| Title: | Model Based Treatment of Missing Data |
|---|---|
| Description: | Contains model-based treatment of missing data for regression models with missing values in covariates or the dependent variable using maximum likelihood or Bayesian estimation (Ibrahim et al., 2005; <doi:10.1198/016214504000001844>; Luedtke, Robitzsch, & West, 2020a, 2020b; <doi:10.1080/00273171.2019.1640104><doi:10.1037/met0000233>). The regression model can be nonlinear (e.g., interaction effects, quadratic effects or B-spline functions). Multilevel models with missing data in predictors are available for Bayesian estimation. Substantive-model compatible multiple imputation can be also conducted. |
| Authors: | Alexander Robitzsch [aut, cre], Oliver Luedtke [aut] |
| Maintainer: | Alexander Robitzsch <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.10-1 |
| Built: | 2024-07-16 07:27:45 UTC |
| Source: | https://github.com/alexanderrobitzsch/mdmb |
Contains model-based treatment of missing data for regression models with missing values in covariates or the dependent variable using maximum likelihood or Bayesian estimation (Ibrahim et al., 2005; <doi:10.1198/016214504000001844>; Luedtke, Robitzsch, & West, 2020a, 2020b; <doi:10.1080/00273171.2019.1640104><doi:10.1037/met0000233>). The regression model can be nonlinear (e.g., interaction effects, quadratic effects or B-spline functions). Multilevel models with missing data in predictors are available for Bayesian estimation. Substantive-model compatible multiple imputation can be also conducted.
The maximum likelihood estimation of regression models with
missing values in covariates is implemented in frm_em.
Available regression models are linear regression, logistic regression,
ordinal probit regression and models with Box-Cox or Yeo-Johnson transformed
normally distributed outcomes. The factorization based regression model
also allow the inclusion of latent variables and measurement error prone
covariates.
Bayesian estimation and multiple imputation of regression models with
missing values in covariates is implemented in frm_fb. The same
regression models like in frm_em can be specified. Moreover,
multilevel models can also be specified with Bayesian estimation.
The function frm_fb allows substantive model compatible multiple imputation.
Alexander Robitzsch [aut, cre], Oliver Luedtke [aut]
Maintainer: Alexander Robitzsch <[email protected]>
Ibrahim, J. G., Chen, M. H., Lipsitz, S. R., & Herring, A. H. (2005). Missing-data methods for generalized linear models: A comparative review. Journal of the American Statistical Association, 100, 332-346.
Luedtke, O., Robitzsch, A., & West, S. (2020a). Analysis of interactions and nonlinear effects with missing data: A factored regression modeling approach using maximum likelihood estimation. Multivariate Behavioral Research, 55(3), 361-381. doi:10.1080/00273171.2019.1640104
Luedtke, O., Robitzsch, A., & West, S. (2020b). Regression models involving nonlinear effects with missing data: A sequential modeling approach using Bayesian estimation. Psychological Methods, 25(2), 157-181. doi:10.1037/met0000233
The EM algorithm for the multivariate normal model is implemented in
norm2::emNorm in the norm2 package.
A corresponding MCMC algorithm can be
found in the norm2::mcmcNorm function.
See the lavaan, OpenMx or sem packages for full information maximum likelihood approaches for handling missing data for multivariate normal distributions, linear regression models, and, more generally, structural equation modeling with missing data.
Structural equation models with missing data can be also estimated with a
two-stage procedure. In a first stage, a mean vector and a covariance matrix
is estimated (possibly with auxiliary variables) and in the second stage,
the structural equation model is estimated on the previously obtained
mean vector and covariance matrix. The procedure is implemented in the
semTools::twostage function in the semTools package.
## ## |\ /||~~\ |\ /||~~\ ## | \/ || || \/ ||--< ## | ||__/ | ||__/ ## ## ## > library(mdmb) ## * mdmb 0.0-13 (2017-01-15) #### ## |\ /||~~\ |\ /||~~\ ## | \/ || || \/ ||--< ## | ||__/ | ||__/ ## ## ## > library(mdmb) ## * mdmb 0.0-13 (2017-01-15) ##
Example datasets for mdmb package.
data(data.mb01) data(data.mb02) data(data.mb03) data(data.mb04) data(data.mb05)data(data.mb01) data(data.mb02) data(data.mb03) data(data.mb04) data(data.mb05)
Dataset data.mb01. Simulated dataset with missing values.
Variables Y, X and Z are continuous.
