sequence

PEM_itermax

Maximum number of iterations for PEM

ll_fct

Name of log-likelihood function

ll_args

Arguments of log-likelihood function

deviance.history

Deviance history, a data frame.

res_ll_fct

Result of maximized log-likelihood function

vars

Vector containing parameter names

eps

Numeric

ic

List

digits_crit

Integer

digits_penalty

Integer


Classification Reliability in a CDM

Description

This function computes the classification accuracy and consistency originally proposed by Cui, Gierl and Chang (2012; see also Wang et al., 2015). The function computes both statistics by estimators of Johnson and Sinharay (2018; see also Sinharay & Johnson, 2019) and simulation based estimation.

Usage

cdm.est.class.accuracy(cdmobj, n.sims=0, version=2)

Arguments

cdmobj

Object of class din or gdina

n.sims

Number of simulated persons. If n.sims=0, then the number of persons in the original data is used as the sample size. In case of missing item responses, for every simulated dataset this sample size is used.

version

Correct classification reliability statistics can be obtained using the default version=2. For backward compatibility, version=1 contains estimators for CDM (<=6.2) which have been shown to be biased (Johnson & Sinharay, 2018).

Details

The item parameters and the probability distribution of latent classes is used as the basis of the simulation. Accuracy and consistency is estimated for both MLE and MAP classification estimators. In addition, classification accuracy measures are available for the separate classification of all skills.

Value

A data frame for MLE, MAP and MAP (Skill 1, ..., Skill KK) classification reliability for the whole latent class pattern and marginal skill classification with following columns:

Pa_est

Classification accuracy (Cui et al., 2012) using the estimator of Johnson and Sinharay, 2018

Pa_sim

Classification accuracy based on simulated data (only for din models)

Pc

Classification consistency (Cui et al., 2012) using the estimator of Johnson and Sinharay, 2018

Pc_sim

Classification consistency based on simulated data (only for din models)

References

Cui, Y., Gierl, M. J., & Chang, H.-H. (2012). Estimating classification consistency and accuracy for cognitive diagnostic assessment. Journal of Educational Measurement, 49, 19-38. doi:10.1111/j.1745-3984.2011.00158.x

Johnson, M. S., & Sinharay, S. (2018). Measures of agreement to assess attribute-level classification accuracy and consistency for cognitive diagnostic assessments. Journal of Educational Measurement, 45(4), 635-664. doi:10.1111/jedm.12196

Sinharay, S., & Johnson, M. S. (2019). Measures of agreement: Reliability, classification accuracy, and classification consistency. In M. von Davier & Y.-S. Lee (Eds.). Handbook of diagnostic classification models (pp. 359-377). Cham: Springer. doi:10.1007/978-3-030-05584-4_17

Wang, W., Song, L., Chen, P., Meng, Y., & Ding, S. (2015). Attribute-level and pattern-level classification consistency and accuracy indices for cognitive diagnostic assessment. Journal of Educational Measurement, 52(4), 457-476. doi:10.1111/jedm.12096

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: DINO data example
#############################################################################

data(sim.dino, package="CDM")
data(sim.qmatrix, package="CDM")

#***
# Model 1: estimate DINO model with din
mod1 <- CDM::din( sim.dino, q.matrix=sim.qmatrix, rule="DINO")
# estimate classification reliability
cdm.est.class.accuracy( mod1, n.sims=5000)

#***
# Model 2: estimate DINO model with gdina
mod2 <- CDM::gdina( sim.dino, q.matrix=sim.qmatrix, rule="DINO")
# estimate classification reliability
cdm.est.class.accuracy( mod2 )

m1 <- mod1$coef[, c("guess", "slip" ) ]
m2 <- mod2$coef
m2 <- cbind( m1, m2[ seq(1,18,2), "est" ],
          1 - m2[ seq(1,18,2), "est" ]  - m2[ seq(2,18,2), "est" ]  )
colnames(m2) <- c("g.M1", "s.M1", "g.M2", "s.M2" )
  ##   > round( m2, 3 )
  ##          g.M1  s.M1  g.M2  s.M2
  ##   Item1 0.109 0.192 0.109 0.191
  ##   Item2 0.073 0.234 0.072 0.234
  ##   Item3 0.139 0.238 0.146 0.238
  ##   Item4 0.124 0.065 0.124 0.009
  ##   Item5 0.125 0.035 0.125 0.037
  ##   Item6 0.214 0.523 0.214 0.529
  ##   Item7 0.193 0.514 0.192 0.514
  ##   Item8 0.246 0.100 0.246 0.100
  ##   Item9 0.201 0.032 0.195 0.032
# Note that s (the slipping parameter) substantially differs for Item4
# for DINO estimation in 'din' and 'gdina'

## End(Not run)

Extract Estimated Item Parameters and Skill Class Distribution Parameters

Description

Extracts the estimated parameters from either din, gdina, gdina or gdm objects.

Usage

## S3 method for class 'din'
coef(object, ...)

## S3 method for class 'gdina'
coef(object, ...)

## S3 method for class 'mcdina'
coef(object, ...)

## S3 method for class 'gdm'
coef(object, ...)

## S3 method for class 'slca'
coef(object, ...)

Arguments

object

An object inheriting from either class din, class gdina, class mcdina, class slca or class gdm.

...

Additional arguments to be passed.

Value

A vector, a matrix or a data frame of the estimated parameters for the fitted model.

See Also

din, gdina, gdm, mcdina, slca

Examples

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

# DINA model
d1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix)
coef(d1)

## Not run: 
# GDINA model
d2 <- CDM::gdina( sim.dina, q.matrix=sim.qmatrix)
coef(d2)

# GDM model
theta.k <- seq(-4,4,len=11)
d3 <- CDM::gdm( sim.dina, irtmodel="2PL", theta.k=theta.k,
            Qmatrix=as.matrix(sim.qmatrix),  centered.latent=TRUE)
coef(d3)

## End(Not run)

Artificial Data: DINA and DINO

Description

Artificial data: dichotomously coded fictitious answers of 400 respondents to 9 items assuming 3 underlying attributes.

Usage

data(sim.dina)
  data(sim.dino)
  data(sim.qmatrix)

Format

The sim.dina and sim.dino data sets include dichotomous answers of N=400N=400 respondents to J=9J=9 items, thus they are 400×9400 \times 9 data matrices. For both data sets K=3K=3 attributes are assumed to underlie the process of responding, stored in sim.qmatrix.

The sim.dina data set is simulated according to the DINA condensation rule, whereas the sim.dino data set is simulated according to the DINO condensation rule. The slipping errors for the items 1 to 9 in both data sets are 0.20, 0.20, 0.20, 0.20, 0.00, 0.50, 0.50, 0.10, 0.03 and the guessing errors are 0.10, 0.125, 0.15, 0.175, 0.2, 0.225, 0.25, 0.275, 0.3. The attributes are assumed to be mastered with expected probabilities of -0.4, 0.2, 0.6, respectively. The correlation of the attributes is 0.3 for attributes 1 and 2, 0.4 for attributes 1 and 3 and 0.1 for attributes 2 and 3.

Example Index

Dataset sim.dina

anova (Examples 1, 2), cdi.kli (Example 1), din (Examples 2, 4, 5), gdina (Example 1), itemfit.sx2 (Example 2), modelfit.cor.din (Example 1)

Dataset sim.dino

cdm.est.class.accuracy (Example 1), din (Example 3), gdina (Examples 2, 3, 4),

References

Rupp, A. A., Templin, J. L., & Henson, R. A. (2010) Diagnostic Measurement: Theory, Methods, and Applications. New York: The Guilford Press.


Several Datasets for the CDM Package

Description

Several datasets for the CDM package

Usage

data(data.cdm01)
data(data.cdm02)
data(data.cdm03)
data(data.cdm04)
data(data.cdm05)
data(data.cdm06)
data(data.cdm07)
data(data.cdm08)
data(data.cdm09)
data(data.cdm10)

Format

References

Chen, H., & Chen, J. (2017). Cognitive diagnostic research on chinese students' English listening skills and implications on skill training. English Language Teaching, 10(12), 107-115. http://dx.doi.org/10.5539/elt.v10n12p107

Chen, J., & de la Torre, J. (2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37, 419-437. http://dx.doi.org/10.1177/0146621613479818

Chen, Y., Li, X., Liu, J., & Ying, Z. (2017). Regularized latent class analysis with application in cognitive diagnosis. Psychometrika, 82, 660-692. http://dx.doi.org/10.1007/s11336-016-9545-6

Chiu, C.-Y., Koehn, H.-F., & Wu, H.-M. (2016). Fitting the reduced RUM with Mplus: A tutorial. International Journal of Testing, 16(4), 331-351. http://dx.doi.org/10.1080/15305058.2016.1148038

Fang, G., Liu, J., & Ying, Z. (2017). On the identifiability of diagnostic classification models. arXiv, 1706.01240. https://arxiv.org/abs/1706.01240

Heller, J. and Wickelmaier, F. (2013). Minimum discrepancy estimation in probabilistic knowledge structures. Electronic Notes in Discrete Mathematics, 42, 49-56.
http://dx.doi.org/10.1016/j.endm.2013.05.145

Kuo, B.-C., Chen, C.-H., & de la Torre, J. (2018). A cognitive diagnosis model for identifying coexisting skills and misconceptions. Applied Psychological Measurement, 42(3), 179-191. http://dx.doi.org/10.1177/0146621617722791

Ma, W., & de la Torre, J. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology, 69(3), 253-275.
https://doi.org/10.1111/bmsp.12070

Philipp, M., Strobl, C., de la Torre, J., & Zeileis, A. (2018). On the estimation of standard errors in cognitive diagnosis models. Journal of Educational and Behavioral Statistics, 43(1), 88-115. http://dx.doi.org/10.3102/1076998617719728

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Reduced RUM model, Chiu et al. (2016)
#############################################################################

data(data.cdm03, package="CDM")
dat <- data.cdm03$data
qmatrix <- data.cdm03$qmatrix

#*** Model 1: Reduced RUM
mod1 <- CDM::gdina( dat, q.matrix=qmatrix[,-1], rule="RRUM" )
summary(mod1)

#*** Model 2: Additive model with identity link function
mod2 <- CDM::gdina( dat, q.matrix=qmatrix[,-1], rule="ACDM" )
summary(mod2)

#*** Model 3: Additive model with logit link function
mod3 <- CDM::gdina( dat, q.matrix=qmatrix[,-1], rule="ACDM", linkfct="logit")
summary(mod3)

#############################################################################
# EXAMPLE 2: GDINA model - probability dataset from the pks package
#############################################################################

data(data.cdm05, package="CDM")
dat <- data.cdm05$data
Q <- data.cdm05$q.matrix

#* estimate model
mod1 <- CDM::gdina( dat, q.matrix=Q )
summary(mod1)

## End(Not run)

Dataset from Book 'Diagnostic Measurement' of Rupp, Templin and Henson (2010)

Description

Dataset from Chapter 9 of the book 'Diagnostic Measurement' (Rupp, Templin & Henson, 2010).

Usage

data(data.dcm)

Format

The format of the data is a list containing the dichotomous item response data data (10000 persons at 7 items) and the Q-matrix q.matrix (7 items and 3 skills):

List of 2
$ data :'data.frame':
..$ id: int [1:10000] 1 2 3 4 5 6 7 8 9 10 ...
..$ D1: num [1:10000] 0 0 0 0 1 0 1 0 0 1 ...
..$ D2: num [1:10000] 0 0 0 0 0 1 1 1 0 1 ...
..$ D3: num [1:10000] 1 0 1 0 1 1 0 0 0 1 ...
..$ D4: num [1:10000] 0 0 1 0 0 1 1 1 0 0 ...
..$ D5: num [1:10000] 1 0 0 0 1 1 1 0 1 0 ...
..$ D6: num [1:10000] 0 0 0 0 1 1 1 0 0 1 ...
..$ D7: num [1:10000] 0 0 0 0 0 1 1 0 1 1 ...
$ q.matrix: num [1:7, 1:3] 1 0 0 1 1 0 1 0 1 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : chr [1:7] "D1" "D2" "D3" "D4" ...
.. ..$ : chr [1:3] "skill1" "skill2" "skill3"

Source

For supplementary material of the Rupp, Templin and Henson book (2010) see http://dcm.coe.uga.edu/.

The dataset was downloaded from http://dcm.coe.uga.edu/supplemental/chapter9.html.

References

Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic Measurement: Theory, Methods, and Applications. New York: The Guilford Press.

Examples

## Not run: 
data(data.dcm, package="CDM")

dat <- data.dcm$data[,-1]
Q <- data.dcm$q.matrix

#*****************************************************
# Model 1: DINA model
#*****************************************************
mod1 <- CDM::din( dat, q.matrix=Q )
summary(mod1)

#--------
# Model 1m: estimate model in mirt package
library(mirt)
library(sirt)

  #** define theta grid of skills
  # use the function skillspace.hierarchy just for convenience
hier <- "skill1 > skill2"
skillspace <- CDM::skillspace.hierarchy( hier, skill.names=colnames(Q) )
Theta <- as.matrix(skillspace$skillspace.complete)
  #** create mirt model
mirtmodel <- mirt::mirt.model("
      skill1=1
      skill2=2
      skill3=3
      (skill1*skill2)=4
      (skill1*skill3)=5
      (skill2*skill3)=6
      (skill1*skill2*skill3)=7
          " )
  #** mirt parameter table
mod.pars <- mirt::mirt( dat, mirtmodel, pars="values")
  # use starting values of .20 for guessing parameter
ind <- which( mod.pars$name=="d" )
mod.pars[ind,"value"] <- stats::qlogis(.20) # guessing parameter on the logit metric
  # use starting values of .80 for anti-slipping parameter
ind <- which( ( mod.pars$name %in% paste0("a",1:20 ) ) & (mod.pars$est) )
mod.pars[ind,"value"] <- stats::qlogis(.80) - stats::qlogis(.20)
mod.pars
  #** prior for the skill space distribution
I <- ncol(dat)
lca_prior <- function(Theta,Etable){
  TP <- nrow(Theta)
  if ( is.null(Etable) ){ prior <- rep( 1/TP, TP ) }
  if ( ! is.null(Etable) ){
    prior <- ( rowSums(Etable[, seq(1,2*I,2)]) + rowSums(Etable[,seq(2,2*I,2)]) )/I
  }
  prior <- prior / sum(prior)
  return(prior)
 }

 #** estimate model in mirt
mod1m <- mirt::mirt(dat, mirtmodel, pars=mod.pars, verbose=TRUE,
            technical=list( customTheta=Theta, customPriorFun=lca_prior) )
  # The number of estimated parameters is incorrect because mirt does not correctly count
  # estimated parameters from the user customized  prior distribution.
mod1m@nest <- as.integer(sum(mod.pars$est) + nrow(Theta) - 1)
  # extract log-likelihood
mod1m@logLik
  # compute AIC and BIC
( AIC <- -2*mod1m@logLik+2*mod1m@nest )
( BIC <- -2*mod1m@logLik+log(mod1m@Data$N)*mod1m@nest )
  #** extract item parameters
cmod1m <- sirt::mirt.wrapper.coef(mod1m)$coef
# compare estimated guessing and slipping parameters
dfr <- data.frame(    "din.guess"=mod1$guess$est,
                  "mirt.guess"=plogis(cmod1m$d), "din.slip"=mod1$slip$est,
                  "mirt.slip"=1-plogis( rowSums( cmod1m[, c("d", paste0("a",1:7) ) ] ) )
                    )
round(t(dfr),3)
  ##               [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]
  ##   din.guess  0.217 0.193 0.189 0.135 0.143 0.135 0.162
  ##   mirt.guess 0.226 0.189 0.184 0.132 0.142 0.132 0.158
  ##   din.slip   0.338 0.331 0.334 0.220 0.222 0.211 0.042
  ##   mirt.slip  0.339 0.333 0.336 0.223 0.225 0.214 0.044

# compare estimated skill class distribution
dfr <- data.frame("din"=mod1$attribute.patt$class.prob,
                    "mirt"=mod1m@Prior[[1]] )
round(t(dfr),3)
  ##         [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]
  ##   din  0.113 0.083 0.094 0.092 0.064 0.059 0.065 0.429
  ##   mirt 0.116 0.074 0.095 0.064 0.095 0.058 0.066 0.433

#** extract estimated classifications
fsc1m <- sirt::mirt.wrapper.fscores( mod1m )
#- estimated reliabilities
fsc1m$EAP.rel
  ##      skill1    skill2    skill3
  ##   0.5479942 0.5362595 0.5357961
#- estimated classfications: EAPs, MLEs and MAPs
head( round(fsc1m$person,3) )
  ##     case     M EAP.skill1 SE.EAP.skill1 EAP.skill2 SE.EAP.skill2 EAP.skill3 SE.EAP.skill3
  ##   1    1 0.286      0.508         0.500      0.067         0.251      0.820         0.384
  ##   2    2 0.000      0.162         0.369      0.191         0.393      0.190         0.392
  ##   3    3 0.286      0.200         0.400      0.211         0.408      0.607         0.489
  ##   4    4 0.000      0.162         0.369      0.191         0.393      0.190         0.392
  ##   5    5 0.571      0.802         0.398      0.267         0.443      0.928         0.258
  ##   6    6 0.857      0.998         0.045      1.000         0.019      1.000         0.020
  ##     MLE.skill1 MLE.skill2 MLE.skill3 MAP.skill1 MAP.skill2 MAP.skill3
  ##   1          1          0          1          1          0          1
  ##   2          0          0          0          0          0          0
  ##   3          0          0          1          0          0          1
  ##   4          0          0          0          0          0          0
  ##   5          1          0          1          1          0          1
  ##   6          1          1          1          1          1          1

#** estimate model fit in mirt
( fit1m <- mirt::M2( mod1m ) )

#*****************************************************
# Model 2: DINO model
#*****************************************************
mod2 <- CDM::din( dat, q.matrix=Q, rule="DINO")
summary(mod2)

#*****************************************************
# Model 3: log-linear model (LCDM): this model is the GDINA model with the
#    logit link function
#*****************************************************
mod3 <- CDM::gdina( dat, q.matrix=Q, link="logit")
summary(mod3)

#*****************************************************
# Model 4: GDINA model with identity link function
#*****************************************************
mod4 <- CDM::gdina( dat, q.matrix=Q )
summary(mod4)

#*****************************************************
# Model 5: GDINA additive model identity link function
#*****************************************************
mod5 <- CDM::gdina( dat, q.matrix=Q, rule="ACDM")
summary(mod5)

#*****************************************************
# Model 6: GDINA additive model logit link function
#*****************************************************
mod6 <- CDM::gdina( dat, q.matrix=Q, link="logit", rule="ACDM")
summary(mod6)

#--------
# Model 6m: GDINA additive model in mirt package
# use data specifications from Model 1m)
  #** create mirt model
mirtmodel <- mirt::mirt.model("
      skill1=1,4,5,7
      skill2=2,4,6,7
      skill3=3,5,6,7
          " )
  #** mirt parameter table
mod.pars <- mirt::mirt( dat, mirtmodel, pars="values")
 #** estimate model in mirt
 # Theta and lca_prior as defined as in Model 1m
mod6m <- mirt::mirt(dat, mirtmodel, pars=mod.pars, verbose=TRUE,
            technical=list( customTheta=Theta, customPriorFun=lca_prior) )
mod6m@nest <- as.integer(sum(mod.pars$est) + nrow(Theta) - 1)
  # extract log-likelihood
mod6m@logLik
  # compute AIC and BIC
( AIC <- -2*mod6m@logLik+2*mod6m@nest )
( BIC <- -2*mod6m@logLik+log(mod6m@Data$N)*mod6m@nest )
  #** skill distribution
  mod6m@Prior[[1]]
  #** extract item parameters
cmod6m <- mirt.wrapper.coef(mod6m)$coef
print(cmod6m,digits=4)
  ##     item    a1    a2    a3       d g u
  ##   1   D1 1.882 0.000 0.000 -0.9330 0 1
  ##   2   D2 0.000 2.049 0.000 -1.0430 0 1
  ##   3   D3 0.000 0.000 2.028 -0.9915 0 1
  ##   4   D4 2.697 2.525 0.000 -2.9925 0 1
  ##   5   D5 2.524 0.000 2.478 -2.7863 0 1
  ##   6   D6 0.000 2.818 2.791 -3.1324 0 1
  ##   7   D7 3.113 2.918 2.785 -4.2794 0 1

#*****************************************************
# Model 7: Reduced RUM model
#*****************************************************
mod7 <- CDM::gdina( dat, q.matrix=Q, rule="RRUM")
summary(mod7)

#*****************************************************
# Model 8: latent class model with 3 classes and 4 sets of starting values
#*****************************************************

#-- Model 8a: randomLCA package
library(randomLCA)
mod8a <- randomLCA::randomLCA( dat, nclass=3, verbose=TRUE, notrials=4)

#-- Model8b: rasch.mirtlc function in sirt package
library(sirt)
mod8b <- sirt::rasch.mirtlc( dat, Nclasses=3, nstarts=4 )
summary(mod8a)
summary(mod8b)

## End(Not run)

DTMR Fraction Data (Bradshaw et al., 2014)

Description

This is a simulated dataset of the DTMR fraction data described in Bradshaw, Izsak, Templin and Jacobson (2014).

Usage

data(data.dtmr)

Format

The format is:

List of 5
$ data : num [1:5000, 1:27] 0 0 0 0 0 1 0 0 1 1 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:27] "M1" "M2" "M3" "M4" ...
$ q.matrix :'data.frame': 27 obs. of 4 variables:
..$ RU : int [1:27] 1 0 0 1 1 0 1 0 0 0 ...
..$ PI : int [1:27] 0 0 1 0 0 1 0 0 0 0 ...
..$ APP: int [1:27] 0 1 0 0 0 0 0 1 1 1 ...
..$ MC : int [1:27] 0 0 0 0 0 0 0 0 0 0 ...
$ skill.distribution:'data.frame': 16 obs. of 5 variables:
..$ RU : int [1:16] 0 0 0 0 0 0 0 0 1 1 ...
..$ PI : int [1:16] 0 0 0 0 1 1 1 1 0 0 ...
..$ APP : int [1:16] 0 0 1 1 0 0 1 1 0 0 ...
..$ MC : int [1:16] 0 1 0 1 0 1 0 1 0 1 ...
..$ freq: int [1:16] 1064 350 280 406 196 126 238 770 14 28 ...
$ itempars :'data.frame': 27 obs. of 7 variables:
..$ item : chr [1:27] "M1" "M2" "M3" "M4" ...
..$ lam0 : num [1:27] -1.12 0.59 -2.07 -1.19 -1.67 -3.81 -0.73 -0.62 -0.09 0.28 ...
..$ RU : num [1:27] 2.24 0 0 0.65 1.52 0 1.2 0 0 0 ...
..$ PI : num [1:27] 0 0 1.7 0 0 2.08 0 0 0 0 ...
..$ APP : num [1:27] 0 1.27 0 0 0 0 0 4.25 2.16 0.87 ...
..$ MC : num [1:27] 0 0 0 0 0 0 0 0 0 0 ...
..$ RU.PI: num [1:27] 0 0 0 0 0 0 0 0 0 0 ...
$ sim_data :function (N, skill.distribution, itempars)
..- attr(*, "srcref")='srcref' int [1:8] 1 13 20 1 13 1 1 20
.. ..- attr(*, "srcfile")=Classes 'srcfilecopy', 'srcfile' <environment: 0x00000000298a8ed0>

The attribute definition are as follows

RU: Referent units

PI: Partitioning and iterating attribute

APP: Appropriateness attribute

MC: Multiplicative Comparison attribute

Source

Simulated dataset according to Bradshaw et al. (2014).

References

Bradshaw, L., Izsak, A., Templin, J., & Jacobson, E. (2014). Diagnosing teachers' understandings of rational numbers: Building a multidimensional test within the diagnostic classification framework. Educational Measurement: Issues and Practice, 33, 2-14.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Model comparisons data.dtmr
#############################################################################

data(data.dtmr, package="CDM")
data <- data.dtmr$data
q.matrix <- data.dtmr$q.matrix
I <- ncol(data)

#*** Model 1: LCDM
# define item wise rules
rule <- rep( "ACDM", I )
names(rule) <- colnames(data)
rule[ c("M14","M17") ] <- "GDINA2"
# estimate model
mod1 <- CDM::gdina( data, q.matrix, linkfct="logit", rule=rule)
summary(mod1)

#*** Model 2: DINA model
mod2 <- CDM::gdina( data, q.matrix, rule="DINA" )
summary(mod2)

#*** Model 3: RRUM model
mod3 <- CDM::gdina( data, q.matrix, rule="RRUM" )
summary(mod3)

#--- model comparisons

# LCDM vs. DINA
anova(mod1,mod2)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -76570.89 153141.8    69 153279.8 153729.5 1726.645 10  0
  ##   1 Model 1 -75707.57 151415.1    79 151573.1 152088.0       NA NA NA

# LCDM vs. RRUM
anova(mod1,mod3)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -75746.13 151492.3    77 151646.3 152148.1 77.10994  2  0
  ##   1 Model 1 -75707.57 151415.1    79 151573.1 152088.0       NA NA NA

#--- model fit
summary( CDM::modelfit.cor.din( mod1 ) )
  ##   Test of Global Model Fit
  ##          type   value       p
  ##   1   max(X2) 7.74382 1.00000
  ##   2 abs(fcor) 0.04056 0.72707
  ##
  ##   Fit Statistics
  ##                       est
  ##   MADcor          0.00959
  ##   SRMSR           0.01217
  ##   MX2             0.75696
  ##   100*MADRESIDCOV 0.20283
  ##   MADQ3           0.02220

#############################################################################
# EXAMPLE 2: Simulating data of structure data.dtmr
#############################################################################

data(data.dtmr, package="CDM")

# draw sample of N=200
set.seed(87)
data.dtmr$sim_data(N=200, skill.distribution=data.dtmr$skill.distribution,
             itempars=data.dtmr$itempars)

## End(Not run)

Dataset ECPE

Description

ECPE dataset from the Templin and Hoffman (2013) tutorial of specifying cognitive diagnostic models in Mplus.

Usage

data(data.ecpe)

Format

The format of the data is a list containing the dichotomous item response data data (2922 persons at 28 items) and the Q-matrix q.matrix (28 items and 3 skills):

List of 2
$ data :'data.frame':
..$ id : int [1:2922] 1 2 3 4 5 6 7 8 9 10 ...
..$ E1 : int [1:2922] 1 1 1 1 1 1 1 0 1 1 ...
..$ E2 : int [1:2922] 1 1 1 1 1 1 1 1 1 1 ...
..$ E3 : int [1:2922] 1 1 1 1 1 1 1 1 1 1 ...
..$ E4 : int [1:2922] 0 1 1 1 1 1 1 1 1 1 ...
[...]
..$ E27: int [1:2922] 1 1 1 1 1 1 1 0 1 1 ...
..$ E28: int [1:2922] 1 1 1 1 1 1 1 1 1 1 ...
$ q.matrix:'data.frame':
..$ skill1: int [1:28] 1 0 1 0 0 0 1 0 0 1 ...
..$ skill2: int [1:28] 1 1 0 0 0 0 0 1 0 0 ...
..$ skill3: int [1:28] 0 0 1 1 1 1 1 0 1 0 ...

The skills are

skill1: Morphosyntactic rules

skill2: Cohesive rules

skill3: Lexical rules.

Details

The dataset has been used in Templin and Hoffman (2013), and Templin and Bradshaw (2014).

Source

The dataset was downloaded from http://psych.unl.edu/jtemplin/teaching/dcm/dcm12ncme/.

References

Templin, J., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79, 317-339.

Templin, J., & Hoffman, L. (2013). Obtaining diagnostic classification model estimates using Mplus. Educational Measurement: Issues and Practice, 32, 37-50.