List of 2 $ complete:'data.frame': 4000 obs. of 3 variables: ..$ X: num [1:4000] -1.08 0.57 -0.32 0.34 1.21 -0.44 -1.07 -0.29 0.76 -1.75 ... ..$ Z: num [1:4000] -0.02 0.26 -1.45 1.24 0.98 -2.36 0.84 -1.08 -0.15 -1.36 ... ..$ Y: num [1:4000] 0.88 1.75 -0.82 -1.81 -1.58 -3.34 -3.35 -0.29 1.47 0.23 ... $ missing :'data.frame': 4000 obs. of 3 variables: ..$ X: num [1:4000] -1.08 0.57 NA NA 1.21 NA -1.07 -0.29 0.76 NA ... ..$ Z: num [1:4000] -0.02 0.26 -1.45 1.24 0.98 -2.36 0.84 -1.08 -0.15 -1.36 ... ..$ Y: num [1:4000] 0.88 1.75 -0.82 -1.81 -1.58 -3.34 -3.35 -0.29 1.47 0.23 ...
Dataset data.mb02. Simulated dataset with missing values. The
variables Z and Y are dichotomous.
List of 2 $ complete:'data.frame': 2000 obs. of 3 variables: ..$ X: num [1:2000] -0.93 0.3 -0.93 0.7 0.52 -1.38 -0.14 0.09 0.23 -1.64 ... ..$ Z: num [1:2000] 1 0 1 0 0 1 1 1 1 1 ... ..$ Y: num [1:2000] 1 1 0 1 1 0 1 0 0 1 ... $ missing :'data.frame': 2000 obs. of 3 variables: ..$ X: num [1:2000] -0.93 0.3 -0.93 0.7 0.52 NA -0.14 0.09 0.23 -1.64 ... ..$ Z: num [1:2000] 1 0 1 NA NA 1 NA 1 1 1 ... ..$ Y: num [1:2000] 1 1 0 1 1 0 1 0 0 1 ...
Dataset data.mb03. This dataset is from Enders, Baraldi & Cham (2014)
and contains three
variables primary school reading (x), primary school learning problems (z)
and middle school reading (y) which all have missing values.
'data.frame': 74 obs. of 3 variables: $ x: num NA NA NA NA NA 8.34 NA 8.36 6.89 8.56 ... $ z: num 8.81 4.5 6.31 4.7 5.1 4 6.11 3.7 6.81 6.31 ... $ y: num 5 5.1 6.3 9 9 9.3 NA 10.7 6.2 NA ...
Dataset data.mb04. This multilevel dataset contains three variables:
level-1 variables y, x and the level-2 variable w.
'data.frame': 500 obs. of 4 variables: $ idcluster: int 1 1 1 1 1 2 2 2 2 2 ... $ x : num NA NA -1.15 -1.65 0.25 ... $ w : num -0.552 -0.552 -0.552 -0.552 -0.552 ... $ y : num NA NA -0.0711 0.7165 -0.1917 ...
Dataset data.mb05. This dataset contains selected (and transformed)
variables of the German PISA 2012 data.
'data.frame': 5001 obs. of 13 variables: $ idschool : num 1001 1001 1001 1001 1001 ... $ idstud : num 1e+05 1e+05 1e+05 1e+05 1e+05 ... $ female : num 1 1 0 0 0 1 1 1 0 0 ... $ books : num NA 3 3 1 NA 2 NA 1 NA 2 ... $ hisced : num NA 6 6 2 NA 2 NA 2 NA 2 ... $ hisei : num NA 30.6 57.7 26.9 NA ... $ hisei10 : num NA 0.257 0.596 0.211 NA ... $ native : num NA NA 1 0 NA 0 NA 1 NA 1 ... $ ANCINTMAT: num NA 0.644 -0.096 1.057 NA ... $ MATHEFF : num NA 0.34 0.54 -0.18 NA 0.15 NA NA NA NA ... $ READ : num -0.25 -0.503 0.421 -1.664 -0.894 ... $ MATH : num -0.565 -0.854 0.384 -0.896 -0.534 ... $ W_FSTUWT : num 140 140 140 140 140 ...