See Also

GDINA::ecpe

Examples

## Not run: 
data(data.ecpe, package="CDM")

dat <- data.ecpe$data[,-1]
Q <- data.ecpe$q.matrix

#*** Model 1: LCDM model
mod1 <- CDM::gdina( dat, q.matrix=Q, link="logit")
summary(mod1)

#*** Model 2: DINA model
mod2 <- CDM::gdina( dat, q.matrix=Q, rule="DINA")
summary(mod2)

# Model comparison using likelihood ratio test
anova(mod1,mod2)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -42841.61 85683.23    63 85809.23 86185.97 206.0359 18  0
  ##   1 Model 1 -42738.60 85477.19    81 85639.19 86123.57       NA NA NA

#*** Model 3: Hierarchical LCDM (HLCDM) | Templin and Bradshaw (2014)
#      Testing a linear hierarchy
hier <- "skill3 > skill2 > skill1"
skill.names <- colnames(Q)
# define skill space with hierarchy
skillspace <- CDM::skillspace.hierarchy( hier, skill.names=skill.names )
skillspace$skillspace.reduced
  ##        skill1 skill2 skill3
  ##   A000      0      0      0
  ##   A001      0      0      1
  ##   A011      0      1      1
  ##   A111      1      1      1
zeroprob.skillclasses <- skillspace$zeroprob.skillclasses

# define user-defined parameters in LCDM: hierarchical LCDM (HLCDM)
Mj.user <- mod1$Mj
# select items with require two attributes
items <- which( rowSums(Q) > 1 )
# modify design matrix for item parameters
for (ii in items){
    m1 <- Mj.user[[ii]]
    Mj.user[[ii]][[1]] <- (m1[[1]])[,-2]
    Mj.user[[ii]][[2]] <- (m1[[2]])[-2]
}

# estimate model
#    note that avoid.zeroprobs is set to TRUE to avoid algorithmic instabilities
mod3 <- CDM::gdina( dat, q.matrix=Q, link="logit",
            zeroprob.skillclasses=zeroprob.skillclasses, Mj=Mj.user,
            avoid.zeroprobs=TRUE )
summary(mod3)

#*****************************************
#** estimate further models

#*** Model 4: RRUM model
mod4 <- CDM::gdina( dat, q.matrix=Q, rule="RRUM")
summary(mod4)
# compare some models
IRT.compareModels(mod1, mod2, mod3, mod4 )

#*** Model 5a: GDINA model with identity link
mod5a <- CDM::gdina( dat, q.matrix=Q, link="identity")
summary(mod5a)
#*** Model 5b: GDINA model with logit link
mod5b <- CDM::gdina( dat, q.matrix=Q, link="logit")
summary(mod5b)
#*** Model 5c: GDINA model with log link
mod5c <- CDM::gdina( dat, q.matrix=Q, link="log")
summary(mod5c)
# compare models
IRT.compareModels(mod5a, mod5b, mod5c)

## End(Not run)

Fraction Subtraction Dataset with Different Subsets of Data and Different Q-Matrices

Description

Contains different sub-datasets of the fraction subtraction data of Tatsuoka with different Q-matrix specifications.

Usage

data(data.fraction1)
data(data.fraction2)
data(data.fraction3)
data(data.fraction4)
data(data.fraction5)

Format

Source

See fraction.subtraction.data for more information about the data source.

References

Chen, Y., Liu, J., Xu, G. and Ying, Z. (2015). Statistical analysis of Q-matrix based diagnostic classification models. Journal of the American Statistical Association, 110(510), 850-866.

de la Torre, J. (2009). DINA model parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34, 115-130.

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

de la Torre, J., & Douglas, J. A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333-353.

de la Torre, J., & Douglas, J. A. (2008). Model evaluation and multiple strategies in cognitive diagnosis: An analysis of fraction subtraction data. Psychometrika, 73, 595-624.

Henson, R. A., Templin, J. T., & Willse, J. T. (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables. Psychometrika, 74, 191-210.

Huo, Y., & de la Torre, J. (2014). Estimating a cognitive diagnostic model for multiple strategies via the EM algorithm. Applied Psychological Measurement, 38, 464-485.

See Also

GDINA::frac20


Dataset data.hr (Ravand et al., 2013)

Description

Dataset data.hr used for illustrating some functionalities of the CDM package (Ravand, Barati, & Widhiarso, 2013).

Usage

data(data.hr)

Format

The format of the dataset is:

List of 2
$ data : num [1:1550, 1:35] 1 0 1 1 1 0 1 1 1 0 ...
$ q.matrix:'data.frame':
..$ Skill1: int [1:35] 0 0 0 0 0 0 1 0 0 0 ...
..$ Skill2: int [1:35] 0 0 0 0 1 0 0 0 0 0 ...
..$ Skill3: int [1:35] 0 1 1 1 1 0 0 1 0 0 ...
..$ Skill4: int [1:35] 1 0 0 0 0 0 0 0 1 1 ...
..$ Skill5: int [1:35] 0 0 0 0 0 1 0 0 1 1 ...

Source

Simulated data according to Ravand et al. (2013).

References

Ravand, H., Barati, H., & Widhiarso, W. (2013). Exploring diagnostic capacity of a high stakes reading comprehension test: A pedagogical demonstration. Iranian Journal of Language Testing, 3(1), 1-27.

Examples

## Not run: 
data(data.hr, package="CDM")

dat <- data.hr$data
Q <- data.hr$q.matrix

#*************
# Model 1: DINA model
mod1 <- CDM::din( dat, q.matrix=Q )
summary(mod1)       # summary

# plot results
plot(mod1)

# inspect coefficients
coef(mod1)

# posterior distribution
posterior <- mod1$posterior
round( posterior[ 1:5, ], 4 )  # first 5 entries

# estimate class probabilities
mod1$attribute.patt

# individual classifications
mod1$pattern[1:5,]   # first 5 entries

#*************
# Model 2: GDINA model
mod2 <- CDM::gdina( dat, q.matrix=Q)
summary(mod2)

#*************
# Model 3: Reduced RUM model
mod3 <- CDM::gdina( dat, q.matrix=Q, rule="RRUM" )
summary(mod3)

#--------
# model comparisons

# DINA vs GDINA
anova( mod1, mod2 )
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   1 Model 1 -31391.27 62782.54   101 62984.54 63524.49 195.9099 20  0
  ##   2 Model 2 -31293.32 62586.63   121 62828.63 63475.50       NA NA NA

# RRUM vs. GDINA
anova( mod2, mod3 )
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -31356.22 62712.43   105 62922.43 63483.76 125.7924 16  0
  ##   1 Model 1 -31293.32 62586.64   121 62828.64 63475.50       NA NA NA

# DINA vs. RRUM
anova(mod1,mod3)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   1 Model 1 -31391.27 62782.54   101 62984.54 63524.49 70.11246  4  0
  ##   2 Model 2 -31356.22 62712.43   105 62922.43 63483.76       NA NA NA

#-------
# model fit

# DINA
fmod1 <- CDM::modelfit.cor.din( mod1, jkunits=0)
summary(fmod1)
  ##   Test of Global Model Fit
  ##          type    value       p
  ##   1   max(X2) 16.35495 0.03125
  ##   2 abs(fcor)  0.10341 0.01416
  ##
  ##   Fit Statistics
  ##                       est
  ##   MADcor          0.01911
  ##   SRMSR           0.02445
  ##   MX2             0.93157
  ##   100*MADRESIDCOV 0.39100
  ##   MADQ3           0.02373

# GDINA
fmod2 <- CDM::modelfit.cor.din( mod2, jkunits=0)
summary(fmod2)
  ##   Test of Global Model Fit
  ##          type   value p
  ##   1   max(X2) 7.73670 1
  ##   2 abs(fcor) 0.07215 1
  ##
  ##   Fit Statistics
  ##                       est
  ##   MADcor          0.01830
  ##   SRMSR           0.02300
  ##   MX2             0.82584
  ##   100*MADRESIDCOV 0.37390
  ##   MADQ3           0.02383

# RRUM
fmod3 <- CDM::modelfit.cor.din( mod3, jkunits=0)
summary(fmod3)
  ##   Test of Global Model Fit
  ##          type    value       p
  ##   1   max(X2) 15.49369 0.04925
  ##   2 abs(fcor)  0.10076 0.02201
  ##
  ##   Fit Statistics
  ##                       est
  ##   MADcor          0.01868
  ##   SRMSR           0.02374
  ##   MX2             0.87999
  ##   100*MADRESIDCOV 0.38409
  ##   MADQ3           0.02416

## End(Not run)

Dataset Jang (2009)

Description

Simulated dataset according to the Jang (2005) L2 reading comprehension study.

Usage

data(data.jang)

Format

The format is:

List of 2
$ data : num [1:1500, 1:37] 1 1 1 1 1 1 1 1 1 1 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:37] "I1" "I2" "I3" "I4" ...
$ q.matrix:'data.frame':
..$ CDV: int [1:37] 1 0 0 1 0 0 0 0 0 0 ...
..$ CIV: int [1:37] 0 0 1 0 0 0 1 0 1 1 ...
..$ SSL: int [1:37] 1 1 1 1 0 0 0 0 0 0 ...
..$ TEI: int [1:37] 0 0 0 0 0 0 0 1 0 0 ...
..$ TIM: int [1:37] 0 0 0 1 1 1 0 0 0 0 ...
..$ INF: int [1:37] 0 1 0 0 0 0 1 0 0 0 ...
..$ NEG: int [1:37] 0 0 0 0 1 0 1 0 0 0 ...
..$ SUM: int [1:37] 0 0 0 0 1 0 0 0 0 0 ...
..$ MCF: int [1:37] 0 0 0 0 0 0 0 0 0 0 ...

Source

Simulated dataset.

References

Jang, E. E. (2009). Cognitive diagnostic assessment of L2 reading comprehension ability: Validity arguments for Fusion Model application to LanguEdge assessment. Language Testing, 26, 31-73.

Examples

## Not run: 
data(data.jang, package="CDM")

data <- data.jang$data
q.matrix <- data.jang$q.matrix

#*** Model 1: Reduced RUM model
mod1 <- CDM::gdina( data, q.matrix, rule="RRUM", conv.crit=.001, increment.factor=1.025 )
summary(mod1)

#*** Model 2: Additive model (identity link)
mod2 <- CDM::gdina( data, q.matrix, rule="ACDM", conv.crit=.001, linkfct="identity" )
summary(mod2)

#*** Model 3: DINA model
mod3 <- CDM::gdina( data, q.matrix, rule="DINA", conv.crit=.001 )
summary(mod3)

anova(mod1,mod2)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   1 Model 1 -30315.03 60630.06   153 60936.06 61748.98 88.29627  0  0
  ##   2 Model 2 -30270.88 60541.76   153 60847.76 61660.68       NA NA NA
anova(mod1,mod3)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -30373.99 60747.97   129 61005.97 61691.38 117.9128 24  0
  ##   1 Model 1 -30315.03 60630.06   153 60936.06 61748.98       NA NA NA

# RRUM
summary( CDM::modelfit.cor.din( mod1, jkunits=0) )
  ##          type    value       p
  ##   1   max(X2) 11.79073 0.39645
  ##   2 abs(fcor)  0.09541 0.07422
  ##                       est
  ##   MADcor          0.01834
  ##   SRMSR           0.02300
  ##   MX2             0.86718
  ##   100*MADRESIDCOV 0.38690
  ##   MADQ3           0.02413

# additive model (identity)
summary( CDM::modelfit.cor.din( mod2, jkunits=0) )
  ##          type   value       p
  ##   1   max(X2) 9.78958 1.00000
  ##   2 abs(fcor) 0.08770 0.22993
  ##                       est
  ##   MADcor          0.01721
  ##   SRMSR           0.02158
  ##   MX2             0.69163
  ##   100*MADRESIDCOV 0.36343
  ##   MADQ3           0.02423

# DINA model
summary( CDM::modelfit.cor.din( mod3, jkunits=0) )
  ##          type    value       p
  ##   1   max(X2) 13.48449 0.16020
  ##   2 abs(fcor)  0.10651 0.01256
  ##                       est
  ##   MADcor          0.01999
  ##   SRMSR           0.02495
  ##   MX2             0.92820
  ##   100*MADRESIDCOV 0.42226
  ##   MADQ3           0.02258

## End(Not run)

MELAB Data (Li, 2011)

Description

This is a simulated dataset according to the MELAB reading study (Li, 2011; Li & Suen, 2013). Li (2011) investigated the Fusion model (RUM model) for calibrating this dataset. The dataset in this package is simulated assuming the reduced RUM model (RRUM).

Usage

data(data.melab)

Format

The format of the dataset is:

List of 3
$ data : num [1:2019, 1:20] 0 1 0 1 1 0 0 0 1 1 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:20] "I1" "I2" "I3" "I4" ...
$ q.matrix :'data.frame':
..$ skill1: int [1:20] 1 1 0 0 1 1 0 1 0 1 ...
..$ skill2: int [1:20] 0 0 0 0 0 0 0 0 0 0 ...
..$ skill3: int [1:20] 0 0 0 1 0 1 1 0 1 0 ...
..$ skill4: int [1:20] 1 0 1 0 1 0 0 1 0 1 ...
$ skill.labels:'data.frame':
..$ skill : Factor w/ 4 levels "skill1","skill2",..: 1 2 3 4
..$ skill.label: Factor w/ 4 levels "connecting and synthesizing",..: 4 3 2 1

Source

Simulated data according to Li (2011).

References

Li, H. (2011). A cognitive diagnostic analysis of the MELAB reading test. Spaan Fellow, 9, 17-46.

Li, H., & Suen, H. K. (2013). Constructing and validating a Q-matrix for cognitive diagnostic analyses of a reading test. Educational Assessment, 18, 1-25.

Examples

## Not run: 
data(data.melab, package="CDM")

data <- data.melab$data
q.matrix <- data.melab$q.matrix

#*** Model 1: Reduced RUM model
mod1 <- CDM::gdina( data, q.matrix, rule="RRUM" )
summary(mod1)

#*** Model 2: GDINA model
mod2 <- CDM::gdina( data, q.matrix, rule="GDINA" )
summary(mod2)

#*** Model 3: DINA model
mod3 <- CDM::gdina( data, q.matrix, rule="DINA" )
summary(mod3)

#*** Model 4: 2PL model
mod4 <- CDM::gdm( data, theta.k=seq(-6,6,len=21), center )
summary(mod4)

#----
# Model comparisons

#*** RRUM vs. GDINA
anova(mod1,mod2)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df       p
  ##   1 Model 1 -20252.74 40505.48    69 40643.48 41030.60 30.88801 18 0.02966
  ##   2 Model 2 -20237.30 40474.59    87 40648.59 41136.69       NA NA      NA

  ##  -> GDINA is not superior to RRUM (according to AIC and BIC)

#*** DINA vs. RRUM
anova(mod1,mod3)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -20332.52 40665.04    55 40775.04 41083.61 159.5566 14  0
  ##   1 Model 1 -20252.74 40505.48    69 40643.48 41030.60       NA NA NA

  ##  -> RRUM fits the data significantly better than the DINA model

#*** RRUM vs. 2PL (use only AIC and BIC for comparison)
anova(mod1,mod4)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -20390.19 40780.38    43 40866.38 41107.62 274.8962 26  0
  ##   1 Model 1 -20252.74 40505.48    69 40643.48 41030.60       NA NA NA

  ## -> RRUM fits the data better than 2PL

#----
# Model fit statistics

# RRUM
fmod1 <- CDM::modelfit.cor.din( mod1, jkunits=0)
summary(fmod1)
  ##   Test of Global Model Fit
  ##          type    value       p
  ##   1   max(X2) 10.10408 0.28109
  ##   2 abs(fcor)  0.06726 0.24023
  ##
  ##   Fit Statistics
  ##                       est
  ##   MADcor          0.01708
  ##   SRMSR           0.02158
  ##   MX2             0.96590
  ##   100*MADRESIDCOV 0.27269
  ##   MADQ3           0.02781

  ##  -> not a significant misfit of the RRUM model

# GDINA
fmod2 <- CDM::modelfit.cor.din( mod2, jkunits=0)
summary(fmod2)
  ##   Test of Global Model Fit
  ##          type    value       p
  ##   1   max(X2) 10.40294 0.23905
  ##   2 abs(fcor)  0.06817 0.20964
  ##
  ##   Fit Statistics
  ##                       est
  ##   MADcor          0.01703
  ##   SRMSR           0.02151
  ##   MX2             0.94468
  ##   100*MADRESIDCOV 0.27105
  ##   MADQ3           0.02713

## End(Not run)

Large-Scale Dataset with Multiple Groups

Description

Large-scale dataset with multiple groups, survey weights and 11 polytomous items.

Usage

data(data.mg)

Format

A data frame with 38243 observations on the following 14 variables.

idstud

Student identifier

group

Group identifier

weight

Survey weight

I1

Item 1

I2

Item 2

I3

Item 3

I4

Item 4

I5

Item 5

I6

Item 6

I7

Item 7

I8

Item 8

I9

Item 9

I10

Item 10

I11

Item 11

Source

Subsample of a large-scale dataset of 11 survey questions.

Examples

## Not run: 
library(psych)
data(dat.mg, package="CDM")
psych::describe( data.mg )
  ##   > psych::describe(data.mg)
  ##          var     n       mean       sd     median    trimmed      mad        min        max
  ##   idstud   1 38243 1039653.91 19309.80 1037899.00 1039927.73 30240.59 1007168.00 1069949.00
  ##   group    2 38243       8.06     4.07       7.00       8.06     5.93       2.00      14.00
  ##   weight   3 38243      28.76    19.25      31.88      27.92    19.12       0.79     191.89
  ##   I1       4 37665       0.88     0.32       1.00       0.98     0.00       0.00       1.00
  ##   I2       5 37639       0.93     0.25       1.00       1.00     0.00       0.00       1.00
  ##   I3       6 37473       0.76     0.43       1.00       0.83     0.00       0.00       1.00
  ##   I4       7 37687       1.88     0.39       2.00       2.00     0.00       0.00       2.00
  ##   I5       8 37638       1.36     0.75       2.00       1.44     0.00       0.00       2.00
  ##   I6       9 37587       1.05     0.82       1.00       1.06     1.48       0.00       2.00
  ##   I7      10 37576       1.55     0.85       2.00       1.57     1.48       0.00       3.00
  ##   I8      11 37044       0.45     0.50       0.00       0.44     0.00       0.00       1.00
  ##   I9      12 37249       0.48     0.50       0.00       0.47     0.00       0.00       1.00
  ##   I10     13 37318       0.63     0.48       1.00       0.66     0.00       0.00       1.00
  ##   I11     14 37412       1.35     0.80       1.00       1.35     1.48       0.00       3.00

## End(Not run)

Dataset for Polytomous GDINA Model

Description

Dataset for the estimation of the polytomous GDINA model.

Usage

data(data.pgdina)

Format

The dataset is a list with the item response data and the Q-matrix. The format is:

List of 2
$ dat : num [1:1000, 1:30] 1 1 1 1 1 0 1 1 1 1 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:30] "I1" "I2" "I3" "I4" ...
$ q.matrix: num [1:30, 1:5] 1 0 0 0 0 1 0 0 0 2 ...

Details

The dataset was simulated by the following R code:

set.seed(89)
# define Q-matrix
Qmatrix <- matrix(c(1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,
1,1,2,0,0,0,0,1,2,0,0,0,0,1,2,0,0,0,0,1,1,2,0,0,0,1,2,2,0,1,0,2,
1,0,0,1,1,0,2,2,0,0,2,1,0,1,0,0,2,2,1,2,0,0,0,0,0,2,0,0,0,0,0,2,
0,0,0,0,0,2,0,0,0,0,0,1,2,0,2,0,0,0,2,0,2,0,0,0,2,0,1,2,0,0,2,0,
0,2,0,0,1,1,0,0,1,1,0,1,1,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1),
nrow=30, ncol=5, byrow=TRUE )
# define covariance matrix between attributes
Sigma <- matrix(c(1,.6,.6,.3,.3,.6,1,.6,.3,.3,.6,.6,1,
.3,.3,.3,.3,.3,1,.8,.3,.3,.3,.8,1), 5,5, byrow=TRUE )
# define thresholds for attributes
q1 <- c( -.5, .9 ) # attributes 1,...,4
q2 <- c(0) # attribute 5
# number of persons
N <- 1000
# simulate latent attributes
alpha1 <- mvrnorm(n=N, mu=rep(0,5), Sigma=Sigma)
alpha <- 0*alpha1
for (aa in 1:4){
alpha[ alpha1[,aa] > q1[1], aa ] <- 1
alpha[ alpha1[,aa] > q1[2], aa ] <- 2
}
aa <- 5 ; alpha[ alpha1[,aa] > q2[1], aa ] <- 1
# define item parameters
guess <- c(.07,.01,.34,.07,.11,.23,.27,.07,.08,.34,.19,.19,.25,.04,.34,
.03,.29,.05,.01,.17,.15,.35,.19,.16,.08,.18,.19,.07,.17,.34)
slip <- c(0,.11,.14,.09,.03,.09,.03,.1,.14,.07,.06,.19,.09,.19,.07,.08,
.16,.18,.16,.02,.11,.12,.16,.14,.18,.01,.18,.14,.05,.18)
# simulate item responses
I <- 30 # number of items
dat <- latresp <- matrix( 0, N, I, byrow=TRUE)
for (ii in 1:I){
# ii <- 2
# latent response matrix
latresp[,ii] <- 1*( rowMeans( alpha >=matrix( Qmatrix[ ii, ], nrow=N,
ncol=5, byrow=TRUE ) )==1 )
# response probability
prob <- ifelse( latresp[,ii]==1, 1-slip[ii], guess[ii] )
# simulate item responses
dat[,ii] <- 1 * ( runif(N ) < prob )
}
colnames(dat) <- paste0("I",1:I)

References

Chen, J., & de la Torre, J. (2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37, 419-437.


PISA 2000 Reading Study (Chen & de la Torre, 2014)

Description

This is a sub-dataset of the PISA 2000 of German students including 26 items of the reading test. The 26 items was analyzed in Chen and de la Torre (2014) and a subset of 20 items was analyzed in Chen and Chen (2016).

Usage

data(data.pisa00R.ct)
data(data.pisa00R.cc)

Format

References

Chen, H., & Chen, J. (2016). Exploring reading comprehension skill relationships through the G-DINA model. Educational Psychology, 36(6), 1049-1064.

Chen, J., & de la Torre, J. (2014). A procedure for diagnostically modeling extant large-scale assessment data: the case of the programme for international student assessment in reading. Psychology, 5(18), 1967-1978.

Examples

#############################################################################
# EXAMPLE 1: PISA items from Chen and de la Torre (2014)
#            dichotomize item responses
#############################################################################

data(data.pisa00R.ct, package="CDM")
dat <- data.pisa00R.ct$data
Q <- data.pisa00R.ct$q.matrix
resp <- dat[, rownames(Q)]

#** extract item-wise maximum
maxK <- apply( resp, 2, max, na.rm=TRUE )
#** dichotomize response data
resp1 <- resp
for (ii in seq(1,ncol(resp)) ){
    resp1[,ii] <- 1 * ( resp[,ii]==maxK[ii] )
}

Dataset SDA6 (Jurich & Bradshaw, 2014)

Description

This is a simulated dataset of the SDA6 study according to informations given in Jurich and Bradshaw (2014).

Usage

data(data.sda6)

Format

The datasets contains 17 items observed at 1710 students.

The format is:

List of 2
$ data : num [1:1710, 1:17] 0 1 0 1 0 0 0 0 1 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:17] "MCM01" "MCM03" "MCM13" "MCM17" ...
$ q.matrix:'data.frame':
..$ CM: int [1:17] 1 1 1 1 0 0 0 0 0 0 ...
..$ II: int [1:17] 0 0 0 0 1 1 1 1 0 0 ...
..$ PP: int [1:17] 0 0 0 0 0 0 0 0 1 1 ...
..$ DG: int [1:17] 0 0 0 0 0 0 0 0 0 0 ...

The meaning of the skills is

CM – Critique Methods

II – Identify Improvements

PP – Protect Participants

DG – Discern Generalizability

Source

Simulated data

References

Jurich, D. P., & Bradshaw, L. P. (2014). An illustration of diagnostic classification modeling in student learning outcomes assessment. International Journal of Testing, 14, 49-72.

Examples

## Not run: 
data(data.sda6, package="CDM")

data <- data.sda6$data
q.matrix <- data.sda6$q.matrix

#*** Model 1a: LCDM with gdina
mod1a <- CDM::gdina( data, q.matrix, rule="ACDM", linkfct="logit",
                  reduced.skillspace=FALSE )
summary(mod1a)

#*** Model 1b: estimate LCDM with gdm
mod1b <- CDM::gdm( data, q.matrix=q.matrix, theta.k=c(0,1) )
summary(mod1b)

#*** Model 2: LCDM with hierarchy II > CM
B <- "II > CM"
ss2 <- CDM::skillspace.hierarchy(B=B, skill.names=colnames(q.matrix ) )
mod2 <- CDM::gdina( data, q.matrix, rule="ACDM", linkfct="logit",
                skillclasses=ss2$skillspace.reduced,
                reduced.skillspace=FALSE )
summary(mod2)

#*** Model 3: LCDM with hierarchy II > CM and DG > CM
B <- "II > CM
      DG > CM"
ss2 <- CDM::skillspace.hierarchy(B=B, skill.names=colnames(q.matrix ) )
mod3 <- CDM::gdina( data, q.matrix, rule="ACDM", linkfct="logit",
               skillclasses=ss2$skillspace.reduced,
               reduced.skillspace=FALSE )
summary(mod3)

# model comparisons
anova(mod1a,mod2)
anova(mod1a,mod3)
# model fit
summary( CDM::modelfit.cor.din(mod1a))
summary( CDM::modelfit.cor.din(mod2) )
summary( CDM::modelfit.cor.din(mod3) )

## End(Not run)

Dataset Student Questionnaire

Description

This dataset contains item responses of students at a scale of cultural activities (act), mathematics self concept (sc) and mathematics joyment (mj).

Usage

data(data.Students)

Format

A data frame with 2400 observations on the following 15 variables.

urban

Urbanization level: 1=town, 0=otherwise

female

A dummy variable for female student

act1

Visit a museum (0=never, 1=once or twice a year, 2=more than twice a year)

act2

Visit a theater or classical concert (0,1,2)

act3

Visit a rock or pop concert (0,1,2)

act4

Visit a cinema (0,1,2)

act5

Visit a public library (0,1,2)

sc1

Item 1 self concept "I am usually good at math." (0=do not agree at all, 1=rather do not agree, 2=rather agree, 3=completely agree)

sc2

Item 2 self concept: "Mathematics is harder for me than many of my classmates." (0,1,2,3) (reversed)

sc3

Item 3 self concept: "I am just not good at math." (0,1,2,3) (reversed)

sc4

Item 4 self concept: "I'm learning fast in math." (0,1,2,3)

mj1

Item 1 mathematics joyment: "I would like more math at school." (0,1,2,3)

mj2

Item 2 mathematics joyment: "I like to learn mathematics." (0,1,2,3)

mj3

Item 3 mathematics joyment: "Math is boring." (0,1,2,3) (reversed)

mj4

Item 4 mathematics joyment: "I like math." (0,1,2,3)

Source

Subsample of students from an Austrian survey of 8th grade students.