Enders, C. K., Baraldi, A. N., & Cham, H. (2014). Estimating interaction effects with incomplete predictor variables. Psychological Methods, 19(1), 39-55. doi:10.1037/a0035314
## Not run: ############################################################################# # EXAMPLE 1: Linear interaction example from Enders et al. (2014) ############################################################################# # load packages library(mdmb) library(mice) library(mitools) library(sandwich) #--- attach example dataset (Enders et al., 2014) from mdmb package data( data.mb03, package="mdmb") dat <- data.mb03 #--- center data which speeds convergence of Bayesian estimation #--- of the imputation model for (vv in 1:3){ M_vv <- mean( dat[,vv], na.rm=TRUE ) dat[,vv] <- dat[,vv] - M_vv } #--- generate initial imputed values withj mice package imp <- mice::mice( dat, m=, maxit=20 ) data_init <- mice::complete(imp, action=1) #--- define number of iterations and number of imputed datasets iter <- 50000; burnin <- 5000 Nimp <- 100 #******* imputation model M3 with quadratic effects # model for dependent variable dep <- list("model"="linreg", "formula"=y ~ x*z + I(x^2) + I(z^2) ) # covariate models ind_x <- list( "model"="linreg", "formula"=x ~ z + I(z^2) ) ind_z <- list( "model"="linreg", "formula"=z ~ 1 ) ind <- list( x=ind_x, z=ind_z) #generate imputations imp <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, burnin=burnin, iter=iter, data_init=data_init, Nimp=Nimp) #--- create list of multiply imputed datasets datlist <- mdmb::frm2datlist(imp) #------------------------------- #--- analyze imputed datasets with mice package # convert into object of class mids imp2 <- miceadds::datlist2mids(datlist) # estimate linear model on multiply imputed datasets mod1 <- with(imp2, stats::lm( y ~ x*z ) ) summary( mice::pool(mod1) ) #------------------------------- #--- analyze imputed datasets using sandwich standard errors results <- list() variances <- list() Nimp <- length(datlist) for (ii in 1:Nimp){ mod_ii <- stats::lm( y ~ x*z, data=datlist[[ii]] ) variances[[ii]] <- sandwich::vcovHC(mod_ii) results[[ii]] <- coef(mod_ii) } mod2 <- mitools::MIcombine(results=results,variances=variances,df.complete=69) summary(mod2) ## End(Not run)## Not run: ############################################################################# # EXAMPLE 1: Linear interaction example from Enders et al. (2014) ############################################################################# # load packages library(mdmb) library(mice) library(mitools) library(sandwich) #--- attach example dataset (Enders et al., 2014) from mdmb package data( data.mb03, package="mdmb") dat <- data.mb03 #--- center data which speeds convergence of Bayesian estimation #--- of the imputation model for (vv in 1:3){ M_vv <- mean( dat[,vv], na.rm=TRUE ) dat[,vv] <- dat[,vv] - M_vv } #--- generate initial imputed values withj mice package imp <- mice::mice( dat, m=, maxit=20 ) data_init <- mice::complete(imp, action=1) #--- define number of iterations and number of imputed datasets iter <- 50000; burnin <- 5000 Nimp <- 100 #******* imputation model M3 with quadratic effects # model for dependent variable dep <- list("model"="linreg", "formula"=y ~ x*z + I(x^2) + I(z^2) ) # covariate models ind_x <- list( "model"="linreg", "formula"=x ~ z + I(z^2) ) ind_z <- list( "model"="linreg", "formula"=z ~ 1 ) ind <- list( x=ind_x, z=ind_z) #generate imputations imp <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, burnin=burnin, iter=iter, data_init=data_init, Nimp=Nimp) #--- create list of multiply imputed datasets datlist <- mdmb::frm2datlist(imp) #------------------------------- #--- analyze imputed datasets with mice package # convert into object of class mids imp2 <- miceadds::datlist2mids(datlist) # estimate linear model on multiply imputed datasets mod1 <- with(imp2, stats::lm( y ~ x*z ) ) summary( mice::pool(mod1) ) #------------------------------- #--- analyze imputed datasets using sandwich standard errors results <- list() variances <- list() Nimp <- length(datlist) for (ii in 1:Nimp){ mod_ii <- stats::lm( y ~ x*z, data=datlist[[ii]] ) variances[[ii]] <- sandwich::vcovHC(mod_ii) results[[ii]] <- coef(mod_ii) } mod2 <- mitools::MIcombine(results=results,variances=variances,df.complete=69) summary(mod2) ## End(Not run)
The function eval_prior_list evaluates several prior distributions
specified in a list. The function eval_prior_list_sumlog
computes the sum of the logarithms of all prior values.
eval_prior_list(par, par_prior, log=FALSE, eps=1e-50) eval_prior_list_sumlog(par, par_prior, use_grad=1)eval_prior_list(par, par_prior, log=FALSE, eps=1e-50) eval_prior_list_sumlog(par, par_prior, use_grad=1)
par |
Parameter vector |
par_prior |
List of prior distributions specified in a list (see Examples) |
log |
Logical indicating whether the logarithm of the prior should be computed |
eps |
Decimal to be added to the prior to avoid computation of the logarithm for values of zero |
use_grad |
Integer value for computation value for gradient |
Vector or a numeric value
############################################################################# # EXAMPLE 1: Evaluation of prior values ############################################################################# # normal distribution b0 <- list( "dnorm", list(mean=0,sd=100) ) # t distribution with one degree of freedom (Cauchy distribution) b1 <- list( "dt", list(df=1) ) # define list of priors beta_prior <- list( b0, b1 ) # parameter value beta <- c( 0.3, 1 ) #-- evaluate priors mdmb::eval_prior_list(par=beta, par_prior=beta_prior) mdmb::eval_prior_list_sumlog(par=beta, par_prior=beta_prior)############################################################################# # EXAMPLE 1: Evaluation of prior values ############################################################################# # normal distribution b0 <- list( "dnorm", list(mean=0,sd=100) ) # t distribution with one degree of freedom (Cauchy distribution) b1 <- list( "dt", list(df=1) ) # define list of priors beta_prior <- list( b0, b1 ) # parameter value beta <- c( 0.3, 1 ) #-- evaluate priors mdmb::eval_prior_list(par=beta, par_prior=beta_prior) mdmb::eval_prior_list_sumlog(par=beta, par_prior=beta_prior)
The factored regression model (FRM) allows the estimation of the linear regression model (with normally distributed residuals) and the generalized logistic regression model (logistic regression for dichotomous outcomes). Missing values in covariates are handled by posing a conditional univariate distribution for each covariate. The approach follows Ibrahim (1990), Ibrahim, Chen, Lipsitz and Herring (2005), Lee and Mitra (2016), and Zhang and Wang (2017) and is applied in Luedtke, Robitzsch, and West (2020a, 2020b). Latent variables and covariates with measurement error or multiple indicators can also be handled within this framework (see Examples 3, 4 and 5).
Missing values are handled by numerical integration in frm_em
(see also Allison, 2012). The user has to specify an integration grid for each
variable (defined in argument nodes for each model).
Standard error estimates in frm_em are obtained by a numerical differentiation of
the Fisher score function (see Jamshidian & Jennrich, 2000).
The function frm_fb employs a fully Bayesian approach with noninformative
prior distribution. This function imputes missing values in the models from
the posterior distributions. Imputed datasets can be extracted by the
function frm2datlist.
The current functionality only support missing values on continuous covariates (accommodating skewness and only positive values), dichotomous covariates and ordinal covariates.
Multilevel models (using model="mlreg") for normally distributed
(outcome="normal") and ordinal variables (outcome="probit")
as well as variables at higher levels (using argument variable_level)
are accommodated. Note that multilevel models can only be fit with frm_fb
and not with frm_em.
The handling of nominal covariates will be included in future mdmb package versions.
# Factored regression model: Numerical integration with EM algorithm frm_em(dat, dep, ind, weights=NULL, verbose=TRUE, maxiter=500, conv_dev=1e-08, conv_parm=1e-05, nodes_control=c(11,5), h=1e-04, use_grad=2, update_model=NULL) ## S3 method for class 'frm_em' coef(object, ...) ## S3 method for class 'frm_em' logLik(object, ...) ## S3 method for class 'frm_em' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'frm_em' vcov(object, ...) # Factored regression model: Fully Bayesian estimation frm_fb(dat, dep, ind, weights=NULL, verbose=TRUE, data_init=NULL, iter=500, burnin=100, Nimp=10, Nsave=3000, refresh=25, acc_bounds=c(.45,.50), print_iter=10, use_gibbs=TRUE, aggregation=TRUE) ## S3 method for class 'frm_fb' coef(object, ...) ## S3 method for class 'frm_fb' plot(x, idparm=NULL, ask=TRUE, ... ) ## S3 method for class 'frm_fb' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'frm_fb' vcov(object, ...) frm2datlist(object, as_mids=FALSE) # create list of imputed datasets# Factored regression model: Numerical integration with EM algorithm frm_em(dat, dep, ind, weights=NULL, verbose=TRUE, maxiter=500, conv_dev=1e-08, conv_parm=1e-05, nodes_control=c(11,5), h=1e-04, use_grad=2, update_model=NULL) ## S3 method for class 'frm_em' coef(object, ...) ## S3 method for class 'frm_em' logLik(object, ...) ## S3 method for class 'frm_em' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'frm_em' vcov(object, ...) # Factored regression model: Fully Bayesian estimation frm_fb(dat, dep, ind, weights=NULL, verbose=TRUE, data_init=NULL, iter=500, burnin=100, Nimp=10, Nsave=3000, refresh=25, acc_bounds=c(.45,.50), print_iter=10, use_gibbs=TRUE, aggregation=TRUE) ## S3 method for class 'frm_fb' coef(object, ...) ## S3 method for class 'frm_fb' plot(x, idparm=NULL, ask=TRUE, ... ) ## S3 method for class 'frm_fb' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'frm_fb' vcov(object, ...) frm2datlist(object, as_mids=FALSE) # create list of imputed datasets
dat |
Data frame |
dep |
List containing model specification for dependent variable. The list has arguments (see Examples)
|
ind |
List containing a list of univariate conditional models for covariates.