Examples

## Not run: 
library(psych)
data(data.Students, package="CDM")
psych::describe(data.Students)
  ##          var    n mean   sd median trimmed  mad min max range  skew kurtosis   se
  ##   urban    1 2400 0.31 0.46    0.0    0.27 0.00   0   1     1  0.81    -1.34 0.01
  ##   female   2 2400 0.51 0.50    1.0    0.51 0.00   0   1     1 -0.03    -2.00 0.01
  ##   act1     3 2248 0.65 0.73    0.5    0.56 0.74   0   2     2  0.64    -0.88 0.02
  ##   act2     4 2230 0.47 0.69    0.0    0.34 0.00   0   2     2  1.13    -0.06 0.01
  ##   act3     5 2218 0.33 0.60    0.0    0.21 0.00   0   2     2  1.62     1.48 0.01
  ##   act4     6 2342 1.35 0.76    2.0    1.44 0.00   0   2     2 -0.69    -0.96 0.02
  ##   act5     7 2223 0.52 0.74    0.0    0.40 0.00   0   2     2  1.05    -0.41 0.02
  ##   sc1      8 2352 0.96 0.80    1.0    0.91 1.48   0   3     3  0.45    -0.39 0.02
  ##   sc2      9 2347 0.90 0.88    1.0    0.81 1.48   0   3     3  0.66    -0.41 0.02
  ##   sc3     10 2335 0.86 0.96    1.0    0.73 1.48   0   3     3  0.84    -0.35 0.02
  ##   sc4     11 2337 1.29 0.90    1.0    1.24 1.48   0   3     3  0.24    -0.71 0.02
  ##   mj1     12 2351 2.26 0.82    2.0    2.37 1.48   0   3     3 -0.94     0.28 0.02
  ##   mj2     13 2345 1.89 0.91    2.0    1.95 1.48   0   3     3 -0.35    -0.80 0.02
  ##   mj3     14 2334 1.47 1.02    1.0    1.47 1.48   0   3     3  0.10    -1.11 0.02
  ##   mj4     15 2346 1.59 0.99    2.0    1.62 1.48   0   3     3 -0.03    -1.06 0.02

## End(Not run)

TIMSS 2003 Mathematics 8th Grade (Su et al., 2013)

Description

This is a dataset with a subset of 23 Mathematics items from TIMSS 2003 items used in Su, Choi, Lee, Choi and McAninch (2013).

Usage

data(data.timss03.G8.su)

Format

The data contains scored item responses (data), the Q-matrix (q.matrix) and further item informations (iteminfo).

The format is

List of 3
$ data :'data.frame':
..$ idstud : num [1:757] 1e+07 1e+07 1e+07 1e+07 1e+07 ...
..$ idbook : num [1:757] 1 1 1 1 1 1 1 1 1 1 ...
..$ M012001 : num [1:757] 0 1 0 0 1 0 1 0 0 0 ...
..$ M012002 : num [1:757] 1 1 0 1 0 0 1 1 1 1 ...
..$ M012004 : num [1:757] 0 1 1 1 1 0 1 1 0 0 ...
[...]
..$ M022234B: num [1:757] 0 0 0 0 0 0 0 0 0 0 ...
..$ M022251 : num [1:757] 0 0 0 0 0 0 0 0 0 0 ...
..$ M032570 : num [1:757] 1 1 0 1 0 0 1 1 1 1 ...
..$ M032643 : num [1:757] 1 0 0 0 0 0 1 1 0 0 ...
$ q.matrix: int [1:23, 1:13] 1 0 0 0 0 0 1 0 0 0 ...
..- attr(*, "dimnames")=List of 2
.. ..$ : chr [1:23] "M012001" "M012002" "M012004" "M012016" ...
.. ..$ : chr [1:13] "S1" "S2" "S3" "S4" ...
$ iteminfo: chr [1:23, 1:9] "M012001" "M012002" "M012004" "M012016" ...
..- attr(*, "dimnames")=List of 2
.. ..$ : NULL
.. ..$ : chr [1:9] "item" "ItemType" "reporting_category" "content" ...

For a detailed description of skills S1, S2, ..., S15 see Su et al. (2013, Table 2).

Source

Subset of US 8th graders (Booklet 1) in the TIMSS 2003 mathematics study

References

Skaggs, G., Wilkins, J. L. M., & Hein, S. F. (2016). Grain size and parameter recovery with TIMSS and the general diagnostic model. International Journal of Testing, 16(4), 310-330.

Su, Y.-L., Choi, K. M., Lee, W.-C., Choi, T., & McAninch, M. (2013). Hierarchical cognitive diagnostic analysis for TIMSS 2003 mathematics. CASMA Research Report 35. Center for Advanced Studies in Measurement and Assessment (CASMA), University of Iowa.

See Also

The TIMSS 2003 dataset for 8th graders (with a larger number of items) was also analyzed in Skaggs, Wilkins and Hein (2016).

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Data Su et al. (2013)
#############################################################################

data(data.timss03.G8.su, package="CDM")
data <- data.timss03.G8.su$data[,-c(1,2)]
q.matrix <- data.timss03.G8.su$q.matrix

#*** Model 1: DINA model with complete skill space of 2^13=8192 skill classes
mod1 <- CDM::din( data, q.matrix )

#*** Model 2: Skill space approximation with 3000 skill classes instead of
#    2^13=8192 classes as in Model 1
ss2 <- CDM::skillspace.approximation( L=3000, K=ncol(q.matrix) )
mod2 <- CDM::din( data, q.matrix, skillclasses=ss2 )

#*** Model 3: DINA model with a hierarchical skill space
#   see Su et al. (2013): Fig. 6
B <- "S1 > S2 > S7 > S8
      S15 > S9
      S3 > S9
      S13 > S4 > S9
      S14 > S5 > S6 > S11"
# Note that S10 and S12 are not included in the dataset contained in this package
skill.names <- colnames(q.matrix)
ss3 <- CDM::skillspace.hierarchy(B=B, skill.names=skill.names)
# The reduced skill space "only" contains 325 skill classes
mod3 <- CDM::din( data, q.matrix, skillclasses=ss3$skillspace.reduced )

## End(Not run)

TIMSS 2007 Mathematics 4th Grade (Lee et al., 2011)

Description

TIMSS 2007 (Grade 4) dataset with 25 mathematics (dichotomized) items used in Lee, Park and Taylan (2011), Park and Lee (2014) and Park, Xing and Lee (2018). The dataset includes a sample of 698 Austrian students.

Usage

data(data.timss07.G4.lee)
data(data.timss07.G4.py)
data(data.timss07.G4.Qdomains)

Format

Source

TIMSS 2007 study, 4th Grade, Austrian sample on booklets 4 and 5

References

Lee, Y. S., Park, Y. S., & Taylan, D. (2011). A cognitive diagnostic modeling of attribute mastery in Massachusetts, Minnesota, and the US national sample using the TIMSS 2007. International Journal of Testing, 11, 144-177.

Park, Y. S., & Lee, Y. S. (2014). An extension of the DINA model using covariates: Examining factors affecting response probability and latent classification. Applied Psychological Measurement, 38(5), 376-390.

Park, Y. S., Xing, K., & Lee, Y. S. (2018). Explanatory cognitive diagnostic models: Incorporating latent and observed predictors. Applied Psychological Measurement, 42(5), 376-392.

Yamaguchi, K., & Okada, K. (2018). Comparison among cognitive diagnostic models for the TIMSS 2007 fourth grade mathematics assessment. PloS ONE, 13(2), e0188691.

See Also

A comparison of several countries based on the 25 items is conducted in Yamaguchi and Okada (2018).

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: DINA model Lee et al. (2011) - 15 skills
#############################################################################

data(data.timss07.G4.lee, package="CDM")
dat <- data.timss07.G4.lee$data
q.matrix <- data.timss07.G4.lee$q.matrix
# extract items
items <- grep( "M0", colnames(dat), value=TRUE )

#*** Model 1: estimate DINA model
mod1 <- CDM::din( dat[,items], q.matrix )
summary(mod1)

#############################################################################
# EXAMPLE 2: DINA models Park and Lee (2014) - 7 skills and 3 skills
#############################################################################

data(data.timss07.G4.lee, package="CDM")
data(data.timss07.G4.py, package="CDM")
data(data.timss07.G4.Qdomains, package="CDM")

dat <- data.timss07.G4.lee$data
q.matrix <- data.timss07.G4.py$q.matrix
items <- rownames(q.matrix)

#*** Model 1: estimate DINA model
mod1 <- CDM::din( dat[,items], q.matrix )
summary(mod1)

#*** Model 2: estimate DINA model with Q-matrix defined by domains
Q <- data.timss07.G4.Qdomains
mod2 <- CDM::din( dat[,items], q.matrix=Q )
summary(mod2)

## End(Not run)

TIMSS 2011 Mathematics 4th Grade Austrian Students

Description

This is the TIMSS 2011 dataset of 4668 Austrian fourth-graders. See George and Robitzsch (2014, 2015, 2018) for publications using the TIMSS 2011 dataset for cognitive diagnosis modeling. The dataset has also been analyzed by Sedat and Arican (2015).

Usage

data(data.timss11.G4.AUT)
data(data.timss11.G4.AUT.part)
data(data.timss11.G4.sa)

Format

References

George, A. C., & Robitzsch, A. (2014). Multiple group cognitive diagnosis models, with an emphasis on differential item functioning. Psychological Test and Assessment Modeling, 56(4), 405-432.

George, A. C., & Robitzsch, A. (2015) Cognitive diagnosis models in R: A didactic. The Quantitative Methods for Psychology, 11, 189-205.

George, A. C., & Robitzsch, A. (2018). Focusing on interactions between content and cognition: A new perspective on gender differences in mathematical sub-competencies. Applied Measurement in Education, 31(1), 79-97.

Sedat, S. E. N., & Arican, M. (2015). A diagnostic comparison of Turkish and Korean students' Mathematics performances on the TIMSS 2011 assessment. Journal of Measurement and Evaluation in Education and Psychology, 6(2), 238-253.


Variance Matrix of a Nonlinear Estimator Using the Delta Method

Description

Computes the variance of a nonlinear parameter using the delta method.

Usage

deltaMethod(derived.pars, est, Sigma, h=1e-05)

Arguments

derived.pars

Vector of derived parameters written in R formula framework (see Examples).

est

Vector of parameter estimates

Sigma

Covariance matrix of parameters

h

Numerical differentiation parameter

Value

coef

Vector of nonlinear parameters

vcov

Covariance matrix of nonlinear parameters

se

Vector of standard errors

A

First derivative of nonlinear transformation

univarTest

Data frame containing univariate summary of nonlinear parameters

WaldTest

Multivariate parameter test for nonlinear parameter

See Also

See car::deltaMethod or msm::deltamethod.

Examples

#############################################################################
# EXAMPLE 1: Nonlinear parameter
#############################################################################

#-- parameter estimate
est <- c( 510.67, 102.57)
names(est) <- c("mu", "sigma")
#-- covariance matrix
Sigma <- matrix( c(5.83, 0.45, 0.45, 3.21 ), nrow=2, ncol=2 )
colnames(Sigma) <- rownames(Sigma) <- names(est)
#-- define derived nonlinear parameters
derived.pars <- list( "d"=~ I( ( mu - 508 ) / sigma ),
                      "dsig"=~ I( sigma / 100 - 1) )

#*** apply delta method
res <- CDM::deltaMethod( derived.pars, est, Sigma )
res

Parameter Estimation for Mixed DINA/DINO Model

Description

din provides parameter estimation for cognitive diagnosis models of the types “DINA”, “DINO” and “mixed DINA and DINO”.

Usage

din(data, q.matrix, skillclasses=NULL,
      conv.crit=0.001, dev.crit=10^(-5), maxit=500,
      constraint.guess=NULL, constraint.slip=NULL,
      guess.init=rep(0.2, ncol(data)), slip.init=guess.init,
      guess.equal=FALSE, slip.equal=FALSE, zeroprob.skillclasses=NULL,
      weights=rep(1, nrow(data)), rule="DINA",
      wgt.overrelax=0, wgtest.overrelax=FALSE, param.history=FALSE,
      seed=0, progress=TRUE, guess.min=0, slip.min=0, guess.max=1, slip.max=1)

## S3 method for class 'din'
print(x, ...)

Arguments

data

A required N×JN \times J data matrix containing the binary responses, 0 or 1, of NN respondents to JJ test items, where 1 denotes a correct response and 0 an incorrect one. The nth row of the matrix represents the binary response pattern of respondent n. NA values are allowed.

q.matrix

A required binary J×KJ \times K containing the attributes not required or required, 0 or 1, to master the items. The jth row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item jj.

skillclasses

An optional matrix for determining the skill space. The argument can be used if a user wants less than 2K2^K skill classes.

conv.crit

A numeric which defines the termination criterion of iterations in the parameter estimation process. Iteration ends if the maximal change in parameter estimates is below this value.

dev.crit

A numeric value which defines the termination criterion of iterations in relative change in deviance.

maxit

An integer which defines the maximum number of iterations in the estimation process.

constraint.guess

An optional matrix of fixed guessing parameters. The first column of this matrix indicates the numbers of the items whose guessing parameters are fixed and the second column the values the guessing parameters are fixed to.

constraint.slip

An optional matrix of fixed slipping parameters. The first column of this matrix indicates the numbers of the items whose slipping parameters are fixed and the second column the values the slipping parameters are fixed to.

guess.init

An optional initial vector of guessing parameters. Guessing parameters are bounded between 0 and 1.

slip.init

An optional initial vector of slipping parameters. Slipping parameters are bounded between 0 and 1.

guess.equal

An optional logical indicating if all guessing parameters are equal to each other. Default is FALSE.

slip.equal

An optional logical indicating if all slipping parameters are equal to each other. Default is FALSE.

zeroprob.skillclasses

An optional vector of integers which indicates which skill classes should have zero probability. Default is NULL (no skill classes with zero probability).

weights

An optional vector of weights for the response pattern. Non-integer weights allow for different sampling schemes.

rule

An optional character string or vector of character strings specifying the model rule that is used. The character strings must be of "DINA" or "DINO". If a vector of character strings is specified, implying an item wise condensation rule, the vector must be of length JJ, which is the number of items. The default is the condensation rule "DINA" for all items.

wgt.overrelax

A parameter which is relevant when an overrelaxation algorithm is used

wgtest.overrelax

A logical which indicates if the overrelexation parameter being estimated during iterations

param.history

A logical which indicates if the parameter history during iterations should be saved. The default is FALSE.

seed

Simulation seed for initial parameters. A value of zero corresponds to deterministic starting values, an integer value different from zero to random initial values with set.seed(seed).

progress

An optional logical indicating whether the function should print the progress of iteration in the estimation process.

guess.min

Minimum value of guessing parameters to be estimated.

slip.min

Minimum value of slipping parameters to be estimated.

guess.max

Maximum value of guessing parameters to be estimated.

slip.max

Maximum value of slipping parameters to be estimated.

x

Object of class din

...

Further arguments to be passed

Details

In the CDM DINA (deterministic-input, noisy-and-gate; de la Torre & Douglas, 2004) and DINO (deterministic-input, noisy-or-gate; Templin & Henson, 2006) models endorsement probabilities are modeled based on guessing and slipping parameters, given the different skill classes. The probability of respondent nn (or corresponding respondents class nn) for solving item jj is calculated as a function of the respondent's latent response ηnj\eta_{nj} and the guessing and slipping rates gjg_j and sjs_j for item jj conditional on the respondent's skill class αn\alpha_n:

P(Xnj=1αn)=gj(1ηnj)(1sj)ηnj.P(X_{nj}=1 | \alpha_n)=g_j^{(1- \eta_{nj})}(1 - s_j) ^{\eta_{nj}}.

The respondent's latent response (class) ηnj\eta_{nj} is a binary number, 0 or 1, where 1 indicates presence of all (rule="DINO") or at least one (rule="DINO") required skill(s) for item jj, respectively.

DINA and DINO parameter estimation is performed by maximization of the marginal likelihood of the data. The a priori distribution of the skill vectors is a uniform distribution. The implementation follows the EM algorithm by de la Torre (2009).

The function din returns an object of the class din (see ‘Value’), for which plot, print, and summary methods are provided; plot.din, print.din, and summary.din, respectively.

Value

coef

Estimated model parameters. Note that only freely estimated parameters are included.

item

A data frame giving for each item condensation rule, the estimated guessing and slipping parameters and their standard errors. All entries are rounded to 3 digits.

guess

A data frame giving the estimated guessing parameters and their standard errors for each item.

slip

A data frame giving the estimated slipping parameters and their standard errors for each item.

IDI

A matrix giving the item discrimination index (IDI; Lee, de la Torre & Park, 2012) for each item jj

IDIj=1sjgj,IDI_j=1 - s_j - g_j,

where a high IDI corresponds to good test items which have both low guessing and slipping rates. Note that a negative IDI indicates violation of the monotonicity condition gj<1sjg_j < 1 - s_j. See din for help.

itemfit.rmsea

The RMSEA item fit index (see itemfit.rmsea).

mean.rmsea

Mean of RMSEA item fit indexes.

loglike

A numeric giving the value of the maximized log likelihood.

AIC

A numeric giving the AIC value of the model.

BIC

A numeric giving the BIC value of the model.

Npars

Number of estimated parameters

posterior

A matrix given the posterior skill distribution for all respondents. The nth row of the matrix gives the probabilities for respondent n to possess any of the 2K2^K skill classes.

like

A matrix giving the values of the maximized likelihood for all respondents.

data

The input matrix of binary response data.

q.matrix

The input matrix of the required attributes.

pattern

A matrix giving the skill classes leading to highest endorsement probability for the respective response pattern (mle.est) with the corresponding posterior class probability (mle.post), the attribute classes having the highest occurrence posterior probability given the response pattern (map.est) with the corresponding posterior class probability (map.post), and the estimated posterior for each response pattern (pattern).

attribute.patt

A data frame giving the estimated occurrence probabilities of the skill classes and the expected frequency of the attribute classes given the model.

skill.patt

A matrix given the population prevalences of the skills.

subj.pattern

A vector of strings indicating the item response pattern for each subject.

attribute.patt.splitted

A dataframe giving the skill class of the respondents.

display

A character giving the model specified under rule.

item.patt.split

A matrix giving the splitted response pattern.

item.patt.freq

A numeric vector given the frequencies of the response pattern in item.patt.split.

seed

Used simulation seed for initial parameters

partable

Parameter table which is used for coef and vcov.

vcov.derived

Design matrix for extended set of parameters in vcov.

converged

Logical indicating whether convergence was achieved.

control

Optimization parameters used in estimation

Note

The calculation of standard errors using sampling weights which represent multistage sampling schemes is not correct. Please use replication methods (like Jackknife) instead.

References

de la Torre, J. (2009). DINA model parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34, 115–130.

de la Torre, J., & Douglas, J. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.

Lee, Y.-S., de la Torre, J., & Park, Y. S. (2012). Relationships between cognitive diagnosis, CTT, and IRT indices: An empirical investigation. Asia Pacific Educational Research, 13, 333-345.

Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic Measurement: Theory, Methods, and Applications. New York: The Guilford Press.

Templin, J., & Henson, R. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287–305.

See Also

plot.din, the S3 method for plotting objects of the class din; print.din, the S3 method for printing objects of the class din; summary.din, the S3 method for summarizing objects of the class din, which creates objects of the class summary.din; din, the main function for DINA and DINO parameter estimation, which creates objects of the class din.

See the gdina function for the estimation of the generalized DINA (GDINA) model.

For assessment of model fit see modelfit.cor.din and anova.din.

See itemfit.sx2 for item fit statistics.

See discrim.index for computing discrimination indices.

See also CDM-package for general information about this package.

See the NPCD::JMLE function in the NPCD package for joint maximum likelihood estimation of the DINA, DINO and NIDA model.

See the dina::DINA_Gibbs function in the dina package for MCMC based estimation of the DINA model.

Examples

#############################################################################
# EXAMPLE 1: Examples based on dataset fractions.subtraction.data
#############################################################################

## dataset fractions.subtraction.data and corresponding Q-Matrix
head(fraction.subtraction.data)
fraction.subtraction.qmatrix

## Misspecification in parameter specification for method CDM::din()
## leads to warnings and terminates estimation procedure. E.g.,

# See Q-Matrix specification
fractions.dina.warning1 <- CDM::din(data=fraction.subtraction.data,
  q.matrix=t(fraction.subtraction.qmatrix))

# See guess.init specification
fractions.dina.warning2 <- CDM::din(data=fraction.subtraction.data,
  q.matrix=fraction.subtraction.qmatrix, guess.init=rep(1.2,
  ncol(fraction.subtraction.data)))

# See rule specification
fractions.dina.warning3 <- CDM::din(data=fraction.subtraction.data,
  q.matrix=fraction.subtraction.qmatrix, rule=c(rep("DINA",
  10), rep("DINO", 9)))

## Parameter estimation of DINA model
# rule="DINA" is default
fractions.dina <- CDM::din(data=fraction.subtraction.data,
  q.matrix=fraction.subtraction.qmatrix, rule="DINA")
attributes(fractions.dina)
str(fractions.dina)

## For instance assessing the guessing parameters through
## assignment
fractions.dina$guess

## corresponding summaries, including IDI,
## most frequent skill classes and information
## criteria AIC and BIC
summary(fractions.dina)

## In particular, assessing detailed summary through assignment
detailed.summary.fs <- summary(fractions.dina)
str(detailed.summary.fs)

## Item discrimination index of item 8 is too low. This is also
## visualized in the first plot
plot(fractions.dina)

## The reason therefore is a high guessing parameter
round(fractions.dina$guess[,1], 2)

## Estimate DINA model with different random initial parameters using seed=1345
fractions.dina1 <- CDM::din(data=fraction.subtraction.data,
  q.matrix=fraction.subtraction.qmatrix, rule="DINA", seed=1345)

## Fix the guessing parameters of items 5, 8 and 9 equal to .20
# define a constraint.guess matrix
constraint.guess <-  matrix(c(5,8,9, rep(0.2, 3)), ncol=2)
fractions.dina.fixed <- CDM::din(data=fraction.subtraction.data,
  q.matrix=fraction.subtraction.qmatrix,
  constraint.guess=constraint.guess)

## The second plot shows the expected (MAP) and observed skill
## probabilities. The third plot visualizes the skill class
## occurrence probabilities; Only the 'top.n.skill.classes' most frequent
## skill classes are labeled; it is obvious that the skill class '11111111'
## (all skills are mastered) is the most probable in this population.
## The fourth plot shows the skill probabilities conditional on response
## patterns; in this population the skills 3 and 6 seem to be
## mastered easier than the others. The fourth plot shows the
## skill probabilities conditional on a specified response
## pattern; it is shown whether a skill is mastered (above
## .5+'uncertainty') unclassifiable (within the boundaries) or
## not mastered (below .5-'uncertainty'). In this case, the
## 527th respondent was chosen; if no response pattern is
## specified, the plot will not be shown (of course)
pattern <- paste(fraction.subtraction.data[527, ], collapse="")
plot(fractions.dina, pattern=pattern, display.nr=4)

#uncertainty=0.1, top.n.skill.classes=6 are default
plot(fractions.dina.fixed, uncertainty=0.1, top.n.skill.classes=6,
  pattern=pattern)

## Not run: 
#############################################################################
# EXAMPLE 2: Examples based on dataset sim.dina
#############################################################################

# DINA Model
d1 <- CDM::din(sim.dina, q.matr=sim.qmatrix, rule="DINA",
  conv.crit=0.01, maxit=500, progress=TRUE)
summary(d1)

# DINA model with hierarchical skill classes (Hierarchical DINA model)
# 1st step:  estimate an initial full model to look at the indexing
#    of skill classes
d0 <- CDM::din(sim.dina, q.matr=sim.qmatrix, maxit=1)
d0$attribute.patt.splitted
#      [,1] [,2] [,3]
# [1,]    0    0    0
# [2,]    1    0    0
# [3,]    0    1    0
# [4,]    0    0    1
# [5,]    1    1    0
# [6,]    1    0    1
# [7,]    0    1    1
# [8,]    1    1    1
#
# In this example, following hierarchical skill classes are only allowed:
# 000, 001, 011, 111
# We define therefore a vector of indices for skill classes with
# zero probabilities (see entries in the rows of the matrix
# d0$attribute.patt.splitted above)
zeroprob.skillclasses <- c(2,3,5,6)     # classes 100, 010, 110, 101
# estimate the hierarchical DINA model
d1a <- CDM::din(sim.dina, q.matr=sim.qmatrix,
          zeroprob.skillclasses=zeroprob.skillclasses )
summary(d1a)

# Mixed DINA and DINO Model
d1b <- CDM::din(sim.dina, q.matr=sim.qmatrix, rule=
          c(rep("DINA", 7), rep("DINO", 2)), conv.crit=0.01,
          maxit=500, progress=FALSE)
summary(d1b)

# DINO Model
d2 <- CDM::din(sim.dina, q.matr=sim.qmatrix, rule="DINO",
  conv.crit=0.01, maxit=500, progress=FALSE)
summary(d2)

# Comparison of DINA and DINO estimates
lapply(list("guessing"=rbind("DINA"=d1$guess[,1],
  "DINO"=d2$guess[,1]), "slipping"=rbind("DINA"=
  d1$slip[,1], "DINO"=d2$slip[,1])), round, 2)

# Comparison of the information criteria
c("DINA"=d1$AIC, "MIXED"=d1b$AIC, "DINO"=d2$AIC)

# following estimates:
d1$coef            # guessing and slipping parameter
d1$guess           # guessing parameter
d1$slip            # slipping parameter
d1$skill.patt      # probabilities for skills
d1$attribute.patt  # skill classes with probabilities
d1$subj.pattern    # pattern per subject

# posterior probabilities for every response pattern
d1$posterior

# Equal guessing parameters
d2a <- CDM::din( data=sim.dina, q.matrix=sim.qmatrix,
            guess.equal=TRUE, slip.equal=FALSE )
d2a$coef

# Equal guessing and slipping parameters
d2b <- CDM::din( data=sim.dina, q.matrix=sim.qmatrix,
            guess.equal=TRUE, slip.equal=TRUE )
d2b$coef

#############################################################################
# EXAMPLE 3: Examples based on dataset sim.dino
#############################################################################

# DINO Estimation
d3 <- CDM::din(sim.dino, q.matr=sim.qmatrix, rule="DINO",
        conv.crit=0.005, progress=FALSE)

# Mixed DINA and DINO Model
d3b <- CDM::din(sim.dino, q.matr=sim.qmatrix,
          rule=c(rep("DINA", 4), rep("DINO", 5)), conv.crit=0.001,
          progress=FALSE)

# DINA Estimation
d4 <- CDM::din(sim.dino, q.matr=sim.qmatrix, rule="DINA",
  conv.crit=0.005, progress=FALSE)

# Comparison of DINA and DINO estimates
lapply(list("guessing"=rbind("DINO"=d3$guess[,1],  "DINA"=d4$guess[,1]),
       "slipping"=rbind("DINO"=d3$slip[,1], "DINA"=d4$slip[,1])), round, 2)

# Comparison of the information criteria
c("DINO"=d3$AIC, "MIXED"=d3b$AIC, "DINA"=d4$AIC)

#############################################################################
# EXAMPLE 4: Example estimation with weights based on dataset sim.dina
#############################################################################

# Here, a weighted maximum likelihood estimation is used
# This could be useful for survey data.

# i.e. first 200 persons have weight 2, the other have weight 1
(weights <- c(rep(2, 200), rep(1, 200)))

d5 <- CDM::din(sim.dina, sim.qmatrix, rule="DINA", conv.crit=
  0.005, weights=weights, progress=FALSE)

# Comparison of the information criteria
c("DINA"=d1$AIC, "WEIGHTS"=d5$AIC)

#############################################################################
# EXAMPLE 5: Example estimation within a balanced incomplete
##           block (BIB) design generated on dataset sim.dina
#############################################################################

# generate BIB data

# The next example shows that the din function works for
# (relatively arbitrary) missing value pattern

# Here, a missing by design is generated in the dataset dinadat.bib
sim.dina.bib <- sim.dina
sim.dina.bib[1:100, 1:3] <- NA
sim.dina.bib[101:300, 4:8] <- NA
sim.dina.bib[301:400, c(1,2,9)] <- NA

d6 <- CDM::din(sim.dina.bib, sim.qmatrix, rule="DINA",
         conv.crit=0.0005, weights=weights, maxit=200)

d7 <- CDM::din(sim.dina.bib, sim.qmatrix, rule="DINO",
         conv.crit=0.005, weights=weights)

# Comparison of DINA and DINO estimates
lapply(list("guessing"=rbind("DINA"=d6$guess[,1],
  "DINO"=d7$guess[,1]), "slipping"=rbind("DINA"=
  d6$slip[,1], "DINO"=d7$slip[,1])), round, 2)

#############################################################################
# EXAMPLE 6: DINA model with attribute hierarchy
#############################################################################

set.seed(987)
# assumed skill distribution: P(000)=P(100)=P(110)=P(111)=.245 and
#     "deviant pattern": P(010)=.02
K <- 3 # number of skills

# define alpha
alpha <- scan()
    0 0 0
    1 0 0
    1 1 0
    1 1 1
    0 1 0

alpha <- matrix( alpha, length(alpha)/K, K, byrow=TRUE )
alpha <- alpha[ c( rep(1:4,each=245), rep(5,20) ),  ]

# define Q-matrix
q.matrix <- scan()
    1 0 0   1 0 0   1 0 0
    0 1 0   0 1 0   0 1 0
    0 0 1   0 1 0   0 0 1
    1 1 0   1 0 1   0 1 1

q.matrix <- matrix( q.matrix, nrow=length(q.matrix)/K, ncol=K, byrow=TRUE )

# simulate DINA data
dat <- CDM::sim.din( alpha=alpha, q.matrix=q.matrix )$dat

#*** Model 1: estimate DINA model | no skill space restriction
mod1 <- CDM::din( dat, q.matrix )

#*** Model 2: DINA model | hierarchy A2 > A3
B <- "A2 > A3"
skill.names <- paste0("A",1:3)
skillspace <- CDM::skillspace.hierarchy( B, skill.names )$skillspace.reduced
mod2 <- CDM::din( dat, q.matrix, skillclasses=skillspace )

#*** Model 3: DINA model | linear hierarchy A1 > A2 > A3
#   This is a misspecied model because due to P(010)=.02 the relation A1>A2
#   does not hold.
B <- "A1 > A2
      A2 > A3"
skill.names <- paste0("A",1:3)
skillspace <- CDM::skillspace.hierarchy( B, skill.names )$skillspace.reduced
mod3 <- CDM::din( dat, q.matrix, skillclasses=skillspace )

#*** Model 4: 2PL model in gdm
mod4 <- CDM::gdm( dat, theta.k=seq(-5,5,len=21),
           decrease.increments=TRUE, skillspace="normal" )
summary(mod4)

anova(mod1,mod2)
  ##       Model   loglike Deviance Npars      AIC      BIC  Chisq df       p
  ##   2 Model 2 -7052.460 14104.92    29 14162.92 14305.24 0.9174  2 0.63211
  ##   1 Model 1 -7052.001 14104.00    31 14166.00 14318.14     NA NA      NA

anova(mod2,mod3)
  ##       Model   loglike Deviance Npars      AIC      BIC    Chisq df       p
  ##   2 Model 2 -7059.058 14118.12    27 14172.12 14304.63 13.19618  2 0.00136
  ##   1 Model 1 -7052.460 14104.92    29 14162.92 14305.24       NA NA      NA

anova(mod2,mod4)
  ##       Model  loglike Deviance Npars      AIC      BIC    Chisq df  p
  ##   2 Model 2 -7220.05 14440.10    24 14488.10 14605.89 335.1805  5  0
  ##   1 Model 1 -7052.46 14104.92    29 14162.92 14305.24       NA NA NA

# compare fit statistics
summary( CDM::modelfit.cor.din( mod2 ) )
summary( CDM::modelfit.cor.din( mod4 ) )

#############################################################################
# EXAMPLE 7: Fitting the basic local independence model (BLIM) with din
#############################################################################

library(pks)
data(DoignonFalmagne7, package="pks")
  ##  str(DoignonFalmagne7)
  ##    $ K  : int [1:9, 1:5] 0 1 0 1 1 1 1 1 1 0 ...
  ##     ..- attr(*, "dimnames")=List of 2
  ##     .. ..$ : chr [1:9] "00000" "10000" "01000" "11000" ...
  ##     .. ..$ : chr [1:5] "a" "b" "c" "d" ...
  ##    $ N.R: Named int [1:32] 80 92 89 3 2 1 89 16 18 10 ...
  ##     ..- attr(*, "names")=chr [1:32] "00000" "10000" "01000" "00100" ...

# The idea is to fit the local independence model with the din function.
# This can be accomplished by specifying a DINO model with
# prespecified skill classes.

# extract dataset
dat <- as.numeric( unlist( sapply( names(DoignonFalmagne7$N.R),
    FUN=function( ll){ strsplit( ll, split="") } ) ) )
dat <- matrix( dat, ncol=5, byrow=TRUE )
colnames(dat) <- colnames(DoignonFalmagne7$K)
rownames(dat) <- names(DoignonFalmagne7$N.R)

# sample weights
weights <- DoignonFalmagne7$N.R

# define Q-matrix
q.matrix <- t(DoignonFalmagne7$K)
v1 <- colnames(q.matrix) <- paste0("S", colnames(q.matrix))
q.matrix <- q.matrix[, - 1] # remove S00000

# define skill classes
SC <- ncol(q.matrix)
skillclasses <- matrix( 0, nrow=SC+1, ncol=SC)
colnames(skillclasses) <- colnames(q.matrix)
rownames(skillclasses) <- v1
skillclasses[ cbind( 2:(SC+1), 1:SC ) ] <- 1

# estimate BLIM with din function
mod1 <- CDM::din(data=dat, q.matrix=q.matrix, skillclasses=skillclasses,
            rule="DINO", weights=weights   )
summary(mod1)
  ##   Item parameters
  ##     item guess  slip   IDI rmsea
  ##   a    a 0.158 0.162 0.680 0.011
  ##   b    b 0.145 0.159 0.696 0.009
  ##   c    c 0.008 0.181 0.811 0.001
  ##   d    d 0.012 0.129 0.859 0.001
  ##   e    e 0.025 0.146 0.828 0.007

# estimate basic local independence model with pks package
mod2 <- pks::blim(K, N.R, method="ML") # maximum likelihood estimation by EM algorithm
mod2
  ##   Error and guessing parameters
  ##         beta      eta
  ##   a 0.164871 0.103065
  ##   b 0.163113 0.095074
  ##   c 0.188839 0.000004
  ##   d 0.079835 0.000003
  ##   e 0.088648 0.019910

## End(Not run)

Identifiability Conditions of the DINA Model

Description

Check necessary and sufficient identifiability conditions of the DINA model according Gu and Xu (xxxx) for a given Q-matrix.

Usage

din_identifiability(q.matrix)

## S3 method for class 'din_identifiability'
summary(object, ...)

Arguments

q.matrix

Q-matrix

object

Object of class din_identifiability

...

Further arguments to be passed

Value

List with values

dina_identified

Logical indicating whether the DINA model is identified

index_single

Condition 1: vector of logicals indicating whether skills are measured by at least one item with a single loading

is_three_items

Condition 2: vector of logicals indicating whether skills are measured by at least three items

submat_distinct

Condition 3: logical indicating whether all columns of the submatrix QQ^\ast are distinct.

References

Gu, Y., & Xu, G. (2018). The sufficient and necessary condition for the identifiability and estimability of the DINA model. Psychometrika, xx(xx), xxx-xxx. https://doi.org/10.1007/s11336-018-9619-8

See Also

See din.equivalent.class for equivalent (i.e., non-distinguishable) skill classes in the DINA model.

Examples

#############################################################################
# EXAMPLE 1: Some examples of Gu and Xu (2019)
#############################################################################

#* Matrix 1 in Equation (5) of Gu & Xu (2019)
Q1 <- diag(3)
Q2 <- matrix( scan(text="1 1 0 1 0 1 1 1 1 1 1 1"), ncol=3, byrow=TRUE)
Q <- rbind(Q1, Q2)

res <- CDM::din_identifiability(q.matrix=Q)
summary(res)

# remove two items
res <- CDM::din_identifiability(q.matrix=Q[-c(2,5),])
summary(res)

#* Matrix 1 in Equation (6) of Gu & Xu (2019)
Q1 <- diag(3)
Q2 <- matrix( c(1,1,1), nrow=4, ncol=3, byrow=TRUE)
Q <- rbind(Q1, Q2)

res <- CDM::din_identifiability(q.matrix=Q)
summary(res)

Deterministic Classification and Joint Maximum Likelihood Estimation of the Mixed DINA/DINO Model

Description

This function allows the estimation of the mixed DINA/DINO model by joint maximum likelihood and a deterministic classification based on ideal latent responses.

Usage

din.deterministic(dat, q.matrix, rule="DINA", method="JML", conv=0.001,
    maxiter=300, increment.factor=1.05, progress=TRUE)

Arguments

dat

Data frame of dichotomous item responses

q.matrix

Q-matrix with binary entries (see din).

rule

The condensation rule (see din).

method

Estimation method. The default is joint maximum likelihood estimation (JML). Other options include an adaptive estimation of guessing and slipping parameters (adaptive) while using these estimated parameters as weights in the individual deviation function and classification based on the Hamming distance (hamming) and the weighted Hamming distance (weighted.hamming) (see Chiu & Douglas, 2013).

conv

Convergence criterion for guessing and slipping parameters

maxiter

Maximum number of iterations

increment.factor

A numeric value of at least one which could help to improve convergence behavior and decreases parameter increments in every iteration. This option is disabled by setting this argument to 1.

progress

An optional logical indicating whether the function should print the progress of iteration in the estimation process.

Value

A list with following entries

attr.est

Estimated attribute patterns

criterion

Criterion of the classification function. For joint maximum likelihood it is the deviance.

guess

Estimated guessing parameters

slip

Estimated slipping parameters

prederror

Average individual prediction error

q.matrix

Used Q-matrix

dat

Used data frame

References

Chiu, C. Y., & Douglas, J. (2013). A nonparametric approach to cognitive diagnosis by proximity to ideal response patterns. Journal of Classification, 30, 225-250.

See Also

For estimating the mixed DINA/DINO model using marginal maximum likelihood estimation see din.

See also the NPCD::JMLE function in the NPCD package for joint maximum likelihood estimation of the DINA or the DINO model.

Examples

#############################################################################
# EXAMPLE 1: 13 items and 3 attributes
#############################################################################

set.seed(679)
N <- 3000
# specify true Q-matrix
q.matrix <- matrix( 0, 13, 3 )
q.matrix[1:3,1] <- 1
q.matrix[4:6,2] <- 1
q.matrix[7:9,3] <- 1
q.matrix[10,] <- c(1,1,0)
q.matrix[11,] <- c(1,0,1)
q.matrix[12,] <- c(0,1,1)
q.matrix[13,] <- c(1,1,1)
q.matrix <- rbind( q.matrix, q.matrix )
colnames(q.matrix) <- paste0("Attr",1:ncol(q.matrix))

# simulate data according to the DINA model
dat <- CDM::sim.din( N=N, q.matrix)$dat

# Joint maximum likelihood estimation (the default: method="JML")
res1 <- CDM::din.deterministic( dat, q.matrix )

# Adaptive estimation of guessing and slipping parameters
res <- CDM::din.deterministic( dat, q.matrix, method="adaptive" )

# Classification using Hamming distance
res <- CDM::din.deterministic( dat, q.matrix, method="hamming" )

# Classification using weighted Hamming distance
res <- CDM::din.deterministic( dat, q.matrix, method="weighted.hamming" )

## Not run: 
#********* load NPCD library for JML estimation
library(NPCD)

# DINA model
res <- NPCD::JMLE( Y=dat[1:100,], Q=q.matrix, model="DINA" )
as.data.frame(res$par.est )   # item parameters
res$alpha.est                 # skill classifications

# RRUM model
res <- NPCD::JMLE( Y=dat[1:100,], Q=q.matrix, model="RRUM" )
as.data.frame(res$par.est )

## End(Not run)

Calculation of Equivalent Skill Classes in the DINA/DINO Model

Description

This function computes indistinguishable skill classes for the DINA and DINO model (Gross & George, 2014; Zhang, DeCarlo & Ying, 2013).

Usage

din.equivalent.class(q.matrix, rule="DINA")

Arguments

q.matrix

The Q-matrix (see din).

rule

The condensation rule. If it is a string, then the rule applies to all items. If it is a vector, then for each item DINA or DINO rule can be chosen.

Value

A list with following entries:

latent.responseM

Matrix of latent responses

latent.response

Latent responses represented as a string

S

Matrix containing all skill classes

gini

Gini coefficient of the frequency distribution of identifiable skill classes which result in the same latent response

skillclasses

Data frame with skill class (skillclass), latent responses (latent.response) and an identifier for distinguishable skill classes (distinguish.class).

References

Gross, J. & George, A. C. (2014). On prerequisite relations between attributes in noncompensatory diagnostic classification. Methodology, 10(3), 100-107.

Zhang, S. S., DeCarlo, L. T., & Ying, Z. (2013). Non-identifiability, equivalence classes, and attribute-specific classification in Q-matrix based cognitive diagnosis models. arXiv preprint, arXiv:1303.0426.

Examples

#############################################################################
# EXAMPLE 1: Equivalency classes for DINA model for fraction subtraction data
#############################################################################

#-- DINA models

data(data.fraction2, package="CDM")

# first Q-matrix
Q1 <- data.fraction2$q.matrix1
m1 <- CDM::din.equivalent.class( q.matrix=Q1, rule="DINA" )
  ## 8 Skill classes | 5  distinguishable skill classes | Gini coefficient=0.3

# second Q-matrix
Q1 <- data.fraction2$q.matrix2
m1 <- CDM::din.equivalent.class( q.matrix=Q1, rule="DINA" )
  ## 32 Skill classes | 9  distinguishable skill classes | Gini coefficient=0.5

# third Q-matrix
Q1 <- data.fraction2$q.matrix3
m1 <- CDM::din.equivalent.class( q.matrix=Q1, rule="DINA" )
  ## 8 Skill classes | 8  distinguishable skill classes | Gini coefficient=0

# original fraction subtraction data
m1 <- CDM::din.equivalent.class( q.matrix=CDM::fraction.subtraction.qmatrix, rule="DINA")
  ## 256 Skill classes | 58  distinguishable skill classes | Gini coefficient=0.659

Q-Matrix Validation (Q-Matrix Modification) for Mixed DINA/DINO Model

Description

Q-matrix entries can be modified by the Q-matrix validation method of de la Torre (2008). After estimating a mixed DINA/DINO model using the din function, item parameters and the item discrimination parameters IDIjIDI_j are recalculated. Q-matrix rows are determined by maximizing the estimated item discrimination index IDIj=1sjgjIDI_j=1-s_j -g_j.

Usage

din.validate.qmatrix(object, IDI_diff=.02, print=TRUE)

Arguments

object

Object of class din

IDI_diff

Minimum difference in IDI values for choosing a new Q-matrix vector

print

An optional logical indicating whether the function should print the progress of iteration in the estimation process.

Value

A list with following entries:

coef.modified

Estimated parameters by applying Q-matrix modifications

coef.modified.short

A shortened matrix of coef.modified. Only Q-matrix rows which increase the IDIIDI are displayed.

q.matrix.prop

The proposed Q-matrix by Q-matrix validation.

References

Chiu, C. Y. (2013). Statistical refinement of the Q-matrix in cognitive diagnosis. Applied Psychological Measurement, 37, 598-618.

de la Torre, J. (2008). An empirically based method of Q-matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45, 343-362.

See Also

The mixed DINA/DINO model can be estimated with din.

See Chiu (2013) for an alternative estimation approach based on residual sum of squares which is implemented NPCD::Qrefine function in the NPCD package.

See the GDINA::Qval function in the GDINA package for extended functionality.

Examples

#############################################################################
# EXAMPLE 1: Detection of a mis-specified Q-matrix
#############################################################################

set.seed(679)
# specify true Q-matrix
q.matrix <- matrix( 0, 12, 3 )
q.matrix[1:3,1] <- 1
q.matrix[4:6,2] <- 1
q.matrix[7:9,3] <- 1
q.matrix[10,] <- c(1,1,0)
q.matrix[11,] <- c(1,0,1)
q.matrix[12,] <- c(0,1,1)
# simulate data
dat <- CDM::sim.din( N=4000, q.matrix)$dat
# incorrectly modify Q-matrix rows 1 and 10
Q1 <- q.matrix
Q1[1,] <- c(1,1,0)
Q1[10,] <- c(1,0,0)
# estimate DINA model
mod <- CDM::din( dat, q.matr=Q1, rule="DINA")
# apply Q-matrix validation
res <- CDM::din.validate.qmatrix( mod )
  ## item itemindex Skill1 Skill2 Skill3 guess  slip   IDI qmatrix.orig IDI.orig delta.IDI max.IDI
  ## I001         1      1      0      0 0.309 0.251 0.440            0    0.431     0.009   0.440
  ## I010        10      1      1      0 0.235 0.329 0.437            0    0.320     0.117   0.437
  ## I010        10      1      1      1 0.296 0.301 0.403            0    0.320     0.083   0.437
  ##
  ##   Proposed Q-matrix:
  ##
  ##          Skill1 Skill2 Skill3
  ##   Item1       1      0      0
  ##   Item2       1      0      0
  ##   Item3       1      0      0
  ##   Item4       0      1      0
  ##   Item5       0      1      0
  ##   Item6       0      1      0
  ##   Item7       0      0      1
  ##   Item8       0      0      1
  ##   Item9       0      0      1
  ##   Item10      1      1      0
  ##   Item11      1      0      1
  ##   Item12      0      1      1

## Not run: 
#*****************
# Q-matrix estimation ('Qrefine') in the NPCD package
# See Chiu (2013, APM).
#*****************

library(NPCD)
Qrefine.out <- NPCD::Qrefine( dat, Q1, gate="AND", max.ite=50)
print(Qrefine.out)
  ##   The modified Q-matrix
  ##           Attribute 1 Attribute 2 Attribute 3
  ##   Item 1            1           0           0
  ##   Item 2            1           0           0
  ##   Item 3            1           0           0
  ##   Item 4            0           1           0
  ##   Item 5            0           1           0
  ##   Item 6            0           1           0
  ##   Item 7            0           0           1
  ##   Item 8            0           0           1
  ##   Item 9            0           0           1
  ##   Item 10           1           1           0
  ##   Item 11           1           0           1
  ##   Item 12           0           1           1
  ##
  ##   The modified entries
  ##        Item Attribute
  ##   [1,]    1         2
  ##   [2,]   10         2

plot(Qrefine.out)

## End(Not run)

Discrimination Indices at Item-Attribute, Item and Test Level

Description

Computes discrimination indices at the probability metric (de la Torre, 2008; Henson, DiBello & Stout, 2018).

Usage

discrim.index(object, ...)

## S3 method for class 'din'
discrim.index(object, ...)

## S3 method for class 'gdina'
discrim.index(object, ...)

## S3 method for class 'mcdina'
discrim.index(object, ...)

## S3 method for class 'discrim.index'
summary(object, file=NULL, digits=3,  ...)

Arguments

object

Object of class din or gdina.

file

Optional file name for a file in which the summary output should be sunk

digits

Number of digits for rounding

...

Further arguments to be passed

Details

If item jj possesses HjH_j categories, the item-attribute specific discrimination for attribute kk according to Henson et al. (2018) is defined as

DIjk=12maxα(h=1HjP(Xj=hα)P(Xj=hα(k)))DI_{jk}=\frac{1}{2} \max_{ \bm{\alpha} } \left( \sum_{h=1}^{H_j} | P(X_j=h| \bm{\alpha} ) - P(X_j=h| \bm{\alpha}^{(-k)} ) | \right )

where α(k)\bm{\alpha}^{(-k)} and α\bm{\alpha} differ only in attribute kk. The index DIjkDI_{jk} can be found as the value discrim_item_attribute. The test-level discrimination index is defined as

DI=1Jj=1JmaxkDIjk\overline{DI}=\frac{1}{J} \sum_{j=1}^J \max_k DI_{jk}

and can be found in discrim_test.

According to de la Torre (2008) and de la Torre, Rossi and van der Ark (2018), the item discrimination index (IDI) is defined as

IDIj=maxα1,α2,hP(Xj=hα1)P(Xj=hα2)IDI_j=\max_{ \bm{\alpha}_1,\bm{\alpha}_2, h} | P(X_j=h| \bm{\alpha}_1 ) - P(X_j=h| \bm{\alpha}_2 ) |

and can be found as idi in the values list.

Value

A list with following entries

discrim_item_attribute

Discrimination indices DIjkDI_{jk} at item level for each attribute

idi

Item discrimination index IDIjIDI_j

discrim_test

Discrimination index at test level

References

de la Torre, J. (2008). An empirically based method of Q-matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45, 343-362.
http://dx.doi.org/10.1111/j.1745-3984.2008.00069.x

de la Torre, J., van der Ark, L. A., & Rossi, G. (2018). Analysis of clinical data from a cognitive diagnosis modeling framework. Measurement and Evaluation in Counseling and Development, 51(4), 281-296. https://doi.org/10.1080/07481756.2017.1327286

Henson, R., DiBello, L., & Stout, B. (2018). A generalized approach to defining item discrimination for DCMs. Measurement: Interdisciplinary Research and Perspectives, 16(1), 18-29.
http://dx.doi.org/10.1080/15366367.2018.1436855

See Also

See cdi.kli for discrimination indices based on the Kullback-Leibler information.

For a fitted model mod in the GDINA package, discrimination indices can be extracted by the method extract(mod,"discrim") (GDINA::extract).

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: DINA and GDINA model
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

#-- fit GDINA and DINA model
mod1 <- CDM::gdina( sim.dina, q.matrix=sim.qmatrix )
mod2 <- CDM::din( sim.dina, q.matrix=sim.qmatrix )

#-- compute discrimination indices
dimod1 <- CDM::discrim.index(mod1)
dimod2 <- CDM::discrim.index(mod2)
summary(dimod1)
summary(dimod2)

## End(Not run)

Test-specific and Item-specific Entropy for Latent Class Models

Description

Computes test-specific and item-specific entropy as test-diagnostic criteria of cognitive diagnostic models (Asparouhov & Muthen, 2014).

Usage

entropy.lca(object)

## S3 method for class 'entropy.lca'
summary(object, digits=2,  ...)

Arguments

object

Object of class din, gdina or mcdina. For the summary method, it is the result of entropy.lca.

digits

Number of digits to round

...

Further arguments to be passed

Value

A list with the data frame entropy as an entry.

References

Asparouhov, T. & Muthen, B. (2014). Variable-specific entropy contribution. Technical Appendix. http://www.statmodel.com/7_3_papers.shtml

See Also

See cdi.kli for test diagnostic indices based on the Kullback-Leibler information and cdm.est.class.accuracy for calculating the classification accuracy.

Examples

#############################################################################
# EXAMPLE 1: Entropy for DINA model
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

# fit DINA Model
mod1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix, rule="DINA")
summary(mod1)
# compute entropy for test and items
emod1 <- CDM::entropy.lca( mod1 )
summary(emod1)

## Not run: 
#############################################################################
# EXAMPLE 2: Entropy for polytomous GDINA model
#############################################################################

data(data.pgdina, package="CDM")

dat <- data.pgdina$dat
q.matrix <- data.pgdina$q.matrix

# pGDINA model with "DINA rule"
mod1 <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA")
summary(mod1)

# compute entropy
emod1 <- CDM::entropy.lca( mod1 )
summary(emod1)

#############################################################################
# EXAMPLE 3: Entropy for MCDINA model
#############################################################################

data(data.cdm02, package="CDM")

dat <- data.cdm02$data
q.matrix <- data.cdm02$q.matrix

# estimate model with polytomous atribute
mod1 <- CDM::mcdina( dat, q.matrix=q.matrix )
summary(mod1)
# computre entropy
emod1 <- CDM::entropy.lca( mod1 )
summary(emod1)

## End(Not run)

Determination of a Statistically Equivalent DINA Model

Description

This function determines a statistically equivalent DINA model given a Q-matrix using the method of von Davier (2014). Thereby, the dimension of the skill space is expanded, but in the reparameterized version, the Q-matrix has a simple structure or the IRT model is no longer be conjuctive (like in DINA) due to a redefinition of the skill space.

Usage

equivalent.dina(q.matrix, reparameterization="B")

Arguments

q.matrix

The Q-matrix (see din)

reparameterization

The used reparameterization (see von Davier, 2014). A and B are possible reparameterizations.

Value

A list with following entries

q.matrix

Original Q-matrix

q.matrix.ast

Reparameterized Q-matrix

alpha

Original skill space

alpha.ast

Reparameterized skill space

References

von Davier, M. (2014). The DINA model as a constrained general diagnostic model: Two variants of a model equivalency. British Journal of Mathematical and Statistical Psychology, 67, 49-71.

Examples

#############################################################################
# EXAMPLE 1: Toy example
#############################################################################

# define a Q-matrix
Q <- matrix( c( 1,0,0,  0,1,0,
        0,0,1,   1,0,1,  1,1,1 ), byrow=TRUE, ncol=3 )
Q <- Q[ rep(1:(nrow(Q)),each=2), ]

# equivalent DINA model (using the default reparameterization B)
res1 <- CDM::equivalent.dina( q.matrix=Q )
res1

# equivalent DINA model (reparametrization A)
res2 <- CDM::equivalent.dina( q.matrix=Q, reparameterization="A")
res2

## Not run: 
#############################################################################
# EXAMPLE 2: Estimation with two equivalent DINA models
#############################################################################

# simulate data
set.seed(789)
D <- ncol(Q)
mean.alpha <- c( -.5, .5, 0  )
r1 <- .5
Sigma.alpha <- matrix( r1, D, D ) + diag(1-r1,D)
dat1 <- CDM::sim.din( N=2000, q.matrix=Q, mean=mean.alpha, Sigma=Sigma.alpha )

# estimate DINA model
mod1 <- CDM::din( dat1$dat, q.matrix=Q )

# estimate equivalent DINA model
mod2 <- CDM::din( dat1$dat, q.matrix=res1$q.matrix.ast, skillclasses=res1$alpha.ast)
# restricted skill space must be defined by using the argument 'skillclasses'

# compare model summaries
summary(mod2)
summary(mod1)

# compare estimated item parameters
cbind( mod2$coef, mod1$coef )

# compare estimated skill class probabilities
round( cbind( mod2$attribute.patt, mod1$attribute.patt ), 4 )


#############################################################################
# EXAMPLE 3: Examples from von Davier (2014)
#############################################################################

# define Q-matrix
Q <- matrix( 0, nrow=8, ncol=3 )
Q[2, ] <- c(1,0,0)
Q[3, ] <- c(0,1,0)
Q[4, ] <- c(1,1,0)
Q[5, ] <- c(0,0,1)
# Q[6, ] <- c(1,0,1)
Q[6, ] <- c(0,0,1)
Q[7, ] <- c(0,1,1)
Q[8, ] <- c(1,1,1)

#- parametrization A
res1 <- CDM::equivalent.dina(q.matrix=Q, reparameterization="A")
res1

#- parametrization B
res2 <- CDM::equivalent.dina(q.matrix=Q, reparameterization="B")
res2

## End(Not run)

Evaluation of Likelihood

Description

The function eval_likelihood evaluates the likelihood given item responses and item response probabilities.

The function prep_data_long_format stores the matrix of item responses in a long format omitted all missing responses.

Usage

eval_likelihood(data, irfprob, prior=NULL, normalization=FALSE, N=NULL)

prep_data_long_format(data)

Arguments

data

Dataset containing item responses in wide format or long format (generated by prep_data_long_format).

irfprob

Array containing item responses probabilities, format see IRT.irfprob

prior

Optional prior (matrix or vector)

normalization

Logical indicating whether posterior should be normalized

N

Number of persons (optional)

Value

Numeric matrix

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Likelihood data.ecpe
#############################################################################

data(data.ecpe, package="CDM")
dat <- data.ecpe$dat[,-1]
Q <- data.ecpe$q.matrix

#*** store data matrix in long format
data_long <- CDM::prep_data_long_format(data)
str(data_long)

#** estimate GDINA model
mod <- CDM::gdina(dat, q.matrix=Q)
summary(mod)

#** extract data, item response functions and prior
data <- CDM::IRT.data(mod)
irfprob <- CDM::IRT.irfprob(mod)
prob_theta <- attr( irfprob, "prob.theta")

#** compute likelihood
lmod <- CDM::eval_likelihood(data=data, irfprob=irfprob)
max( abs( lmod - CDM::IRT.likelihood(mod) ))

#** compute posterior
pmod <- CDM::eval_likelihood(data=data, irfprob=irfprob, prior=prob.theta,
            normalization=TRUE)
max( abs( pmod - CDM::IRT.posterior(mod) ))

## End(Not run)

Fraction Subtraction Data

Description

Tatsuoka's (1984) fraction subtraction data set is comprised of responses to J=20J=20 fraction subtraction test items from N=536N=536 middle school students.

Usage

data(fraction.subtraction.data)

Format

The fraction.subtraction.data data frame consists of 536 rows and 20 columns, representing the responses of the N=536N=536 students to each of the J=20J=20 test items. Each row in the data set corresponds to the responses of a particular student. Thereby a "1" denotes that a correct response was recorded, while "0" denotes an incorrect response. The other way round, each column corresponds to all responses to a particular item.

Details

The items used for the fraction subtraction test originally appeared in Tatsuoka (1984) and are published in Tatsuoka (2002). They can also be found in DeCarlo (2011). All test items are based on 8 attributes (e.g. convert a whole number to a fraction, separate a whole number from a fraction or simplify before subtracting). The complete list of skills can be found in fraction.subtraction.qmatrix.

Source

The Royal Statistical Society Datasets Website, Series C, Applied Statistics, Data analytic methods for latent partially ordered classification models:
URL: http://www.blackwellpublishing.com/rss/Volumes/Cv51p2_read2.htm

References

DeCarlo, L. T. (2011). On the analysis of fraction subtraction data: The DINA Model, classification, latent class sizes, and the Q-Matrix. Applied Psychological Measurement, 35, 8–26.

Tatsuoka, C. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society, Series C, Applied Statistics, 51, 337–350.

Tatsuoka, K. (1984). Analysis of errors in fraction addition and subtraction problems. Final Report for NIE-G-81-0002, University of Illinois, Urbana-Champaign.

See Also

fraction.subtraction.qmatrix for the corresponding Q-matrix.


Fraction Subtraction Q-Matrix

Description

The Q-Matrix corresponding to Tatsuoka (1984) fraction subtraction data set.

Usage

data(fraction.subtraction.qmatrix)

Format

The fraction.subtraction.qmatrix data frame consists of J=20J=20 rows and K=8K=8 columns, specifying the attributes that are believed to be involved in solving the items. Each row in the data frame represents an item and the entries in the row indicate whether an attribute is needed to master the item (denoted by a "1") or not (denoted by a "0"). The attributes for the fraction subtraction data set are the following:

alpha1

convert a whole number to a fraction,

alpha2

separate a whole number from a fraction,

alpha3

simplify before subtracting,

alpha4

find a common denominator,

alpha5

borrow from whole number part,

alpha6

column borrow to subtract the second numerator from the first,

alpha7

subtract numerators,

alpha8

reduce answers to simplest form.

Details

This Q-matrix can be found in DeCarlo (2011). It is the same used by de la Torre and Douglas (2004).

Source

DeCarlo, L. T. (2011). On the analysis of fraction subtraction data: The DINA Model, classification, latent class sizes, and the Q-Matrix. Applied Psychological Measurement, 35, 8–26.

References

de la Torre, J. and Douglas, J. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.

Tatsuoka, C. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society, Series C, Applied Statistics, 51, 337–350.

Tatsuoka, K. (1984) Analysis of errors in fraction addition and subtraction problems. Final Report for NIE-G-81-0002, University of Illinois, Urbana-Champaign.


Generalized Distance Discriminating Method

Description

Performs the generalized distance discriminating method (GDD; Sun, Xin, Zhang, & de la Torre, 2013) for dichotomous data which is a method for classifying students into skill profiles based on a preliminary unidimensional calibration.

Usage

gdd(data, q.matrix, theta, b, a, skillclasses=NULL)

Arguments

data

Data frame with N×JN \times J item responses

q.matrix

The Q-matrix

theta

Estimated person ability

b

Estimated item intercept from a 2PL model (see Details)

a

Estimated item slope from a 2PL model (see Details)

skillclasses

Optional matrix of skill classes used for estimation

Details

Note that the parameters in the arguments follow the item response model

logitP(Xnj=1θn)=bj+ajθnlogit P( X_{nj}=1 | \theta_n )=b_j + a_j \theta_n

which is employed in the gdm function.

Value

A list with following entries

skillclass.est

Estimated skill class

distmatrix

Distances for every person and every skill class

skillspace

Used skill space for estimation

theta

Used person parameter estimate

References

Sun, J., Xin, T., Zhang, S., & de la Torre, J. (2013). A polytomous extension of the generalized distance discriminating method. Applied Psychological Measurement, 37, 503-521.

Examples

#############################################################################
# EXAMPLE 1: GDD for sim.dina
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

data <- sim.dina
q.matrix <- sim.qmatrix

# estimate 1PL (use irtmodel="2PL" for 2PL estimation)
mod <- CDM::gdm( data, irtmodel="1PL", theta.k=seq(-6,6,len=21),
                    decrease.increments=TRUE, conv=.001, globconv=.001)
# extract item parameters in parametrization b + a*theta
b <- mod$b[,1]
a <- mod$a[,,1]
# extract person parameter estimate
theta <- mod$person$EAP.F1

# generalized distance discriminating method
res <- CDM::gdd( data, q.matrix, theta=theta, b=b, a=a )

Estimating the Generalized DINA (GDINA) Model

Description

This function implements the generalized DINA model for dichotomous attributes (GDINA; de la Torre, 2011) and polytomous attributes (pGDINA; Chen & de la Torre, 2013, 2018). In addition, multiple group estimation is also possible using the gdina function. This function also allows for the estimation of a higher order GDINA model (de la Torre & Douglas, 2004). Polytomous item responses are treated by specifying a sequential GDINA model (Ma & de la Torre, 2016; Tutz, 1997). The simulataneous modeling of skills and misconceptions (bugs) can be also estimated within the GDINA framework (see Kuo, Chen & de la Torre, 2018; see argument rule).

The estimation can also be conducted by posing monotonocity constraints (Hong, Chang, & Tsai, 2016) using the argument mono.constr. Moreover, regularization methods SCAD, lasso, ridge, SCAD-L2 and truncated L1L_1 penalty (TLP) for item parameters can be employed (Xu & Shang, 2018).

Normally distributed priors can be specified for item parameters (item intercepts and item slopes). Note that (for convenience) the prior specification holds simultaneously for all items.

Usage

gdina(data, q.matrix, skillclasses=NULL, conv.crit=0.0001, dev.crit=.1,  maxit=1000,
    linkfct="identity", Mj=NULL, group=NULL, invariance=TRUE,method=NULL,
    delta.init=NULL, delta.fixed=NULL, delta.designmatrix=NULL,
    delta.basispar.lower=NULL, delta.basispar.upper=NULL, delta.basispar.init=NULL,
    zeroprob.skillclasses=NULL, attr.prob.init=NULL, attr.prob.fixed=NULL,
    reduced.skillspace=NULL, reduced.skillspace.method=2, HOGDINA=-1, Z.skillspace=NULL,
    weights=rep(1, nrow(data)), rule="GDINA", bugs=NULL, regular_lam=0,
    regular_type="none", regular_alpha=NA, regular_tau=NA, regular_weights=NULL,
    mono.constr=FALSE, prior_intercepts=NULL, prior_slopes=NULL, progress=TRUE,
    progress.item=FALSE, mstep_iter=10, mstep_conv=1E-4, increment.factor=1.01,
    fac.oldxsi=0, max.increment=.3, avoid.zeroprobs=FALSE, seed=0,
    save.devmin=TRUE, calc.se=TRUE, se_version=1, PEM=TRUE, PEM_itermax=maxit,
    cd=FALSE, cd_steps=1, mono_maxiter=10, freq_weights=FALSE, optimizer="CDM", ...)

## S3 method for class 'gdina'
summary(object, digits=4, file=NULL,  ...)

## S3 method for class 'gdina'
plot(x, ask=FALSE,  ...)

## S3 method for class 'gdina'
print(x,  ...)

Arguments

data

A required N×JN \times J data matrix containing integer responses, 0, 1, ..., K. Polytomous item responses are treated by the sequential GDINA model. NA values are allowed.

q.matrix

A required integer J×KJ \times K matrix containing attributes not required or required, 0 or 1, to master the items in case of dichotomous attributes or integers in case of polytomous attributes. For polytomous item responses the Q-matrix must also include the item name and item category, see Example 11.

skillclasses

An optional matrix for determining the skill space. The argument can be used if a user wants less than 2K2^K skill classes.

conv.crit

Convergence criterion for maximum absolute change in item parameters

dev.crit

Convergence criterion for maximum absolute change in deviance

maxit

Maximum number of iterations

linkfct

A string which indicates the link function for the GDINA model. Options are "identity" (identity link), "logit" (logit link) and "log" (log link). The default is the "identity" link. Note that the link function is chosen for the whole model (i.e. for all items).

Mj

A list of design matrices and labels for each item. The definition of Mj follows the definition of MjM_j in de la Torre (2011). Please study the value Mj of the function in default analysis. See Example 3.

group

A vector of group identifiers for multiple group estimation. Default is NULL (no multiple group estimation).

invariance

Logical indicating whether invariance of item parameters is assumed for multiple group models. If a subset of items should be treated as noninvariant, then invariance can be a vector of item names.

method

Estimation method for item parameters (see) (de la Torre, 2011). The default "WLS" weights probabilities attribute classes by a weighting matrix WjW_j of expected frequencies, whereas the method "ULS" perform unweighted least squares estimation on expected frequencies. The method "ML" directly maximizes the log-likelihood function. The "ML" method is a bit slower but can be much more stable, especially in the case of the RRUM model. Only for the RRUM model, the default is changed to method="ML" if not specified otherwise.

delta.init

List with initial δ\delta parameters

delta.fixed

List with fixed δ\delta parameters. For free estimated parameters NA must be declared.

delta.designmatrix

A design matrix for restrictions on delta. See Example 4.

delta.basispar.lower

Lower bounds for delta basis parameters.

delta.basispar.upper

Upper bounds for delta basis parameters.

delta.basispar.init

An optional vector of starting values for the basis parameters of delta. This argument only applies when using a designmatrix for delta, i.e. delta.designmatrix is not NULL.

zeroprob.skillclasses

An optional vector of integers which indicates which skill classes should have zero probability. Default is NULL (no skill classes with zero probability).

attr.prob.init

Initial probabilities of skill distribution.

attr.prob.fixed

Vector or matrix with fixed probabilities of skill distribution.

reduced.skillspace

A logical which indicates if the latent class skill space dimension should be reduced (see Xu & von Davier, 2008). The default is NULL which applies skill space reduction for more than four skills. The dimensional reduction is only well defined for more than three skills. If the argument zeroprob.skillclasses is not NULL, then reduced.skillspace is set to FALSE.

reduced.skillspace.method

Computation method for skill space reduction in case of reduced.skillspace=TRUE. The default is 2 which is computationally more efficient but introduced in CDM 2.6. For reasons of compatibility of former CDM versions (\le 2.5), reduced.skillspace.method=1 uses the older implemented method. In case of non-convergence with the new method, please try the older method.

HOGDINA

Values of -1, 0 or 1 indicating if a higher order GDINA model (see Details) should be estimated. The default value of -1 corresponds to the case that no higher order factor is assumed to exist. A value of 0 corresponds to independent attributes. A value of 1 assumes the existence of a higher order factor.

Z.skillspace

A user specified design matrix for the skill space reduction as described in Xu and von Davier (2008). See in the Examples section for applications. See Example 6.

weights

An optional vector of sample weights.

rule

A string or a vector of itemwise condensation rules. Allowed entries are GDINA, DINA, DINO, ACDM (additive cognitive diagnostic model) and RRUM (reduced reparametrized unified model, RRUM, see Details). The rule GDINA1 applies only main effects in the GDINA model which is equivalent to ACDM. The rule GDINA2 applies to all main effects and second-order interactions of the attributes. If some item is specified as RRUM, then for all the items the reduced RUM will be estimated which means that the log link function and the ACDM condensation rule is used. In the output, the entry rrum.params contains the parameters transformed in the RUM parametrization. If rule is a string, the condensation rule applies to all items. If rule is a vector, condensation rules can be specified itemwise. The default is GDINA for all items.

bugs

Character vector indicating which columns in the Q-matrix refer to bugs (misconceptions). This is only available if some rule is set to "SISM". Note that bugs must be included as last columns in the Q-matrix.

regular_lam

Regularization parameter λ\lambda

regular_type

Type of regularization. Can be scad (SCAD penalty), lasso (lasso penalty), ridge (ridge penalty), elnet (elastic net), scadL2 (SCAD-L2L_2; Zeng & Xie, 2014), tlp (truncated L1L_1 penalty; Xu & Shang, 2018; Shen, Pan, & Zhu, 2012), mcp (MCP penalty; Zhang, 2010) or none (no regularization).

regular_alpha

Regularization parameter α\alpha (applicable for elastic net or SCAD-L2.

regular_tau

Regularization parameter τ\tau for truncated L1L_1 penalty.

regular_weights

Optional list of item parameter weights used for penalties in regularized estimation (see Example 13)

mono.constr

Logical indicating whether monotonicity constraints should be fulfilled in estimation (implemented by the increasing penalty method; see Nash, 2014, p. 156).

prior_intercepts

Vector with mean and standard deviation for prior of random intercepts (applies to all items)

prior_slopes

Vector with mean and standard deviation for prior of random slopes (applies to all items and all parameters)

progress

An optional logical indicating whether the function should print the progress of iteration in the estimation process.

progress.item

An optional logical indicating whether item wise progress should be displayed

mstep_iter

Number of iterations in M-step if method="ML".

mstep_conv

Convergence criterion in M-step if method="ML".

increment.factor

A factor larger than 1 (say 1.1) to control maximum increments in item parameters. This parameter can be used in case of nonconvergence.

fac.oldxsi

A convergence acceleration factor between 0 and 1 which defines the weight of previously estimated values in current parameter updates.

max.increment

Maximum size of change in increments in M steps of EM algorithm when method="ML" is used.

avoid.zeroprobs

An optional logical indicating whether for estimating item parameters probabilities occur. Especially if not a skill classes are used, it is recommended to switch the argument to TRUE.

seed

Simulation seed for initial parameters. A value of zero corresponds to deterministic starting values, an integer value different from zero to random initial values with set.seed(seed).

save.devmin

An optional logical indicating whether intermediate estimates should be saved corresponding to minimal deviance. Setting the argument to FALSE could help for preventing working memory overflow.

calc.se

Optional logical indicating whether standard errors should be calculated.

se_version

Integer for calculation method of standard errors. se_version=1 is based on the observed log likelihood and included since CDM 5.1 and is the default. Comparability with previous CDM versions can be obtained with se_version=0.

PEM

Logical indicating whether the P-EM acceleration should be applied (Berlinet & Roland, 2012).

PEM_itermax

Number of iterations in which the P-EM method should be applied.

cd

Logical indicating whether coordinate descent algorithm should be used.

cd_steps

Number of steps for each parameter in coordinate descent algorithm

mono_maxiter

Maximum number of iterations for fulfilling the monotonicity constraint

freq_weights

Logical indicating whether frequency weights should be used. Default is FALSE.

optimizer

String indicating which optimizer should be used in M-step estimation in case of method="ML". The internal optimizer of CDM can be requested by optimizer="CDM". The optimization with stats::optim can be requested by optimizer="optim". For the RRUM model, it is always chosen optimizer="optim".

object

A required object of class gdina, obtained from a call to the function gdina.

digits

Number of digits after decimal separator to display.

file

Optional file name for a file in which summary should be sinked.

x

A required object of class gdina

ask

A logical indicating whether every separate item should be displayed in plot.gdina

...

Optional parameters to be passed to or from other methods will be ignored.

Details

The estimation is based on an EM algorithm as described in de la Torre (2011). Item parameters are contained in the delta vector which is a list where the jjth entry corresponds to item parameters of the jjth item.

The following description refers to the case of dichotomous attributes. For using polytomous attributes see Chen and de la Torre (2013) and Example 7 for a definition of the Q-matrix. In this case, Qik=lQ_{ik}=l means that the iith item requires the mastery (at least) of level ll of attribute kk.

Assume that two skills α1\alpha_1 and α2\alpha_2 are required for mastering item jj. Then the GDINA model can be written as

g[P(Xnj=1αn)]=δj0+δj1αn1+δj2αn2+δj12αn1αn2g [ P( X_{nj}=1 | \alpha_n ) ]=\delta_{j0} + \delta_{j1} \alpha_{n1} + \delta_{j2} \alpha_{n2} + \delta_{j12} \alpha_{n1} \alpha_{n2}

which is a two-way GDINA-model (the rule="GDINA2" specification) with a link function gg (which can be the identity, logit or logarithmic link). If the specification ACDM is chosen, then δj12=0\delta_{j12}=0. The DINA model (rule="DINA") assumes δj1=δj2=0\delta_{j1}=\delta_{j2}=0.

For the reduced RUM model (rule="RRUM"), the item response model is

P(Xnj=1αn)=πiri11αi1ri21αi2P(X_{nj}=1 | \alpha_n )=\pi_i^\ast \cdot r_{i1}^{1-\alpha_{i1} } \cdot r_{i2}^{1-\alpha_{i2} }

From this equation, it is obvious, that this model is equivalent to an additive model (rule="ACDM") with a logarithmic link function (linkfct="log").

If a reduced skillspace (reduced.skillspace=TRUE) is employed, then the logarithm of probability distribution of the attributes is modeled as a log-linear model:

logP[(αn1,αn2,,αnK)]=γ0+kγkαnk+k<lγklαnkαnl\log P[ ( \alpha_{n1}, \alpha_{n2}, \ldots, \alpha_{nK} ) ] =\gamma_0 + \sum_k \gamma_k \alpha_{nk} + \sum_{k < l} \gamma_{kl} \alpha_{nk} \alpha_{nl}

If a higher order DINA model is assumed (HOGDINA=1), then a higher order factor θn\theta_n for the attributes is assumed:

P(αnk=1θn)=Φ(akθn+bk)P( \alpha_{nk}=1 | \theta_n )=\Phi ( a_k \theta_n + b_k )

For HOGDINA=0, all attributes αnk\alpha_{nk} are assumed to be independent of each other:

P[(αn1,αn2,,αnK)]=kP(αnk)P[ ( \alpha_{n1}, \alpha_{n2}, \ldots, \alpha_{nK} ) ] =\prod_k P( \alpha_{nk} )

Note that the noncompensatory reduced RUM (NC-RRUM) according to Rupp and Templin (2008) is the GDINA model with the arguments rule="ACDM" and linkfct="log". NC-RRUM can also be obtained by choosing rule="RRUM".

The compensatory RUM (C-RRUM) can be obtained by using the arguments rule="ACDM" and linkfct="logit".

The cognitive diagnosis model for identifying skills and misconceptions (SISM; Kuo, Chen & de la Torre, 2018) can be estimated with rule="SISM" (see Example 12).

The gdina function internally parameterizes the GDINA model as

g[P(Xnj=1αn)]=Mj(αn)δjg [ P( X_{nj}=1 | \alpha_n ) ]=\bm{M}_j ( \alpha _n ) \bm{\delta}_j

with item-specific design matrices Mj(αn)\bm{M}_j (\alpha _n ) and item parameters δj\bm{\delta}_j. Only those attributes are modelled which correspond to non-zero entries in the Q-matrix. Because the Q-matrix (in q.matrix) and the design matrices (in M_j; see Example 3) can be specified by the user, several cognitive diagnosis models can be estimated. Therefore, some additional extensions of the DINA model can also be estimated using the gdina function. These models include the DINA model with multiple strategies (Huo & de la Torre, 2014)

Value

An object of class gdina with following entries

coef

Data frame of item parameters

delta

List with basis item parameters

se.delta

Standard errors of basis item parameters

probitem

Data frame with model implied conditional item probabilities P(Xi=1α)P(X_i=1 | \bm{\alpha}). These probabilities are displayed in plot.gdina.

itemfit.rmsea

The RMSEA item fit index (see itemfit.rmsea).

mean.rmsea

Mean of RMSEA item fit indexes.

loglike

Log-likelihood

deviance

Deviance

G

Number of groups

N

Sample size

AIC

AIC

BIC

BIC

CAIC

CAIC

Npars

Total number of parameters

Nipar

Number of item parameters

Nskillpar

Number of parameters for skill class distribution

Nskillclasses

Number of skill classes

varmat.delta

Covariance matrix of δ\delta item parameters

posterior

Individual posterior distribution

like

Individual likelihood

data

Original data

q.matrix

Used Q-matrix

pattern

Individual patterns, individual MLE and MAP classifications and their corresponding probabilities

attribute.patt

Probabilities of skill classes

skill.patt

Marginal skill probabilities

subj.pattern

Individual subject pattern

attribute.patt.splitted

Splitted attribute pattern

pjk

Array of item response probabilities

Mj

Design matrix MjM_j in GDINA algorithm (see de la Torre, 2011)

Aj

Design matrix AjA_j in GDINA algorithm (see de la Torre, 2011)

rule

Used condensation rules

linkfct

Used link function

delta.designmatrix

Designmatrix for item parameters

reduced.skillspace

A logical if skillspace reduction was performed

Z.skillspace

Design matrix for skillspace reduction

beta

Parameters δ\delta for skill class representation

covbeta

Standard errors of δ\delta parameters

iter

Number of iterations

rrum.params

Parameters in the parametrization of the reduced RUM model if rule="RRUM".

group.stat

Group statistics (sample sizes, group labels)

HOGDINA

The used value of HOGDINA

mono.constr

Monotonicity constraint

regularization

Logical indicating whether regularization is used

regular_lam

Regularization parameter

numb_bound_mono

Number of items with parameters at boundary of monotonicity constraints

numb_regular_pars

Number of regularized item parameters

delta_regularized

List indicating which item parameters are regularized

cd_algorithm

Logical indicating whether coordinate descent algorithm is used

cd_steps

Number of steps for each parameter in coordinate descent algorithm

seed

Used simulation seed

a.attr

Attribute parameters aka_k in case of HOGDINA>=0

b.attr

Attribute parameters bkb_k in case of HOGDINA>=0

attr.rf

Attribute response functions. This matrix contains all aka_k and bkb_k parameters

converged

Logical indicating whether convergence was achieved.

control

Optimization parameters used in estimation

partable

Parameter table for gdina function

polychor

Group-wise matrices with polychoric correlations

sequential

Logical indicating whether a sequential GDINA model is applied for polytomous item responses

...

Further values

Note

The function din does not allow for multiple group estimation. Use this gdina function instead and choose the appropriate rule="DINA" as an argument.

Standard error calculation in analyses which use sample weights or designmatrix for delta parameters (delta.designmatrix!=NULL) is not yet correctly implemented. Please use replication methods instead.

References

Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122-4137.

Chen, J., & de la Torre, J. (2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37, 419-437.

Chen, J., & de la Torre, J. (2018). Introducing the general polytomous diagnosis modeling framework. Frontiers in Psychology | Quantitative Psychology and Measurement, 9(1474).

de la Torre, J., & Douglas, J. A. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333-353.

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

Hong, C. Y., Chang, Y. W., & Tsai, R. C. (2016). Estimation of generalized DINA model with order restrictions. Journal of Classification, 33(3), 460-484.

Huo, Y., de la Torre, J. (2014). Estimating a cognitive diagnostic model for multiple strategies via the EM algorithm. Applied Psychological Measurement, 38, 464-485.

Kuo, B.-C., Chen, C.-H., & de la Torre, J. (2018). A cognitive diagnosis model for identifying coexisting skills and misconceptions. Applied Psychological Measurement, 42(3), 179-191.

Ma, W., & de la Torre, J. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology, 69(3), 253-275.

Nash, J. C. (2014). Nonlinear parameter optimization using R tools. West Sussex: Wiley.

Rupp, A. A., & Templin, J. (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement: Interdisciplinary Research and Perspectives, 6, 219-262.

Shen, X., Pan, W., & Zhu, Y. (2012). Likelihood-based selection and sharp parameter estimation. Journal of the American Statistical Association, 107, 223-232.

Tutz, G. (1997). Sequential models for ordered responses. In W. van der Linden & R. K. Hambleton. Handbook of modern item response theory (pp. 139-152). New York: Springer.

Xu, G., & Shang, Z. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 523, 1284-1295.

Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS.

Zeng, L., & Xie, J. (2014). Group variable selection via SCAD-L2L_2. Statistics, 48, 49-66.

Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics, 38, 894-942.

See Also

See also the din function (for DINA and DINO estimation).

For assessment of model fit see modelfit.cor.din and anova.gdina.

See itemfit.sx2 for item fit statistics.

See sim.gdina for simulating the GDINA model.

See gdina.wald for a Wald test for testing the DINA and ACDM rules at the item-level.

See gdina.dif for assessing differential item functioning.

See discrim.index for computing discrimination indices.

See the GDINA::GDINA function in the GDINA package for similar functionality.

Examples

#############################################################################
# EXAMPLE 1: Simulated DINA data | different condensation rules
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

dat <- sim.dina
Q <- sim.qmatrix

#***
# Model 1: estimation of the GDINA model (identity link)
mod1 <- CDM::gdina( data=dat,  q.matrix=Q)
summary(mod1)
plot(mod1) # apply plot function

## Not run: 
# Model 1a: estimate model with different simulation seed
mod1a <- CDM::gdina( data=dat,  q.matrix=Q, seed=9089)
summary(mod1a)

# Model 1b: estimate model with some fixed delta parameters
delta.fixed <- as.list( rep(NA,9) )        # List for parameters of 9 items
delta.fixed[[2]] <- c( 0, .15, .15, .45 )
delta.fixed[[6]] <- c( .25, .25 )
mod1b <- CDM::gdina( data=dat,  q.matrix=Q, delta.fixed=delta.fixed)
summary(mod1b)

# Model 1c: fix all delta parameters to previously fitted model
mod1c <- CDM::gdina( data=dat,  q.matrix=Q, delta.fixed=mod1$delta)
summary(mod1c)

# Model 1d: estimate GDINA model with GDINA package
mod1d <- GDINA::GDINA( dat=dat, Q=Q, model="GDINA" )
summary(mod1d)
# extract item parameters
GDINA::itemparm(mod1d)
GDINA::itemparm(mod1d, what="delta")
# compare likelihood
logLik(mod1)
logLik(mod1d)

#***
# Model 2: estimation of the DINA model with gdina function
mod2 <- CDM::gdina( data=dat,  q.matrix=Q, rule="DINA")
summary(mod2)
plot(mod2)

#***
# Model 2b: compare results with din function
mod2b <- CDM::din( data=dat,  q.matrix=Q, rule="DINA")
summary(mod2b)

# Model 2: estimation of the DINO model with gdina function
mod3 <- CDM::gdina( data=dat,  q.matrix=Q, rule="DINO")
summary(mod3)

#***
# Model 4: DINA model with logit link
mod4 <- CDM::gdina( data=dat, q.matrix=Q, rule="DINA", linkfct="logit" )
summary(mod4)

#***
# Model 5: DINA model log link
mod5 <- CDM::gdina( data=dat, q.matrix=Q, rule="DINA", linkfct="log")
summary(mod5)

#***
# Model 6: RRUM model
mod6 <- CDM::gdina( data=dat, q.matrix=Q, rule="RRUM")
summary(mod6)

#***
# Model 7: Higher order GDINA model
mod7 <- CDM::gdina( data=dat, q.matrix=Q, HOGDINA=1)
summary(mod7)

#***
# Model 8: GDINA model with independent attributes
mod8 <- CDM::gdina( data=dat, q.matrix=Q, HOGDINA=0)
summary(mod8)

#***
# Model 9: Estimating the GDINA model with monotonicity constraints
mod9 <- CDM::gdina( data=dat, q.matrix=Q, rule="GDINA",
              mono.constr=TRUE, linkfct="logit")
summary(mod9)

#***
# Model 10: Estimating the ACDM model with SCAD penalty and regularization
#           parameter of .05
mod10 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM",
                linkfct="logit", regular_type="scad", regular_lam=.05 )
summary(mod10)

#***
# Model 11: Estimation of GDINA model with prior distributions

# N(0,10^2) prior for item intercepts
prior_intercepts <- c(0,10)
# N(0,1^2) prior for item slopes
prior_slopes <- c(0,1)
# estimate model
mod11 <- CDM::gdina( data=dat, q.matrix=Q, rule="GDINA",
              prior_intercepts=prior_intercepts, prior_slopes=prior_slopes)
summary(mod11)

#############################################################################
# EXAMPLE 2: Simulated DINO data
#    additive cognitive diagnosis model with different link functions
#############################################################################

data(sim.dino, package="CDM")
data(sim.matrix, package="CDM")

dat <- sim.dino
Q <- sim.qmatrix

#***
# Model 1: additive cognitive diagnosis model (ACDM; identity link)
mod1 <- CDM::gdina( data=dat, q.matrix=Q,  rule="ACDM")
summary(mod1)

#***
# Model 2: ACDM logit link
mod2 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", linkfct="logit")
summary(mod2)

#***
# Model 3: ACDM log link
mod3 <- CDM::gdina( data=dat, q.matrix=Q, rule="ACDM", linkfct="log")
summary(mod3)

#***
# Model 4: Different condensation rules per item
I <- 9      # number of items
rule <- rep( "GDINA", I )
rule[1] <- "DINO"   # 1st item: DINO model
rule[7] <- "GDINA2" # 7th item: GDINA model with first- and second-order interactions
rule[8] <- "ACDM"   # 8ht item: additive CDM
rule[9] <- "DINA"   # 9th item: DINA model
mod4 <- CDM::gdina( data=dat, q.matrix=Q, rule=rule )
summary(mod4)

#############################################################################
# EXAMPLE 3: Model with user-specified design matrices
#############################################################################

data(sim.dino, package="CDM")
data(sim.qmatrix, package="CDM")

dat <- sim.dino
Q <- sim.qmatrix

# do a preliminary analysis and modify obtained design matrices
mod0 <- CDM::gdina( data=dat,  q.matrix=Q,  maxit=1)

# extract default design matrices
Mj <- mod0$Mj
Mj.user <- Mj   # these user defined design matrices are modified.
#~~~  For the second item, the following model should hold
#     X1 ~ V2 + V2*V3
mj <- Mj[[2]][[1]]
mj.lab <- Mj[[2]][[2]]
mj <- mj[,-3]
mj.lab <- mj.lab[-3]
Mj.user[[2]] <- list( mj, mj.lab )
#    [[1]]
#        [,1] [,2] [,3]
#    [1,]    1    0    0
#    [2,]    1    1    0
#    [3,]    1    0    0
#    [4,]    1    1    1
#    [[2]]
#    [1] "0"   "1"   "1-2"
#~~~  For the eight item an equality constraint should hold
#     X8 ~ a*V2 + a*V3 + V2*V3
mj <- Mj[[8]][[1]]
mj.lab <- Mj[[8]][[2]]
mj[,2] <- mj[,2] + mj[,3]
mj <- mj[,-3]
mj.lab <- c("0", "1=2", "1-2" )
Mj.user[[8]] <- list( mj, mj.lab )
Mj.user[[8]]
  ##   [[1]]
  ##        [,1] [,2] [,3]
  ##   [1,]    1    0    0
  ##   [2,]    1    1    0
  ##   [3,]    1    1    0
  ##   [4,]    1    2    1
  ##
  ##   [[2]]
  ##   [1] "0"   "1=2" "1-2"
mod <- CDM::gdina( data=dat,  q.matrix=Q,
                    Mj=Mj.user,  maxit=200 )
summary(mod)

#############################################################################
# EXAMPLE 4: Design matrix for delta parameters
#############################################################################

data(sim.dino, package="CDM")
data(sim.qmatrix, package="CDM")

#~~~ estimate an initial model
mod0 <- CDM::gdina( data=dat,  q.matrix=Q, rule="ACDM", maxit=1)
# extract coefficients
c0 <- mod0$coef
I <- 9  # number of items
delta.designmatrix <- matrix( 0, nrow=nrow(c0), ncol=nrow(c0) )
diag( delta.designmatrix) <- 1
# set intercept of item 1 and item 3 equal to each other
delta.designmatrix[ 7, 1 ] <- 1 ; delta.designmatrix[,7] <- 0
# set loading of V1 of item1 and item 3 equal
delta.designmatrix[ 8, 2 ] <- 1 ; delta.designmatrix[,8] <- 0
delta.designmatrix <- delta.designmatrix[, -c(7:8) ]
                # exclude original parameters with indices 7 and 8

#***
# Model 1: ACDM with designmatrix
mod1 <- CDM::gdina( data=dat,  q.matrix=Q,  rule="ACDM",
            delta.designmatrix=delta.designmatrix )
summary(mod1)

#***
# Model 2: Same model, but with logit link instead of identity link function
mod2 <- CDM::gdina( data=dat,  q.matrix=Q,  rule="ACDM",
            delta.designmatrix=delta.designmatrix, linkfct="logit")
summary(mod2)

#############################################################################
# EXAMPLE 5: Multiple group estimation
#############################################################################

# simulate data
set.seed(9279)
N1 <- 200 ; N2 <- 100   # group sizes
I <- 10                 # number of items
q.matrix <- matrix(0,I,2)   # create Q-matrix
q.matrix[1:7,1] <- 1 ; q.matrix[ 5:10,2] <- 1
# simulate first group
dat1 <- CDM::sim.din(N1, q.matrix=q.matrix, mean=c(0,0) )$dat
# simulate second group
dat2 <- CDM::sim.din(N2, q.matrix=q.matrix, mean=c(-.3, -.7) )$dat
# merge data
dat <- rbind( dat1, dat2 )
# group indicator
group <- c( rep(1,N1), rep(2,N2) )

# estimate GDINA model with multiple groups assuming invariant item parameters
mod1 <- CDM::gdina( data=dat, q.matrix=q.matrix,  group=group)
summary(mod1)

# estimate DINA model with multiple groups assuming invariant item parameters
mod2 <- CDM::gdina( data=dat, q.matrix=q.matrix, group=group, rule="DINA")
summary(mod2)

# estimate GDINA model with noninvariant item parameters
mod3 <- CDM::gdina( data=dat, q.matrix=q.matrix,  group=group, invariance=FALSE)
summary(mod3)

# estimate GDINA model with some invariant item parameters (I001, I006, I008)
mod4 <- CDM::gdina( data=dat, q.matrix=q.matrix,  group=group,
            invariance=c("I001", "I006","I008") )

#--- model comparison
IRT.compareModels(mod1,mod2,mod3,mod4)

# estimate GDINA model with non-invariant item parameters except for the
# items I001, I006, I008
mod5 <- CDM::gdina( data=dat, q.matrix=q.matrix,  group=group,
            invariance=setdiff( colnames(dat), c("I001", "I006","I008") ) )

#############################################################################
# EXAMPLE 6: User specified reduced skill space
#############################################################################

#   Some correlations between attributes should be set to zero.
q.matrix <- expand.grid( c(0,1), c(0,1), c(0,1), c(0,1) )
colnames(q.matrix) <- colnames( paste("Attr", 1:4,sep=""))
q.matrix <- q.matrix[ -1, ]
Sigma <- matrix( .5, nrow=4, ncol=4 )
diag(Sigma) <- 1
Sigma[3,2] <- Sigma[2,3] <- 0 # set correlation of attribute A2 and A3 to zero
dat <- CDM::sim.din( N=1000, q.matrix=q.matrix, Sigma=Sigma)$dat

#~~~ Step 1: initial estimation
mod1a <- CDM::gdina( data=dat, q.matrix=q.matrix, maxit=1, rule="DINA")
# estimate also "full" model
mod1 <- CDM::gdina( data=dat, q.matrix=q.matrix, rule="DINA")

#~~~ Step 2: modify designmatrix for reduced skillspace
Z.skillspace <- data.frame( mod1a$Z.skillspace )
# set correlations of A2/A4 and A3/A4 to zero
vars <- c("A2_A3","A2_A4")
for (vv in vars){ Z.skillspace[,vv] <- NULL }

#~~~ Step 3: estimate model with reduced skillspace
mod2 <- CDM::gdina( data=dat, q.matrix=q.matrix,
              Z.skillspace=Z.skillspace, rule="DINA")

#~~~ eliminate all covariances
Z.skillspace <- data.frame( mod1$Z.skillspace )
colnames(Z.skillspace)
Z.skillspace <- Z.skillspace[, -grep( "_", colnames(Z.skillspace),fixed=TRUE)]
colnames(Z.skillspace)

mod3 <- CDM::gdina( data=dat, q.matrix=q.matrix,
               Z.skillspace=Z.skillspace, rule="DINA")
summary(mod1)
summary(mod2)
summary(mod3)

#############################################################################
# EXAMPLE 7: Polytomous GDINA model (Chen & de la Torre, 2013)
#############################################################################

data(data.pgdina, package="CDM")

dat <- data.pgdina$dat
q.matrix <- data.pgdina$q.matrix

# pGDINA model with "DINA rule"
mod1 <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA")
summary(mod1)
# no reduced skill space
mod1a <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA",reduced.skillspace=FALSE)
summary(mod1)

# pGDINA model with "GDINA rule"
mod2 <- CDM::gdina( dat, q.matrix=q.matrix, rule="GDINA")
summary(mod2)

#############################################################################
# EXAMPLE 8: Fraction subtraction data: DINA and HO-DINA model
#############################################################################

data(fraction.subtraction.data, package="CDM")
data(fraction.subtraction.qmatrix, package="CDM")

dat <- fraction.subtraction.data
Q <- fraction.subtraction.qmatrix

# Model 1: DINA model
mod1 <- CDM::gdina( dat, q.matrix=Q, rule="DINA")
summary(mod1)

# Model 2: HO-DINA model
mod2 <- CDM::gdina( dat, q.matrix=Q, HOGDINA=1, rule="DINA")
summary(mod2)

#############################################################################
# EXAMPLE 9: Skill space approximation data.jang
#############################################################################

data(data.jang, package="CDM")

data <- data.jang$data
q.matrix <- data.jang$q.matrix

#*** Model 1: Reduced RUM model
mod1 <- CDM::gdina( data, q.matrix, rule="RRUM", conv.crit=.001, maxit=500 )

#*** Model 2: Reduced RUM model with skill space approximation
# use 300 instead of 2^9=512 skill classes
skillspace <- CDM::skillspace.approximation( L=300, K=ncol(q.matrix) )
mod2 <- CDM::gdina( data, q.matrix, rule="RRUM", conv.crit=.001,
            skillclasses=skillspace )
  ##   > logLik(mod1)
  ##   'log Lik.' -30318.08 (df=153)
  ##   > logLik(mod2)
  ##   'log Lik.' -30326.52 (df=153)

#############################################################################
# EXAMPLE 10: CDM with a linear hierarchy
#############################################################################
# This model is equivalent to a unidimensional IRT model with an ordered
# ordinal latent trait and is actually a probabilistic Guttman model.
set.seed(789)

# define 3 competency levels
alpha <- scan()
   0 0 0   1 0 0   1 1 0   1 1 1

# define skill class distribution
K <- 3
skillspace <- alpha <- matrix( alpha, K + 1,  K, byrow=TRUE )
alpha <- alpha[ rep(  1:4,  c(300,300,200,200) ), ]
# P(000)=P(100)=.3, P(110)=P(111)=.2
# define Q-matrix
Q <- scan()
    1 0 0   1 1 0   1 1 1

Q <- matrix( Q, nrow=K,  ncol=K, byrow=TRUE )
Q <- Q[ rep(1:K, each=4 ), ]
colnames(skillspace) <- colnames(Q) <- paste0("A",1:K)
I <- nrow(Q)

# define guessing and slipping parameters
guess <- stats::runif( I, 0, .3 )
slip <- stats::runif( I, 0, .2 )
# simulate data
dat <- CDM::sim.din( q.matrix=Q, alpha=alpha, slip=slip, guess=guess )$dat

#*** Model 1: DINA model with linear hierarchy
mod1 <- CDM::din( dat, q.matrix=Q, rule="DINA",  skillclasses=skillspace )
summary(mod1)

#*** Model 2: pGDINA model with 3 levels
#    The multidimensional CDM with a linear hierarchy is a unidimensional
#    polytomous GDINA model.
Q2 <- matrix( rowSums(Q), nrow=I, ncol=1 )
mod2 <- CDM::gdina( dat, q.matrix=Q2, rule="DINA" )
summary(mod2)

#*** Model 3: estimate probabilistic Guttman model in sirt
#    Proctor, C. H. (1970). A probabilistic formulation and statistical
#    analysis for Guttman scaling. Psychometrika, 35, 73-78.
library(sirt)
mod3 <- sirt::prob.guttman( dat, itemlevel=Q2[,1] )
summary(mod3)
# -> The three models result in nearly equivalent fit.

#############################################################################
# EXAMPLE 11: Sequential GDINA model (Ma & de la Torre, 2016)
#############################################################################

data(data.cdm04, package="CDM")

#** attach dataset
dat <- data.cdm04$data    # polytomous item responses
q.matrix1 <- data.cdm04$q.matrix1
q.matrix2 <- data.cdm04$q.matrix2

#-- DINA model with first Q-matrix
mod1 <- CDM::gdina( dat, q.matrix=q.matrix1, rule="DINA")
summary(mod1)
#-- DINA model with second Q-matrix
mod2 <- CDM::gdina( dat, q.matrix=q.matrix2, rule="DINA")
#-- GDINA model
mod3 <- CDM::gdina( dat, q.matrix=q.matrix2, rule="GDINA")

#** model comparison
IRT.compareModels(mod1,mod2,mod3)

#############################################################################
# EXAMPLE 12: Simulataneous modeling of skills and misconceptions (Kuo et al., 2018)
#############################################################################

data(data.cdm08, package="CDM")
dat <- data.cdm08$data
q.matrix <- data.cdm08$q.matrix

#*** estimate model
mod <- CDM::gdina( dat0, q.matrix, rule="SISM", bugs=colnames(q.matrix)[5:7] )
summary(mod)

#############################################################################
# EXAMPLE 13: Regularized estimation in GDINA model data.dtmr
#############################################################################

data(data.dtmr, package="CDM")
dat <- data.dtmr$data
q.matrix <- data.dtmr$q.matrix

#***** LASSO regularization with lambda parameter of .02
mod1 <- CDM::gdina(dat, q.matrix=q.matrix, rule="GDINA", regular_lam=.02,
                  regular_type="lasso")
summary(mod1)
mod$delta_regularized

#***** using starting values from previuos estimation
delta.init <- mod1$delta
attr.prob.init <- mod1$attr.prob
mod2 <- CDM::gdina(dat, q.matrix=q.matrix, rule="GDINA", regular_lam=.02, regular_type="lasso",
                delta.init=delta.init, attr.prob.init=attr.prob.init)
summary(mod2)

#***** final estimation fixing regularized estimates to zero and estimate all other
#***** item parameters unregularized
regular_weights <- mod2$delta_regularized
delta.init <- mod2$delta
attr.prob.init <- mod2$attr.prob

mod3 <- CDM::gdina(dat, q.matrix=q.matrix, rule="GDINA", regular_lam=1E5, regular_type="lasso",
                delta.init=delta.init, attr.prob.init=attr.prob.init,
                regular_weights=regular_weights)
summary(mod3)

## End(Not run)

Differential Item Functioning in the GDINA Model

Description

This function assesses item-wise differential item functioning in the GDINA model by using the Wald test (de la Torre, 2011; Hou, de la Torre & Nandakumar, 2014). It is necessary that a multiple group GDINA model is previously fitted.

Usage

gdina.dif(object)

## S3 method for class 'gdina.dif'
summary(object, ...)

Arguments

object

Object of class gdina

...

Further arguments to be passed

Details

The p values are also calculated by a Holm adjustment for multiple comparisons (see p.holm in output difstats).

In the case of two groups, an effect size of differential item functioning (labeled as UA (unsigned area) in difstats value) is defined as the weighted absolute difference of item response functions. The DIF measure for item jj is defined as

UAj=lw(αl)P(Xj=1αl,G=1)P(Xj=1αl,G=2)UA_j=\sum_l w( \alpha_l ) | P( X_j=1 | \alpha_l, G=1 ) - P( X_j=1 | \alpha_l, G=2 ) |

where w(αl)=[P(αlG=1)+P(αlG=2)]/2w( \alpha_l )=[ P( \alpha_l | G=1 ) + P( \alpha_l | G=2 ) ]/2.

Value

A list with following entries

difstats

Data frame containing results of item-wise Wald tests

coef

Data frame containing all (group-wise) item parameters

delta_all

List of δ\delta vectors containing all item parameters

varmat_all

List of covariance matrices of all δ\delta item parameters

prob.exp.group

List with groups and items containing expected latent class sizes and expected probabilities for each group and each item. Based on this information, effect sizes of differential item functioning can be calculated.

References

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

Hou, L., de la Torre, J., & Nandakumar, R. (2014). Differential item functioning assessment in cognitive diagnostic modeling: Application of the Wald test to investigate DIF in the DINA model. Journal of Educational Measurement, 51, 98-125.

See Also

See the GDINA::dif function in the GDINA package for similar functionality.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: DIF for DINA simulated data
#############################################################################

# simulate some data
set.seed(976)
N <- 2000    # number of persons in a group
I <- 9       # number of items
q.matrix <- matrix( 0, 9,2 )
q.matrix[1:3,1] <- 1
q.matrix[4:6,2] <- 1
q.matrix[7:9,c(1,2)] <- 1
# simulate first group
guess <- rep( .2, I )
slip <- rep(.1, I)
dat1 <- CDM::sim.din( N=N, q.matrix=q.matrix, guess=guess, slip=slip,
               mean=c(0,0) )$dat
# simulate second group with some DIF items (items 1, 7 and 8)
guess[ c(1,7)] <- c(.3, .35 )
slip[8] <- .25
dat2 <- CDM::sim.din( N=N, q.matrix=q.matrix, guess=guess, slip=slip,
               mean=c(0.4,.25) )$dat
group <- rep(1:2, each=N )
dat <- rbind( dat1, dat2 )

#*** estimate multiple group GDINA model
mod1 <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA", group=group )
summary(mod1)

#*** assess differential item functioning
dmod1 <- CDM::gdina.dif( mod1)
summary(dmod1)
  ##     item      X2 df      p p.holm     UA
  ##   1 I001 10.1711  2 0.0062 0.0495 0.0428
  ##   2 I002  1.9933  2 0.3691 1.0000 0.0276
  ##   3 I003  0.0313  2 0.9845 1.0000 0.0040
  ##   4 I004  0.0290  2 0.9856 1.0000 0.0044
  ##   5 I005  2.3230  2 0.3130 1.0000 0.0142
  ##   6 I006  1.8330  2 0.3999 1.0000 0.0159
  ##   7 I007 40.6851  2 0.0000 0.0000 0.1184
  ##   8 I008  6.7912  2 0.0335 0.2346 0.0710
  ##   9 I009  1.1538  2 0.5616 1.0000 0.0180

## End(Not run)

Wald Statistic for Item Fit of the DINA and ACDM Rule for GDINA Model

Description

This function tests with a Wald test for the GDINA model whether a DINA or a ACDM condensation rule leads to a sufficient item fit compared to the saturated GDINA rule (de la Torre & Lee, 2013). The Wald test is accompanied by the RMSEA fit and weighted and unweighted distance measures (wgtdist, uwgtdist), see Details (compare Ma, Iaconangelo, & de la Torre, 2016).

Usage

gdina.wald(object)

## S3 method for class 'gdina.wald'
summary(object, digits=3,
    vars=c("X2", "p", "sig", "RMSEA", "wgtdist"),  ...)

Arguments

object

A fitted gdina model

digits

Number of digits after decimal used for rounding.

vars

Vector including variables which should be displayed in summary. See the output stats.

...

Further arguments to be passed

Details

Let Pj(αl)P_j( \alpha _l) the estimated item response function for the GDINA model and P^j(αl)\hat{P}_j( \alpha _l) the item response model for the approximated model (DINA, DINO or ACDM). The unweighted distance uwgtdist as a measure of misfit is defined as

uwgtdist=12Kl(Pj(αl)P^j(αl))2uwgtdist=\frac{1}{2^K} \sum_l ( P_j( \alpha _l) - \hat{P}_j( \alpha _l) )^2

The weighted distance wgtdist measures the discrepancy with respected to the probabilities wl=P(αl)w_l=P( \alpha_l) of estimated skill classes

wgtdist=lwl(Pj(αl)P^j(αl))2wgtdist=\sum_l w_l (P_j( \alpha _l) - \hat{P}_j( \alpha _l) )^2

Value

stats

Data frame with Wald statistic for every item, corresponding p values and a RMSEA fit statistic

References

de la Torre, J., & Lee, Y. S. (2013). Evaluating the Wald test for item-level comparison of saturated and reduced models in cognitive diagnosis. Journal of Educational Measurement, 50, 355-373.

Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection, and attribute classification. Applied Psychological Measurement, 40(3), 200-217.

See Also

See the GDINA::modelcomp function in the GDINA package for similar functionality.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Wald test for DINA simulated data sim.dina
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

# Model 1: estimate GDINA model
mod1 <- CDM::gdina( sim.dina, q.matrix=sim.qmatrix,  rule="GDINA")
summary(mod1)

# perform Wald test
res1 <- CDM::gdina.wald( mod1 )
summary(res1)
# -> results show that all but one item fit according to the DINA rule

# select some output
summary(res1, vars=c("wgtdist", "p") )

## End(Not run)

General Diagnostic Model

Description

This function estimates the general diagnostic model (von Davier, 2008; Xu & von Davier, 2008) which handles multidimensional item response models with ordered discrete or continuous latent variables for polytomous item responses.

Usage

gdm( data, theta.k, irtmodel="2PL", group=NULL, weights=rep(1, nrow(data)),
    Qmatrix=NULL, thetaDes=NULL, skillspace="loglinear",
    b.constraint=NULL, a.constraint=NULL,
    mean.constraint=NULL, Sigma.constraint=NULL, delta.designmatrix=NULL,
    standardized.latent=FALSE, centered.latent=FALSE,
    centerintercepts=FALSE, centerslopes=FALSE,
    maxiter=1000,  conv=1e-5, globconv=1e-5, msteps=4, convM=.0005,
    decrease.increments=FALSE, use.freqpatt=FALSE, progress=TRUE,
    PEM=FALSE, PEM_itermax=maxiter, ...)

## S3 method for class 'gdm'
summary(object, file=NULL, ...)

## S3 method for class 'gdm'
print(x, ...)

## S3 method for class 'gdm'
plot(x, perstype="EAP", group=1, barwidth=.1, histcol=1,
       cexcor=3, pchpers=16, cexpers=.7, ... )

Arguments

data

An N×IN \times I matrix of polytomous item responses with categories k=0,1,...,Kk=0,1,...,K

theta.k

In the one-dimensional case it must be a vector. For multidimensional models it has to be a list of skill vectors if the theta grid differs between dimensions. If not, a vector input can be supplied. If an estimated skillspace (skillspace="est" should be estimated, a vector or a matrix theta.k will be used as initial values of the estimated θ\bold{\theta} grid.

irtmodel

The default 2PL corresponds to the model where item slopes on dimensions are equal for all item categories. If item-category slopes should be estimated, use 2PLcat. If no item slopes should be estimated then 1PL can be selected. Note that fixed item slopes can be specified in the Q-matrix (argument Qmatrix).

group

An optional vector of group identifiers for multiple group estimation. For plot.gdm it is an integer indicating which group should be used for plotting.

weights

An optional vector of sample weights

Qmatrix

An optional array of dimension I×D×KI \times D \times K which indicates pre-specified item loadings on dimensions. The default for category kk is the score kk, i.e. the scoring in the (generalized) partial credit model.

thetaDes

A design matrix for specifying nonlinear item response functions (see Example 1, Models 4 and 5)

skillspace

The parametric assumption of the skillspace. If skillspace="normal" then a univariate or multivariate normal distribution is assumed. The default "loglinear" corresponds to log-linear smoothing of the skillspace distribution (Xu & von Davier, 2008). If skillspace="full", then all probabilities of the skill space are nonparametrically estimated. If skillspace="est", then the θ\bold{\theta} distribution vectors will be estimated (see Details and Examples 4 and 5; Bartolucci, 2007).

b.constraint

In this optional matrix with CbC_b rows and three columns, CbC_b item intercepts bikb_{ik} can be fixed. 1st column: item index, 2nd column: category index, 3rd column: fixed item thresholds

a.constraint

In this optional matrix with CaC_a rows and four columns, CaC_a item intercepts aidka_{idk} can be fixed. 1st column: item index, 2nd column: dimension index, 3rd column: category index, 4th column: fixed item slopes

mean.constraint

A C×3C \times 3 matrix for constraining CC means in the normal distribution assumption (skillspace="normal"). 1st column: Dimension, 2nd column: Group, 3rd column: Value

Sigma.constraint

A C×4C \times 4 matrix for constraining CC covariances in the normal distribution assumption (skillspace="normal"). 1st column: Dimension 1, 2nd column: Dimension 2, 3rd column: Group, 4th column: Value

delta.designmatrix

The design matrix of δ\delta parameters for the reduced skillspace estimation (see Xu & von Davier, 2008)

standardized.latent

A logical indicating whether in a uni- or multidimensional model all latent variables of the first group should be normally distributed and standardized. The default is FALSE.

centered.latent

A logical indicating whether in a uni- or multidimensional model all latent variables of the first group should be normally distributed and do have zero means? The default is FALSE.

centerintercepts

A logical indicating whether intercepts should be centered to have a mean of 0 for all dimensions. This argument does not (yet) work properly for varying numbers of item categories.

centerslopes

A logical indicating whether item slopes should be centered to have a mean of 1 for all dimensions. This argument only works for irtmodel="2PL". The default is FALSE.

maxiter

Maximum number of iterations

conv

Convergence criterion for item parameters and distribution parameters

globconv

Global deviance convergence criterion

msteps

Maximum number of M steps in estimating bb and aa item parameters. The default is to use 4 M steps.

convM

Convergence criterion in M step

decrease.increments

Should in the M step the increments of aa and bb parameters decrease during iterations? The default is FALSE. If there is an increase in deviance during estimation, setting decrease.increments to TRUE is recommended.

use.freqpatt

A logical indicating whether frequencies of unique item response patterns should be used. In case of large data set use.freqpatt=TRUE can speed calculations (depending on the problem). Note that in this case, not all person parameters are calculated as usual in the output.

progress

An optional logical indicating whether the function should print the progress of iteration in the estimation process.

PEM

Logical indicating whether the P-EM acceleration should be applied (Berlinet & Roland, 2012).

PEM_itermax

Number of iterations in which the P-EM method should be applied.

object

A required object of class gdm

file

Optional file name for a file in which summary should be sinked.

x

A required object of class gdm

perstype

Person parameter estimate type. Can be either "EAP", "MAP" or "MLE".

barwidth

Bar width in plot.gdm

histcol

Color of histogram bars in plot.gdm

cexcor

Font size for print of correlation in plot.gdm

pchpers

Point type for scatter plot of person parameters in plot.gdm

cexpers

Point size for scatter plot of person parameters in plot.gdm

...

Optional parameters to be passed to or from other methods will be ignored.

Details

Case irtmodel="1PL":
Equal item slopes of 1 are assumed in this model. Therefore, it corresponds to a generalized multidimensional Rasch model.

logitP(Xnj=kθn)=bj0+dqjdkθndlogit P( X_{nj}=k | \theta_n )=b_{j0} + \sum_d q_{jdk} \theta_{nd}

The Q-matrix entries qjdkq_{jdk} are pre-specified by the user.

Case irtmodel="2PL":
For each item and each dimension, different item slopes ajda_{jd} are estimated:

logitP(Xnj=kθn)=bj0+dajdqjdkθndlogit P( X_{nj}=k | \theta_n )=b_{j0} + \sum_d a_{jd} q_{jdk} \theta_{nd}

Case irtmodel="2PLcat":
For each item, each dimension and each category, different item slopes ajdka_{jdk} are estimated:

logitP(Xnj=kθn)=bj0+dajdkqjdkθndlogit P( X_{nj}=k | \theta_n )=b_{j0} + \sum_d a_{jdk} q_{jdk} \theta_{nd}

Note that this model can be generalized to include terms of any transformation tht_h of the θn\theta_n vector (e.g. quadratic terms, step functions or interaction) such that the model can be formulated as

logitP(Xnj=kθn)=bj0+hajhkqjhkth(θn)logit P( X_{nj}=k | \theta_n )=b_{j0} + \sum_h a_{jhk} q_{jhk} t_h( \theta_{n} )

In general, the number of functions t1,...,tHt_1, ..., t_H will be larger than the θ\theta dimension of DD.

The estimation follows an EM algorithm as described in von Davier and Yamamoto (2004) and von Davier (2008).

In case of skillspace="est", the θ\bold{\theta} vectors (the grid of the theta distribution) are estimated (Bartolucci, 2007; Bacci, Bartolucci & Gnaldi, 2012). This model is called a multidimensional latent class item response model.

Value

An object of class gdm. The list contains the following entries:

item

Data frame with item parameters

person

Data frame with person parameters: EAP denotes the mean of the individual posterior distribution, SE.EAP the corresponding standard error, MLE the maximum likelihood estimate at theta.k and MAP the mode of the posterior distribution

EAP.rel

Reliability of the EAP

deviance

Deviance

ic

Information criteria, number of estimated parameters

b

Item intercepts bjkb_{jk}

se.b

Standard error of item intercepts bjkb_{jk}

a

Item slopes ajda_{jd} resp. ajdka_{jdk}

se.a

Standard error of item slopes ajda_{jd} resp. ajdka_{jdk}

itemfit.rmsea

The RMSEA item fit index (see itemfit.rmsea). This entry comes as a list with total and group-wise item fit statistics.

mean.rmsea

Mean of RMSEA item fit indexes.

Qmatrix

Used Q-matrix

pi.k

Trait distribution

mean.trait

Means of trait distribution

sd.trait

Standard deviations of trait distribution

skewness.trait

Skewnesses of trait distribution

correlation.trait

List of correlation matrices of trait distribution corresponding to each group

pjk

Item response probabilities evaluated at grid theta.k

n.ik

An array of expected counts ncikgn_{cikg} of ability class cc at item ii at category kk in group gg

G

Number of groups

D

Number of dimension of θ\bold{\theta}

I

Number of items

N

Number of persons

delta

Parameter estimates for skillspace representation

covdelta

Covariance matrix of parameter estimates for skillspace representation

data

Original data frame

group.stat

Group statistics (sample sizes, group labels)

p.xi.aj

Individual likelihood

posterior

Individual posterior distribution

skill.levels

Number of skill levels per dimension

K.item

Maximal category per item

theta.k

Used theta design or estimated theta trait distribution in case of skillspace="est"

thetaDes

Used theta design for item responses

se.theta.k

Estimated standard errors of theta.k if it is estimated

time

Info about computation time

skillspace

Used skillspace parametrization

iter

Number of iterations

converged

Logical indicating whether convergence was achieved.

object

Object of class gdm

x

Object of class gdm

perstype

Person paramter estimate type. Can be either "EAP", "MAP" or "MLE".

group

Group which should be used for plot.gdm

barwidth

Bar width in plot.gdm

histcol

Color of histogram bars in plot.gdm

cexcor

Font size for print of correlation in plot.gdm

pchpers

Point type for scatter plot of person parameters in plot.gdm

cexpers

Point size for scatter plot of person parameters in plot.gdm

...

Optional parameters to be passed to or from other methods will be ignored.

References

Bacci, S., Bartolucci, F., & Gnaldi, M. (2012). A class of multidimensional latent class IRT models for ordinal polytomous item responses. arXiv preprint, arXiv:1201.4667.

Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141-157.

Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122-4137.

von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287-307.

von Davier, M., & Yamamoto, K. (2004). Partially observed mixtures of IRT models: An extension of the generalized partial-credit model. Applied Psychological Measurement, 28, 389-406.

Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS.

See Also

Cognitive diagnostic models for dichotomous data can be estimated with din (DINA or DINO model) or gdina (GDINA model, which contains many CDMs as special cases).

For assessment of model fit see modelfit.cor.din and anova.gdm.

See itemfit.sx2 for item fit statistics.

For the estimation of the multidimensional latent class item response model see the MultiLCIRT package and sirt package (function sirt::rasch.mirtlc).

Examples

#############################################################################
# EXAMPLE 1: Fraction Dataset 1
#      Unidimensional Models for dichotomous data
#############################################################################

data(data.fraction1, package="CDM")
dat <- data.fraction1$data
theta.k <- seq( -6, 6, len=15 )   # discretized ability

#***
# Model 1: Rasch model (normal distribution)
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
               centered.latent=TRUE)
summary(mod1)
plot(mod1)

#***
# Model 2: Rasch model (log-linear smoothing)
# set the item difficulty of the 8th item to zero
b.constraint <- matrix( c(8,1,0), 1, 3 )
mod2 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
               skillspace="loglinear", b.constraint=b.constraint  )
summary(mod2)

#***
# Model 3: 2PL model
mod3 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
               skillspace="normal", standardized.latent=TRUE  )
summary(mod3)

## Not run: 
#***
# Model 4: include quadratic term in item response function
#   using the argument decrease.increments=TRUE leads to a more
#   stable estimate
thetaDes <- cbind( theta.k, theta.k^2 )
colnames(thetaDes) <- c( "F1", "F1q" )
mod4 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
          thetaDes=thetaDes, skillspace="normal",
          standardized.latent=TRUE, decrease.increments=TRUE)
summary(mod4)

#***
# Model 5: step function for ICC
#          two different probabilities theta < 0 and theta > 0
thetaDes <- matrix( 1*(theta.k>0), ncol=1 )
colnames(thetaDes) <- c( "Fgrm1" )
mod5 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
          thetaDes=thetaDes, skillspace="normal" )
summary(mod5)

#***
# Model 6: DINA model with din function
mod6 <- CDM::din( dat, q.matrix=matrix( 1, nrow=ncol(dat),ncol=1 ) )
summary(mod6)

#***
# Model 7: Estimating a version of the DINA model with gdm
theta.k <- c(-.5,.5)
mod7 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, skillspace="loglinear" )
summary(mod7)

#############################################################################
# EXAMPLE 2: Cultural Activities - data.Students
#      Unidimensional Models for polytomous data
#############################################################################

data(data.Students, package="CDM")
dat <- data.Students

dat <- dat[, grep( "act", colnames(dat) ) ]
theta.k <- seq( -4, 4, len=11 )   # discretized ability

#***
# Model 1: Partial Credit Model (PCM)
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
           centered.latent=TRUE)
summary(mod1)
plot(mod1)

#***
# Model 1b: PCM using frequency patterns
mod1b <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
           centered.latent=TRUE, use.freqpatt=TRUE)
summary(mod1b)

#***
# Model 2: PCM with two groups
mod2 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
            group=CDM::data.Students$urban + 1, skillspace="normal",
            centered.latent=TRUE)
summary(mod2)

#***
# Model 3: PCM with loglinear smoothing
b.constraint <- matrix( c(1,2,0), ncol=3 )
mod3 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,
    skillspace="loglinear", b.constraint=b.constraint )
summary(mod3)

#***
# Model 4: Model with pre-specified item weights in Q-matrix
Qmatrix <- array( 1, dim=c(5,1,2) )
Qmatrix[,1,2] <- 2     # default is score 2 for category 2
# now change the scoring of category 2:
Qmatrix[c(2,4),1,1] <- .74
Qmatrix[c(2,4),1,2] <- 2.3
# for items 2 and 4 the score for category 1 is .74 and for category 2 it is 2.3
mod4 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
           skillspace="normal", centered.latent=TRUE)
summary(mod4)

#***
# Model 5: Generalized partial credit model
mod5 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,
          skillspace="normal", standardized.latent=TRUE )
summary(mod5)

#***
# Model 6: Item-category slope estimation
mod6 <- CDM::gdm( dat, irtmodel="2PLcat", theta.k=theta.k,  skillspace="normal",
                 standardized.latent=TRUE, decrease.increments=TRUE)
summary(mod6)

#***
# Models 7: items with different number of categories
dat0 <- dat
dat0[ paste(dat0[,1])==2, 1 ] <- 1 # 1st item has only two categories
dat0[ paste(dat0[,3])==2, 3 ] <- 1 # 3rd item has only two categories

# Model 7a: PCM
mod7a <- CDM::gdm( dat0, irtmodel="1PL", theta.k=theta.k,  centered.latent=TRUE )
summary(mod7a)

# Model 7b: Item category slopes
mod7b <- CDM::gdm( dat0, irtmodel="2PLcat", theta.k=theta.k,
                 standardized.latent=TRUE, decrease.increments=TRUE )
summary(mod7b)

#############################################################################
# EXAMPLE 3: Fraction Dataset 2
#      Multidimensional Models for dichotomous data
#############################################################################

data(data.fraction2, package="CDM")
dat <- data.fraction2$data
Qmatrix <- data.fraction2$q.matrix3

#***
# Model 1: One-dimensional Rasch model
theta.k <- seq( -4, 4, len=11 )   # discretized ability
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k,  centered.latent=TRUE)
summary(mod1)
plot(mod1)

#***
# Model 2: One-dimensional 2PL model
mod2 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, standardized.latent=TRUE)
summary(mod2)
plot(mod2)

#***
# Model 3: 3-dimensional Rasch Model (normal distribution)
mod3 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
            centered.latent=TRUE,  globconv=5*1E-3, conv=1E-4 )
summary(mod3)

#***
# Model 4: 3-dimensional Rasch model (loglinear smoothing)
# set some item parameters of items 4,1 and 2 to zero
b.constraint <- cbind( c(4,1,2), 1, 0 )
mod4 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
              b.constraint=b.constraint, skillspace="loglinear" )
summary(mod4)

#***
# Model 5: define a different theta grid for each dimension
theta.k <- list( "Dim1"=seq( -5, 5, len=11 ),
                 "Dim2"=seq(-5,5,len=8),
                 "Dim3"=seq( -3,3,len=6) )
mod5 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
                 b.constraint=b.constraint,  skillspace="loglinear")
summary(mod5)

#***
# Model 6: multdimensional 2PL model (normal distribution)
theta.k <- seq( -5, 5, len=13 )
a.constraint <- cbind( c(8,1,3), 1:3, 1, 1 ) # fix some slopes to 1
mod6 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,  Qmatrix=Qmatrix,
            centered.latent=TRUE, a.constraint=a.constraint, decrease.increments=TRUE,
            skillspace="normal")
summary(mod6)

#***
# Model 7: multdimensional 2PL model (loglinear distribution)
a.constraint <- cbind( c(8,1,3), 1:3, 1, 1 )
b.constraint <- cbind( c(8,1,3), 1, 0 )
mod7 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k,  Qmatrix=Qmatrix,
               b.constraint=b.constraint,  a.constraint=a.constraint,
               decrease.increments=FALSE, skillspace="loglinear")
summary(mod7)

#############################################################################
# EXAMPLE 4: Unidimensional latent class 1PL IRT model
#############################################################################

# simulate data
set.seed(754)
I <- 20     # number of items
N <- 2000   # number of persons
theta <- c( -2, 0, 1, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8)  )
b <- seq(-2,2,len=I)
library(sirt)   # use function sim.raschtype from sirt package
dat <- sirt::sim.raschtype( theta=theta, b=b )

theta.k <- seq(-1, 1, len=4)      # initial vector of theta
# estimate model
mod1 <- CDM::gdm( dat, theta.k=theta.k, skillspace="est", irtmodel="1PL",
            centerintercepts=TRUE, maxiter=200)
summary(mod1)
  ##   Estimated Skill Distribution
  ##         F1    pi.k
  ##   1 -1.988 0.24813
  ##   2 -0.055 0.23313
  ##   3  0.940 0.40059
  ##   4  2.000 0.11816

#############################################################################
# EXAMPLE 5: Multidimensional latent class IRT model
#############################################################################

# We simulate a two-dimensional IRT model in which theta vectors
# are observed at a fixed discrete grid (see below).

# simulate data
set.seed(754)
I <- 13     # number of items
N <- 2400   # number of persons

# simulate Dimension 1 at 4 discrete theta points
theta <- c( -2, 0, 1, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8)  )
b <- seq(-2,2,len=I)
library(sirt)  # use simulation function from sirt package
dat1 <- sirt::sim.raschtype( theta=theta, b=b )
# simulate Dimension 2 at 4 discrete theta points
theta <- c( -3, 0, 1.5, 2 )
theta <- rep( theta, c(N/4,N/4, 3*N/8, N/8)  )
dat2 <- sirt::sim.raschtype( theta=theta, b=b )
colnames(dat2) <- gsub( "I", "U", colnames(dat2))
dat <- cbind( dat1, dat2 )

# define Q-matrix
Qmatrix <- matrix(0,2*I,2)
Qmatrix[ cbind( 1:(2*I), rep(1:2, each=I) ) ] <- 1

theta.k <- seq(-1, 1, len=4)      # initial matrix
theta.k <- cbind( theta.k, theta.k )
colnames(theta.k) <- c("Dim1","Dim2")

# estimate model
mod2 <- CDM::gdm( dat, theta.k=theta.k, skillspace="est", irtmodel="1PL",
              Qmatrix=Qmatrix, centerintercepts=TRUE)
summary(mod2)
  ##   Estimated Skill Distribution
  ##     theta.k.Dim1 theta.k.Dim2    pi.k
  ##   1       -2.022       -3.035 0.25010
  ##   2        0.016        0.053 0.24794
  ##   3        0.956        1.525 0.36401
  ##   4        1.958        1.919 0.13795

#############################################################################
# EXAMPLE 6: Large-scale dataset data.mg
#############################################################################

data(data.mg, package="CDM")
dat <- data.mg[, paste0("I", 1:11 ) ]
theta.k <- seq(-6,6,len=21)

#***
# Model 1: Generalized partial credit model with multiple groups
mod1 <- CDM::gdm( dat, irtmodel="2PL", theta.k=theta.k, group=CDM::data.mg$group,
              skillspace="normal", standardized.latent=TRUE)
summary(mod1)

## End(Not run)

Ideal Response Pattern

Description

This function computes the ideal response pattern which is the latent item response ηlj=k=1Kαlk\eta_{lj}=\prod_{k=1}^K \alpha_{lk} for a person with skill profile ll at item jj.

Usage

ideal.response.pattern(q.matrix, skillspace=NULL, rule="DINA")

Arguments

q.matrix

The Q-matrix

skillspace

An optional skill space matrix. If it is not provided, then all skill classes are used for creating an ideal response pattern.

rule

Chosen condensation rule for the CDM. Can be "DINA" or "DINO".

Value

A list with following entries

idealresp

A matrix with ideal response patterns

skillspace

Used skill space

Examples

#############################################################################
# EXAMPLE 1: Ideal response pattern sim.qmatrix
#############################################################################

data(sim.qmatrix, package="CDM")
q.matrix <- sim.qmatrix

#- ideal response pattern for DINA model
CDM::ideal.response.pattern(q.matrix)

#- ideal response pattern for DINO model
CDM::ideal.response.pattern( q.matrix, rule="DINO" )

# compute ideal responses for a reduced skill space
skillspace <- matrix( c( 0,1,0,
                         1,1,0 ), 2,3, byrow=TRUE )
CDM::ideal.response.pattern( q.matrix, skillspace=skillspace)

Helper Function for Conducting Likelihood Ratio Tests

Description

This is a helper function for conducting likelihood ratio tests and can be generally used for objects for which the logLik method is defined.

Usage

IRT.anova(object, ...)

Arguments

object

Object for which the logLik method is defined.

...

A further object to be passed

See Also

See also IRT.compareModels for model comparisons of several models.

See also as anova.din.


Individual Classification for Fitted Models

Description

Computes individual classifications based on a fitted model.

Usage

IRT.classify(object, type="MLE")

Arguments

object

Fitted model for which methods IRT.likelihood and IRT.posterior are defined.

type

Type of classification: "MLE" (maximum likelihood estimate) or "MAP" (maximum of posterior distribution)

Value

List with entries

class_theta

Individual classification

class_index

Class index of individual classification

class_maxval

Maximum value corresponding to individual classification

See Also

See IRT.factor.scores for similar functionality.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Individual classification data.ecpe
#############################################################################

data(data.ecpe, package="CDM")
dat <- data.ecpe$dat[,-1]
Q <- data.ecpe$q.matrix

#** estimate GDINA model
mod <- CDM::gdina(dat, q.matrix=Q)
summary(mod)

#** classify individuals
cmod <- CDM::IRT.classify(mod)
str(cmod)

## End(Not run)

Comparisons of Several Models

Description

Performs model comparisons based on information criteria and likelihood ratio test. This function allows all objects for which the logLik (stats) S3 method is defined. The output of IRT.modelfit can also be used as input for this function.

Usage

IRT.compareModels(object, ...)

## S3 method for class 'IRT.compareModels'
summary(object, extended=TRUE, ...)

Arguments

object

Object

extended

Optional logical indicating whether all or or only a subset of fit statistics should be printed.

...

Further objects to be passed.

Value

A list with following entries

IC

Data frame with information criteria

LRtest

Data frame with all (useful) pairwise likelihood ratio tests

See Also

The function is based on IRT.IC.

For comparing two models see anova.din.

For computing absolute model fit see IRT.modelfit.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Model comparison sim.dina dataset
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

dat <- sim.dina
q.matrix <- sim.qmatrix

#*** Model 0: DINA model with equal guessing and slipping parameters
mod0 <- CDM::din( dat, q.matrix, guess.equal=TRUE, slip.equal=TRUE )
summary(mod0)

#*** Model 1: DINA model
mod1 <- CDM::din( dat, q.matrix )
summary(mod1)

#*** Model 2: DINO model
mod2 <- CDM::din( dat, q.matrix, rule="DINO")
summary(mod2)

#*** Model 3: Additive GDINA model
mod3 <- CDM::gdina( dat, q.matrix, rule="ACDM")
summary(mod3)

#*** Model 4: GDINA model
mod4 <- CDM::gdina( dat, q.matrix, rule="GDINA")
summary(mod4)

# model comparisons
res <- CDM::IRT.compareModels( mod0, mod1, mod2, mod3, mod4 )
res
  ##   > res
  ##   $IC
  ##     Model   loglike Deviance Npars Nobs      AIC      BIC     AIC3     AICc     CAIC
  ##   1  mod0 -2176.482 4352.963     9  400 4370.963 4406.886 4379.963 4371.425 4415.886
  ##   2  mod1 -2042.378 4084.756    25  400 4134.756 4234.543 4159.756 4138.232 4259.543
  ##   3  mod2 -2086.805 4173.610    25  400 4223.610 4323.396 4248.610 4227.086 4348.396
  ##   4  mod3 -2048.233 4096.466    32  400 4160.466 4288.193 4192.466 4166.221 4320.193
  ##   5  mod4 -2026.633 4053.266    41  400 4135.266 4298.917 4176.266 4144.887 4339.917
  ##
# -> The DINA model (mod1) performed best in terms of AIC.
  ##   $LRtest
  ##     Model1 Model2      Chi2 df            p
  ##   1   mod0   mod1 268.20713 16 0.000000e+00
  ##   2   mod0   mod2 179.35362 16 0.000000e+00
  ##   3   mod0   mod3 256.49745 23 0.000000e+00
  ##   4   mod0   mod4 299.69671 32 0.000000e+00
  ##   5   mod1   mod3 -11.70967  7 1.000000e+00
  ##   6   mod1   mod4  31.48959 16 1.164415e-02
  ##   7   mod2   mod3  77.14383  7 5.262457e-14
  ##   8   mod2   mod4 120.34309 16 0.000000e+00
  ##   9   mod3   mod4  43.19926  9 1.981445e-06
  ##
# -> The GDINA model (mod4) was superior to the other models in terms
#    of the likelihood ratio test.

# get an overview with summary
summary(res)
summary(res,extended=FALSE)

#*******************
# applying model comparison for objects of class IRT.modelfit

# compute model fit statistics
fmod0 <- CDM::IRT.modelfit(mod0)
fmod1 <- CDM::IRT.modelfit(mod1)
fmod4 <- CDM::IRT.modelfit(mod4)

# model comparison
res <- CDM::IRT.compareModels( fmod0, fmod1,  fmod4 )
res
  ##   $IC
  ##        Model   loglike Deviance Npars Nobs      AIC      BIC     AIC3
  ##   mod0  mod0 -2176.482 4352.963     9  400 4370.963 4406.886 4379.963
  ##   mod1  mod1 -2042.378 4084.756    25  400 4134.756 4234.543 4159.756
  ##   mod4  mod4 -2026.633 4053.266    41  400 4135.266 4298.917 4176.266
  ##            AICc     CAIC      maxX2   p_maxX2     MADcor      SRMSR
  ##   mod0 4371.425 4415.886 118.172707 0.0000000 0.09172287 0.10941300
  ##   mod1 4138.232 4259.543   8.728248 0.1127943 0.03025354 0.03979948
  ##   mod4 4144.887 4339.917   2.397241 1.0000000 0.02284029 0.02989669
  ##        X100.MADRESIDCOV      MADQ3     MADaQ3
  ##   mod0        1.9749936 0.08840892 0.08353917
  ##   mod1        0.6713952 0.06184332 0.05923058
  ##   mod4        0.5148707 0.07477337 0.07145600
  ##
  ##   $LRtest
  ##     Model1 Model2      Chi2 df          p
  ##   1   mod0   mod1 268.20713 16 0.00000000
  ##   2   mod0   mod4 299.69671 32 0.00000000
  ##   3   mod1   mod4  31.48959 16 0.01164415

## End(Not run)

S3 Method for Extracting Used Item Response Dataset

Description

This S3 method extracts the used dataset with item responses.

Usage

IRT.data(object, ...)

## S3 method for class 'din'
IRT.data(object, ...)

## S3 method for class 'gdina'
IRT.data(object, ...)

## S3 method for class 'gdm'
IRT.data(object, ...)

## S3 method for class 'mcdina'
IRT.data(object, ...)

## S3 method for class 'reglca'
IRT.data(object, ...)

## S3 method for class 'slca'
IRT.data(object, ...)

Arguments

object

Object of classes din, gdina, mcdina, gdm, slca, reglca.

...

More arguments to be passed.

Value

A matrix (or data frame) with item responses and group identifier and weights vector as attributes.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Several models for sim.dina data
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

dat <- sim.dina
q.matrix <- sim.qmatrix

#--- Model 1: GDINA model
mod1 <- CDM::gdina( data=dat, q.matrix=q.matrix)
summary(mod1)
dmod1 <- CDM::IRT.data(mod1)
str(dmod1)

#--- Model 2: DINA model
mod2 <- CDM::din( data=dat, q.matrix=q.matrix)
summary(mod2)
dmod2 <- CDM::IRT.data(mod2)

#--- Model 3: Rasch model with gdm function
mod3 <- CDM::gdm( data=dat, irtmodel="1PL", theta.k=seq(-4,4,length=11),
                centered.latent=TRUE )
summary(mod3)
dmod3 <- CDM::IRT.data(mod3)

#--- Model 4: Latent class model with two classes

dat <- sim.dina
I <- ncol(dat)

# define design matrices
TP <- 2     # two classes
# The idea is that latent classes refer to two different "dimensions".
# Items load on latent class indicators 1 and 2, see below.
Xdes <- array(0, dim=c(I,2,2,2*I) )
items <- colnames(dat)
dimnames(Xdes)[[4]] <- c(paste0( colnames(dat), "Class", 1),
          paste0( colnames(dat), "Class", 2) )
    # items, categories, classes, parameters
# probabilities for correct solution
for (ii in 1:I){
    Xdes[ ii, 2, 1, ii ] <- 1    # probabilities class 1
    Xdes[ ii, 2, 2, ii+I ] <- 1  # probabilities class 2
}
# estimate model
mod4 <- CDM::slca( dat, Xdes=Xdes)
summary(mod4)
dmod4 <- CDM::IRT.data(mod4)

## End(Not run)

S3 Method for Extracting Expected Counts

Description

This S3 method extracts expected counts from model output.

Usage

IRT.expectedCounts(object, ...)

## S3 method for class 'din'
IRT.expectedCounts(object, ...)

## S3 method for class 'gdina'
IRT.expectedCounts(object, ...)

## S3 method for class 'gdm'
IRT.expectedCounts(object, ...)

## S3 method for class 'mcdina'
IRT.expectedCounts(object, ...)

## S3 method for class 'slca'
IRT.expectedCounts(object, ...)

## S3 method for class 'reglca'
IRT.expectedCounts(object, ...)

Arguments

object

Object of classes din, gdina, mcdina, gdm or slca.

...

More arguments to be passed.

Value

An array with expected counts. The dimensions are items, categories, latent classes and groups.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Expected counts gdm function
#############################################################################

data(data.fraction1, package="CDM")
dat <- data.fraction1$data
theta.k <- seq( -6, 6, len=11 )   # discretized ability

#--- Model 1: Rasch model
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
               centered.latent=TRUE )
emod1 <- CDM::IRT.expectedCounts(mod1)
str(emod1)

#############################################################################
# EXAMPLE 2: Expected counts gdina function
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

#--- Model 1: estimation of the GDINA model
mod1 <- CDM::gdina( data=sim.dina, q.matrix=sim.qmatrix)
summary(mod1)
emod1 <- CDM::IRT.expectedCounts(mod1)
str(emod1)

#--- Model 2: GDINA model with two groups
mod2 <- CDM::gdina( data=CDM::sim.dina, q.matrix=CDM::sim.qmatrix,
                   group=rep(1:2, each=200) )
summary(mod2)
emod2 <- CDM::IRT.expectedCounts( mod2 )
str(emod2)

## End(Not run)

S3 Methods for Extracting Factor Scores (Person Classifications)

Description

This S3 method extracts factor scores or skill classifications.

Usage

IRT.factor.scores(object, ...)

## S3 method for class 'din'
IRT.factor.scores(object, type="MLE", ...)

## S3 method for class 'gdina'
IRT.factor.scores(object, type="MLE", ...)

## S3 method for class 'mcdina'
IRT.factor.scores(object, type="MLE", ...)

## S3 method for class 'gdm'
IRT.factor.scores(object, type="EAP", ...)

## S3 method for class 'slca'
IRT.factor.scores(object, type="MLE", ...)

Arguments

object

Object of classes din, gdina, mcdina, gdm or slca.

type

Type of estimated factor score. This can be "MLE", "MAP" or "EAP". The type EAP cannot be used for objects of class slca.

...

More arguments to be passed.

Value

A matrix or a vector with classified scores.

See Also

For extracting the individual likelihood or the individual posterior see IRT.likelihood or IRT.posterior.

Examples

#############################################################################
# EXAMPLE 1: Extracting factor scores in the DINA model
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

# estimate DINA model
mod1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix)
summary(mod1)
# MLE
fsc1a <- CDM::IRT.factor.scores(mod1)
# MAP
fsc1b <- CDM::IRT.factor.scores(mod1, type="MAP")
# EAP
fsc1c <- CDM::IRT.factor.scores(mod1, type="EAP")
# compare classification for skill 1
stats::xtabs( ~ fsc1a[,1] + fsc1b[,1] )
graphics::boxplot( fsc1c[,1] ~ fsc1a[,1] )

S3 Method for Computing Observed and Expected Frequencies of Univariate and Bivariate Marginals

Description

This S3 method computes observed and expected frequencies for univariate and bivariate distributions.

Usage

IRT.frequencies(object, ...)

IRT_frequencies_default(data, post, probs, weights=NULL)

IRT_frequencies_wrapper(object, ...)

## S3 method for class 'din'
IRT.frequencies(object, ...)

## S3 method for class 'gdina'
IRT.frequencies(object, ...)

## S3 method for class 'mcdina'
IRT.frequencies(object, ...)

## S3 method for class 'gdm'
IRT.frequencies(object, ...)

## S3 method for class 'slca'
IRT.frequencies(object, ...)

Arguments

object

Object of classes din, gdina, mcdina, gdm or slca.

...

More arguments to be passed.

data

Item response data as extracted by IRT.data

post

Individual posterior distribution as extracted by IRT.posterior

probs

Individual posterior distribution as extracted by IRT.irfprob

weights

Optional vector of weights as included as the attribute weights in IRT.data

Value

List with following entries

uni_obs

Univariate observed distribution

uni_exp

Univariate expected distribution

M_obs

Univariate observed means

M_exp

Univariate expected means

SD_obs

Univariate observed standard deviations

SD_exp

Univariate expected standard deviations

biv_obs

Bivariate observed frequencies

biv_exp

Bivariate expected frequencies

biv_N

Bivariate sample size

cov_obs

Observed covariances

cov_cor

Expected covariances

cor_obs

Observed correlations

cor_exp

Expected correlations

chisq

Chi square statistic of local independence

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Usage IRT.frequencies
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

# estimate GDINA model
mod1 <- CDM::gdina( data=sim.dina,  q.matrix=sim.qmatrix)
summary(mod1)

# direct usage of IRT.frequencies
fres1 <- CDM::IRT.frequencies(mod1)

# use of the default function with input data
data <- CDM::IRT.data(object)
post <- CDM::IRT.posterior(object)
probs <- CDM::IRT.irfprob(object)
fres2 <- CDM::IRT_frequencies_default(data=data, post=post, probs=probs)

## End(Not run)

Information Criteria

Description

Computes several information criteria for objects which do have the logLik (stats) S3 method (e.g. din, gdina, gdm, ...) .

Usage

IRT.IC(object)

Arguments

object

Objects which do have the logLik (stats) S3 method.

Value

A vector with deviance and several information criteria.

See Also

See also anova.din for model comparisons. A general method is defined in IRT.compareModels.

Examples

#############################################################################
# EXAMPLE 1: DINA example information criteria
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

#*** Model 1: DINA model
mod1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix )
summary(mod1)
IRT.IC(mod1)

S3 Methods for Extracting Item Response Functions

Description

This S3 method extracts item response functions evaluated at a grid of abilities (skills). Item response functions can be plotted using the IRT.irfprobPlot function.

Usage

IRT.irfprob(object, ...)

## S3 method for class 'din'
IRT.irfprob(object, ...)

## S3 method for class 'gdina'
IRT.irfprob(object, ...)

## S3 method for class 'gdm'
IRT.irfprob(object, ...)

## S3 method for class 'mcdina'
IRT.irfprob(object, ...)

## S3 method for class 'reglca'
IRT.irfprob(object, ...)

## S3 method for class 'slca'
IRT.irfprob(object, ...)

Arguments

object

Object of classes din, gdina, mcdina, gdm, slca, reglca.

...

More arguments to be passed.

Value

An array with item response probabilities (items ×\times categories ×\times skill classes [×\times group]) and attributes

theta

Uni- or multidimensional skill space (theta grid in item response models).

prob.theta

Probability distribution of theta

skillspace

Design matrix and estimated parameters for skill space distribution (only for IRT.posterior.slca)

G

Number of groups

See Also

Plot functions for item response curves: IRT.irfprobPlot.

For extracting the individual likelihood or posterior see IRT.likelihood or IRT.posterior.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Extracting item response functions mcdina model
#############################################################################

data(data.cdm02, package="CDM")

dat <- data.cdm02$data
q.matrix <- data.cdm02$q.matrix

#-- estimate model
mod1 <- CDM::mcdina( dat, q.matrix=q.matrix)
#-- extract item response functions
prmod1 <- CDM::IRT.irfprob(mod1)
str(prmod1)

## End(Not run)

Plot Item Response Functions

Description

This function plots item response functions for fitted item response models for which the IRT.irfprob method is defined.

Usage

IRT.irfprobPlot( object, items=NULL, min.theta=-4, max.theta=4, cumul=FALSE,
     smooth=TRUE, ask=TRUE,  n.theta=40, package="lattice",... )

Arguments

object

Fitted item response model for which the IRT.irfprob method is defined

items

Vector of indices of selected items.

min.theta

Minimum theta to be displayed.

max.theta

Maximum theta to be displayed.

cumul

Optional logical indicating whether cumulated item response functions P(Xkθ)P( X \ge k | \theta ) should be displayed.

smooth

Optional logical indicating whether item response functions should be smoothed for plotting.

ask

Logical for asking for a new plot.

n.theta

Number of theta points if smooth=TRUE is chosen.

package

String indicating which package should be used for plotting the item response curves. Options are "lattice" or "graphics".

...

More arguments to be passed for the plot in lattice.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Plot item response functions from a unidimensional model
#############################################################################

data(data.Students, package="CDM")

dat <- data.Students
resp <- dat[, paste0("sc",1:4) ]
resp[ paste(resp[,1])==3,1] <-  2
psych::describe(resp)

#--- Model 1: PCM in CDM::gdm
theta.k <- seq( -5, 5, len=21 )
mod1 <- CDM::gdm( dat=resp, irtmodel="1PL", theta.k=theta.k, skillspace="normal",
           centered.latent=TRUE)
summary(mod1)

# plot
IRT.irfprobPlot( mod1 )
# plot in graphics package (which comes with R base version)
IRT.irfprobPlot( mod1, package="graphics")
# plot first and third item and do not smooth discretized item response
# functions in IRT.irfprob
IRT.irfprobPlot( mod1, items=c(1,3), smooth=FALSE )
# cumulated IRF
IRT.irfprobPlot( mod1, cumul=TRUE )

#############################################################################
# EXAMPLE 2: Fitted mutidimensional model with gdm
#############################################################################

dat <- CDM::data.fraction2$data
Qmatrix <- CDM::data.fraction2$q.matrix3

# Model 1: 3-dimensional Rasch Model (normal distribution)
theta.k <- seq( -4, 4, len=11 )   # discretized ability
mod1 <- CDM::gdm( dat, irtmodel="1PL", theta.k=theta.k, Qmatrix=Qmatrix,
              centered.latent=TRUE, maxiter=10 )
summary(mod1)

# unsmoothed curves
IRT.irfprobPlot(mod1, smooth=FALSE)
# smoothed curves
IRT.irfprobPlot(mod1)

## End(Not run)

S3 Methods for Computing Item Fit

Description

This S3 method computes some selected item fit statistic.

Usage

IRT.itemfit(object, ...)

## S3 method for class 'din'
IRT.itemfit(object, method="RMSEA", ...)

## S3 method for class 'gdina'
IRT.itemfit(object, method="RMSEA", ...)

## S3 method for class 'gdm'
IRT.itemfit(object, method="RMSEA", ...)

## S3 method for class 'reglca'
IRT.itemfit(object, method="RMSEA", ...)

## S3 method for class 'slca'
IRT.itemfit(object, method="RMSEA", ...)

Arguments

object

Object of classes din, gdina, gdm, slca, reglca.

method

Method for computing item fit statistic. Until now, only method="RMSEA" (see itemfit.rmsea) can be used.

...

More arguments to be passed.

Value

Vector or data frame with item fit statistics.

See Also

For extracting the individual likelihood or posterior see IRT.likelihood or IRT.posterior.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: DINA model item fit
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

# estimate model
mod1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix)
# compute item fit
IRT.itemfit( mod1 )

## End(Not run)

Jackknifing an Item Response Model

Description

This function performs a Jackknife procedure for estimating standard errors for an item response model. The replication design must be defined by IRT.repDesign. Model fit is also assessed via Jackknife.

Statistical inference for derived parameters is performed by IRT.derivedParameters with a fitted object of class IRT.jackknife and a list with defining formulas.

Usage

IRT.jackknife(object,repDesign, ... )

IRT.derivedParameters(jkobject, derived.parameters )

## S3 method for class 'gdina'
IRT.jackknife(object, repDesign, ...)

## S3 method for class 'IRT.jackknife'
coef(object, bias.corr=FALSE, ...)

## S3 method for class 'IRT.jackknife'
vcov(object, ...)

Arguments

object

Objects for which S3 method IRT.jackknife is defined.

repDesign

Replication design generated by IRT.repDesign.

jkobject

Object of class IRT.jackknife.

derived.parameters

List with defined derived parameters (see Example 2, Model 2).

bias.corr

Optional logical indicating whether a bias correction should be employed.

...

Further arguments to be passed.

Value

List with following entries

jpartable

Parameter table with Jackknife estimates

parsM

Matrix with replicated statistics

vcov

Variance covariance matrix of parameters

Examples

## Not run: 
library(BIFIEsurvey)

#############################################################################
# EXAMPLE 1: Multiple group DINA model with TIMSS data | Cluster sample
#############################################################################

data(data.timss11.G4.AUT.part, package="CDM")

dat <- data.timss11.G4.AUT.part$data
q.matrix <- data.timss11.G4.AUT.part$q.matrix2
# extract items
items <- paste(q.matrix$item)

# generate replicate design
rdes <- CDM::IRT.repDesign( data=dat,  wgt="TOTWGT", jktype="JK_TIMSS",
                   jkzone="JKCZONE", jkrep="JKCREP" )

#--- Model 1: fit multiple group GDINA model
mod1 <- CDM::gdina( dat[,items], q.matrix=q.matrix[,-1],
            weights=dat$TOTWGT, group=dat$female +1  )
# jackknife Model 1
jmod1 <- CDM::IRT.jackknife( object=mod1, repDesign=rdes )
summary(jmod1)
coef(jmod1)
vcov(jmod1)

#############################################################################
# EXAMPLE 2: DINA model | Simple random sampling
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")
dat <- sim.dina
q.matrix <- sim.qmatrix

# generate replicate design with 50 jackknife zones (50 random groups)
rdes <- CDM::IRT.repDesign( data=dat,  jktype="JK_RANDOM", ngr=50 )

#--- Model 1: DINA model
mod1 <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINA")
summary(mod1)
# jackknife DINA model
jmod1 <- CDM::IRT.jackknife( object=mod1, repDesign=rdes )
summary(jmod1)

#--- Model 2: DINO model
mod2 <- CDM::gdina( dat, q.matrix=q.matrix, rule="DINO")
summary(mod2)
# jackknife DINA model
jmod2 <- CDM::IRT.jackknife( object=mod2, repDesign=rdes )
summary(jmod2)
IRT.compareModels( mod1, mod2 )

# statistical inference for derived parameters
derived.parameters <- list( "skill1"=~ 0 + I(prob_skillV1_lev1_group1),
    "skilldiff12"=~ 0 + I( prob_skillV2_lev1_group1 - prob_skillV1_lev1_group1 ),
    "skilldiff13"=~ 0 + I( prob_skillV3_lev1_group1 - prob_skillV1_lev1_group1 )
                    )
jmod2a <- CDM::IRT.derivedParameters( jmod2, derived.parameters=derived.parameters )
summary(jmod2a)
coef(jmod2a)

## End(Not run)

S3 Methods for Extracting of the Individual Likelihood and the Individual Posterior

Description

Functions for extracting the individual likelihood and individual posterior distribution.

Usage

IRT.likelihood(object, ...)

IRT.posterior(object, ...)

## S3 method for class 'din'
IRT.likelihood(object, ...)
## S3 method for class 'din'
IRT.posterior(object, ...)

## S3 method for class 'gdina'
IRT.likelihood(object, ...)
## S3 method for class 'gdina'
IRT.posterior(object, ...)

## S3 method for class 'gdm'
IRT.likelihood(object, ...)
## S3 method for class 'gdm'
IRT.posterior(object, ...)

## S3 method for class 'mcdina'
IRT.likelihood(object, ...)
## S3 method for class 'mcdina'
IRT.posterior(object, ...)

## S3 method for class 'reglca'
IRT.likelihood(object, ...)
## S3 method for class 'reglca'
IRT.posterior(object, ...)

## S3 method for class 'slca'
IRT.likelihood(object, ...)
## S3 method for class 'slca'
IRT.posterior(object, ...)

Arguments

object

Object of classes din, gdina, mcdina, gdm, slca, reglca.

...

More arguments to be passed.

Value

For both functions IRT.likelihood and IRT.posterior, it is a matrix with attributes

theta

Uni- or multidimensional skill space (theta grid in item response models).

prob.theta

Probability distribution of theta

skillspace

Design matrix and estimated parameters for skill space distribution (only for IRT.posterior.slca)

G

Number of groups

See Also

GDINA::indlogLik, GDINA::indlogPost

Examples

#############################################################################
# EXAMPLE 1: Extracting likelihood and posterior from a DINA model
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

#*** estimate model
mod1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix, rule="DINA")
#*** extract likelihood
likemod1 <- CDM::IRT.likelihood(mod1)
str(likemod1)
# extract theta
attr(likemod1, "theta" )
#*** extract posterior
pomod1 <- CDM::IRT.posterior( mod1 )
str(pomod1)

S3 Method for Computation of Marginal Posterior Distribution

Description

Computes marginal posterior distributions for fitted models in the CDM package.

Usage

IRT.marginal_posterior(object, dim, remove_zeroprobs=TRUE, ...)

## S3 method for class 'din'
IRT.marginal_posterior(object, dim, remove_zeroprobs=TRUE, ...)
## S3 method for class 'gdina'
IRT.marginal_posterior(object, dim, remove_zeroprobs=TRUE, ...)
## S3 method for class 'mcdina'
IRT.marginal_posterior(object, dim, remove_zeroprobs=TRUE, ...)

Arguments

object

Object of class din, gdina, mcdina

dim

Numeric or character vector indicating dimensions of posterior distribution which should be marginalized

remove_zeroprobs

Logical indicating whether classes with zero probabilities should be removed

...

Further arguments to be passed

Value

List with entries

marg_post

Marginal posterior distribution

map

MAP estimate (individual classification)

theta

Skill classes

See Also

IRT.posterior

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Dataset with three hierarchical skills
#############################################################################

# simulated data with hierarchical skills:
# skill A with 4 levels, skill B with 2 levels and skill C with 3 levels

data(data.cdm10, package="CDM"")
dat <- data.cdm10$data
Q <- data.cdm10$q.matrix
print(Q)

# define hierarchical skill structure
B <- "A1 > A2 > A3
      C1 > C2"
skill_space <- CDM::skillspace.hierarchy(B=B, skill.names=colnames(Q))
zeroprob.skillclasses <- skill_space$zeroprob.skillclasses

# estimate DINA model
mod1 <- CDM::gdina( dat, q.matrix=Q, zeroprob.skillclasses=zeroprob.skillclasses, rule="DINA")
summary(mod1)

# classification for skill A
res <- CDM::IRT.marginal_posterior(object=mod1, dim=c("A1","A2","A3") )
table(res$map)

# classification for skill B
res <- CDM::IRT.marginal_posterior(object=mod1, dim=c("B") )
table(res$map)

# classification for skill C
res <- CDM::IRT.marginal_posterior(object=mod1, dim=c("C1","C2") )
table(res$map)

## End(Not run)

S3 Methods for Assessing Model Fit

Description

This S3 method assesses global (absolute) model fit using the methods described in modelfit.cor.din.

Usage

IRT.modelfit(object, ...)

## S3 method for class 'din'
IRT.modelfit(object, ...)
## S3 method for class 'gdina'
IRT.modelfit(object, ...)

## S3 method for class 'IRT.modelfit.din'
summary(object, ...)
## S3 method for class 'IRT.modelfit.gdina'
summary(object, ...)

Arguments

object

Object of classes din or gdina.

...

More arguments to be passed.

Value

See output of modelfit.cor.din.

See Also

For extracting the individual likelihood or posterior see IRT.likelihood or IRT.posterior.

The model fit of objects of class gdm can be obtained by using the TAM::tam.modelfit.IRT function in the TAM package.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Absolute model fit
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

#*** Model 1: DINA model for DINA simulated data
mod1 <- CDM::din( sim.dina, q.matrix=sim.qmatrix, rule="DINA" )
fmod1 <- CDM::IRT.modelfit( mod1 )
summary(fmod1)
  ##  Test of Global Model Fit
  ##         type value     p
  ##  1   max(X2) 8.728 0.113
  ##  2 abs(fcor) 0.143 0.080
  ##
  ##  Fit Statistics
  ##                    est
  ##  MADcor          0.030
  ##  SRMSR           0.040
  ##  100*MADRESIDCOV 0.671
  ##  MADQ3           0.062
  ##  MADaQ3          0.059

#*** Model 2: GDINA model
mod2 <- CDM::gdina( sim.dina, q.matrix=sim.qmatrix, rule="GDINA" )
fmod2 <- CDM::IRT.modelfit( mod2 )
summary(fmod2)
  ##  Test of Global Model Fit
  ##         type value p
  ##  1   max(X2) 2.397 1
  ##  2 abs(fcor) 0.078 1
  ##
  ##  Fit Statistics
  ##                    est
  ##  MADcor          0.023
  ##  SRMSR           0.030
  ##  100*MADRESIDCOV 0.515
  ##  MADQ3           0.075
  ##  MADaQ3          0.071

## End(Not run)

S3 Method for Extracting a Parameter Table

Description

S3 method which extracts a parameter table.

Usage

IRT.parameterTable(object, ...)

Arguments

object

Object of model classes

...

More arguments to be passed.

Value

A parameter table


Generation of a Replicate Design for IRT.jackknife

Description

This function generates a Jackknife replicate design which is necessary to use the IRT.jackknife function. The function is a wrapper to BIFIE.data.jack in the BIFIEsurvey package.

Usage

IRT.repDesign(data, wgt=NULL, jktype="JK_TIMSS", jkzone=NULL, jkrep=NULL,
   jkfac=NULL, fayfac=1, wgtrep="W_FSTR", ngr=100, Nboot=200, seed=.Random.seed)

Arguments

data

Dataset which must contain weights and item responses

wgt

Vector with sample weights

jktype

Type of jackknife procedure for creating the BIFIE.data object. jktype="JK_TIMSS" refers to TIMSS/PIRLS datasets. The type "JK_GROUP" creates jackknife weights based on a user defined grouping, the type "JK_RANDOM" creates random groups. The number of random groups can be defined in ngr. The argument type="RW_PISA" extracts the replicated design with balanced repeated replicate weights from PISA datasets into objects of class IRT.repDesign. Bootstrap samples can be obtained by type="BOOT".

jkzone

Variable name for jackknife zones. If jktype="JK_TIMSS", then jkzone="JKZONE". However, this default can be overwritten.

jkrep

Variable name containing Jackknife replicates

jkfac

Factor for multiplying jackknife replicate weights. If jktype="JK_TIMSS", then jkfac=2.

fayfac

Fay factor. For Jackknife, the default is 1. For a Bootstrap with RR samples with replacement, the Fay factor is 1/R1/R.

wgtrep

Already available replicate design

ngr

Number of groups

Nboot

Number of bootstrap samples

seed

Random seed

Value

A list with following entries

wgt

Vector with weights

wgtrep

Matrix containing the replicate design

fayfac

Fay factor needed for Jackknife calculations

See Also

See IRT.jackknife for further examples.

See the BIFIE.data.jack function in the BIFIEsurvey package.

Examples

## Not run: 
# load the BIFIEsurvey package
library(BIFIEsurvey)

#############################################################################
# EXAMPLE 1: Design with Jackknife replicate weights in TIMSS
#############################################################################

data(data.timss11.G4.AUT, package="CDM")
dat <- CDM::data.timss11.G4.AUT$data
# generate design
rdes <- CDM::IRT.repDesign( data=dat,  wgt="TOTWGT", jktype="JK_TIMSS",
             jkzone="JKCZONE", jkrep="JKCREP" )
str(rdes)

#############################################################################
# EXAMPLE 2: Bootstrap resampling
#############################################################################

data(sim.qmatrix, package="CDM")
q.matrix <- CDM::sim.qmatrix

# simulate data according to the DINA model
dat <- CDM::sim.din(N=2000, q.matrix=q.matrix )$dat

# bootstrap with 300 random samples
rdes <- CDM::IRT.repDesign( data=dat, jktype="BOOT", Nboot=300 )

## End(Not run)

Root Mean Square Deviation (RMSD) Item Fit Statistic

Description

Computed the item fit statistics root mean square deviation (RMSD), mean absolute deviation (MAD) and mean deviation (MD). See Oliveri and von Davier (2011) for details.

The RMSD statistics was denoted as the RMSEA statistic in older publications, see itemfit.rmsea.

If multiple groups are defined in the model object, a weighted item fit statistic (WRMSD; Yamamoto, Khorramdel, & von Davier, 2013; von Davier, Weeks, Chen, Allen & van der Velden, 2013) is additionally computed.

Usage

IRT.RMSD(object)

## S3 method for class 'IRT.RMSD'
summary(object, file=NULL, digits=3, ...)

## core computation function
IRT_RMSD_calc_rmsd( n.ik, pi.k, probs, eps=1E-30 )

Arguments

object

Object for which the methods IRT.expectedCounts and IRT.irfprob can be applied.

n.ik

Expected counts

pi.k

Probabilities trait distribution

probs

Item response probabilities

eps

Numerical constant avoiding division by zero

digits

Number of digits used for rounding

file

Optional file name for a file in which summary should be sinked.

...

Optional parameters to be passed.

Details

The RMSD and MD statistics are in operational use in PISA studies since PISA 2015. These fit statistics can also be used for investigating uniform and nonuniform differential item functioning.

Value

List with entries

RMSD

Item-wise and group-wise RMSD statistic

RMSD_bc

Item-wise and group-wise RMSD statistic with analytical bias correction

MAD

Item-wise and group-wise MAD statistic

MD

Item-wise and group-wise MD statistic

chisquare_stat

Item-wise and group-wise χ2\chi^2 statistic

...

Further values

References

Oliveri, M. E., & von Davier, M. (2011). Investigation of model fit and score scale comparability in international assessments. Psychological Test and Assessment Modeling, 53, 315-333.

von Davier, M., Weeks, J., Chen, H., Allen, J., & van der Velden, R. (2013). Creating simple and complex derived variables and validation of background questionnaire data. In OECD (Eds.). Technical Report of the Survey of Adults Skills (PIAAC) (Ch. 20). Paris: OECD.

Yamamoto, K., Khorramdel, L., & von Davier, M. (2013). Scaling PIAAC cognitive data. In OECD (Eds.). Technical Report of the Survey of Adults Skills (PIAAC) (Ch. 17). Paris: OECD.

See Also

itemfit.rmsea

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: data.read | 1PL model in TAM
#############################################################################

data(data.read, package="sirt")
dat <- data.read

#*** Model 1: 1PL model
mod1 <- TAM::tam.mml( resp=dat )
summary(mod1)

# item fit statistics
imod1 <- CDM::IRT.RMSD(mod1)
summary(imod1)

#############################################################################
# EXAMPLE 2: data.math| RMSD and MD statistic for assessing DIF
#############################################################################

data(data.math, package="sirt")
dat <- data.math$data
items <- grep("M[A-Z]", colnames(dat), value=TRUE )

#-- fit multiple group Rasch model
mod <- TAM::tam.mml( dat[,items], group=dat$female )
summary(mod)

#-- fit statistics
rmod <- CDM::IRT.RMSD(mod)
summary(rmod)

#############################################################################
# EXAMPLE 3: RMSD statistic DINA model
#############################################################################

data(sim.dina)
data(sim.qmatrix)
dat <- sim.dina
Q <- sim.qmatrix

#-- fit DINA model
mod1 <- CDM::gdina( dat, q.matrix=Q, rule="DINA" )
summary(mod1)

#-- compute RMSD fit statistic
rmod1 <- CDM::IRT.RMSD(mod1)
summary(rmod1)

## End(Not run)

Create Dataset with Group-Specific Items

Description

Creates a dataset with group-specific items which can be used for multiple group comparisons.

Usage

item_by_group(dat, group, invariant=NULL, rm.empty=TRUE)

Arguments

dat

Dataset with item responses

group

Vector of group identifiers

invariant

Optional vector of variables which should not be made group-specific, i.e. which should be treated as invariant across groups.

rm.empty

Logical indicating whether empty columns should be removed

Value

Extended dataset with item responses

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Create dataset with group-specific item responses
#############################################################################

data(data.mg, package="CDM")
dat <- data.mg

#-- create dataset with group-specific item responses
dat0 <- CDM::item_by_group( dat=dat[,paste0("I",1:5)], group=dat$group )

#-- summary statistics
summary(dat0)
colnames(dat0)

#-- set some items to invariant
invariant_items <- c("I1","I4")
dat1 <- CDM::item_by_group( dat=dat[,paste0("I",1:5)], group=dat$group,
            invariant=invariant_items)
colnames(dat1)

## End(Not run)

RMSEA Item Fit

Description

This function estimates a chi squared based measure of item fit in cognitive diagnosis models similar to the RMSEA itemfit implemented in mdltm (von Davier, 2005; cited in Kunina-Habenicht, Rupp & Wilhelm, 2009).

The RMSEA statistic is also called as the RMSD statistic, see IRT.RMSD.

Usage

itemfit.rmsea(n.ik, pi.k, probs, itemnames=NULL)

Arguments

n.ik

An array of four dimensions: Classes x items x categories x groups

pi.k

An array of two dimensions: Classes x groups

probs

An array of three dimensions: Classes x items x categories

itemnames

An optional vector of item names. Default is NULL.

Details

For item jj, the RMSEA itemfit in this function is calculated as follows:

RMSEAj=kcπ(θc)(Pj(θc)njkcNjc)2RMSEA_j=\sqrt{ \sum_k \sum_c \pi ( \bold{\theta}_c) \left( P_j ( \bold{\theta}_c ) - \frac{n_{jkc}}{N_{jc}} \right)^2 }

where cc denotes the class of the skill vector θ\bold{\theta}, kk is the item category, π(θc)\pi ( \bold{\theta}_c) is the estimated class probability of θc\bold{\theta}_c, PjP_j is the estimated item response function, njkcn_{jkc} is the expected number of students with skill θc\bold{\theta}_c on item jj in category kk and NjcN_{jc} is the expected number of students with skill θc\bold{\theta}_c on item jj.

Value

A list with two entries:

rmsea

Vector of RMSEA item statistics

rmsea.groups

Matrix of group-wise RMSEA item statistics

References

Kunina-Habenicht, O., Rupp, A. A., & Wilhelm, O. (2009). A practical illustration of multidimensional diagnostic skills profiling: Comparing results from confirmatory factor analysis and diagnostic classification models. Studies in Educational Evaluation, 35, 64–70.

von Davier, M. (2005). A general diagnostic model applied to language testing data. ETS Research Report RR-05-16. ETS, Princeton, NJ: ETS.

See Also

This function is used in din, gdina and gdm.


S-X2 Item Fit Statistic for Dichotomous Data

Description

Computes the S-X2 item fit statistic (Orlando & Thissen; 2000, 2003) for dichotomous data. Note that completely observed data is necessary for applying this function.

Usage

itemfit.sx2(object, Eik_min=1, progress=TRUE)

## S3 method for class 'itemfit.sx2'
summary(object, ...)

## S3 method for class 'itemfit.sx2'
plot(x, ask=TRUE, ...)

Arguments

object

Object of class din, gdina, gdm, sirt::rasch.mml, sirt::smirt or TAM::tam.mml

x

Object of class din, gdina, gdm, sirt::rasch.mml, sirt::smirt or TAM::tam.mml

Eik_min

The minimum expected cell size for merging score groups.

progress

An optional logical indicating whether progress should be displayed.

ask

An optional logical indicating whether every item should be separately displayed.

...

Further arguments to be passed

Details

The S-X2 item fit statistic compares observed and expected proportions OjkO_{jk} and EjkE_{jk} for item jj and each score group kk and forms a chi-square distributed statistic

SXj2=k=1J1Nk(OjkEjk)2Ejk(1Ejk)S-X_j^2=\sum_{k=1}^{J-1} N_k \frac{ ( O_{jk} - E_{jk} )^2 } { E_{jk} ( 1 - E_{jk} ) }

The degrees of freedom are J1PjJ-1-P_j where PjP_j