See |
weights |
Optional vector of sampling weights |
verbose |
Logical indicating whether convergence progress should be displayed. |
maxiter |
Maximum number of iterations |
conv_dev |
Convergence criterion for deviance |
conv_parm |
Convergence criterion for regression coefficients |
nodes_control |
Control arguments if nodes are not provided by the user. The first value denote the number of nodes, while the second value denotes the spread of the node distribution defined as the factor of the standard deviation of the observed data. |
h |
Step width for numerical differentiation for calculating the covariance matrix |
use_grad |
Computation method for gradient in |
update_model |
Optional proviously fitted model can be used as an input |
data_init |
Initial values for dataset |
iter |
Number of iterations |
burnin |
Number of burnin iterations |
Nimp |
Number of imputed datasets |
Nsave |
(Maximum) Number of values to be saved for MCMC chain |
refresh |
Number of imputations after which proposal distribution should be updated in Metropolis-Hastings step |
acc_bounds |
Bounds for acceptance rates for parameters |
print_iter |
Number of imputation after which iteration progress should be displayed |
use_gibbs |
Logical indicating whether Gibbs sampling instead of
Metropolis-Hastings sampling should be used. Can be only
applied for |
aggregation |
Logical indicating whether complete dataset should be
used for computing the predictive distribution of missing values.
|
object |
Object of corresponding class |
x |
Object of corresponding class |
digits |
Number of digits in |
file |
File to which the |
idparm |
Indices for parameters to be plotted |
ask |
Logical indicating whether the user is asked before new plot |
as_mids |
Logical indicating whether multiply imputed datasets
should be converted into objects of class |
... |
Further arguments to be passed |
The function allows for fitting a factored regression model. Consider the case
of three variables , and . A factored regression model
consists of a sequence of univariate conditional models ,
and such that the joint distribution can be factorized as
Each of the three variables can contain missing values. Missing values are integrated out by posing a distributional assumption for each variable with missing values.
For frm_em it is a list containing the following values
coefs |
Estimated coefficients |
vcov |
Covariance matrix |
partable |
Summary parameter table |
all_coefs |
List with all estimated coefficients |
ll |
Log likelihood value |
like |
Individual likelihood |
dat |
Data frame with included latent values for each variable with missing values |
se |
Standard errors for coefficients |
info |
Information matrix |
conv |
Convergence indicator |
iter |
Number of iterations |
ic |
Information criteria |
ind0 |
List with model specifications including |
predictorMatrix |
Predictor matrix |
variablesMatrix |
Matrix containing all variables appearing in statistical models |
desc_vars |
Descriptive statistics of variables |
model_results |
Results from fitted models |
The output for frm_fb contains particular additional values
tech_summary |
Summary table with informations about MCMC algorithm |
values_coda |
Sampled parameter values saved as class |
parms_mcmc |
Object containing informations of sampled parameters |
imputations_mcmc |
Object containing informations of imputed datasets |
The coef and vcov methods can be used to extract coefficients and
the corresponding covariance matrix, respectively. Standard errors for a fitted
object mod can be extracted by making use of the survey
package and the statement survey::SE(mod).
Alexander Robitzsch
Allison, P. D. (2012). Handling missing data by maximum likelihood. SAS Global Forum 2012.
Bartlett, J. W., & Morris, T. P. (2015) Multiple imputation of covariates by substantive-model compatible fully conditional specification. Stata Journal, 15(2), 437-456.
Bartlett, J. W., Seaman, S. R., White, I. R., Carpenter, J. R., & Alzheimer's Disease Neuroimaging Initiative (2015). Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model. Statistical Methods in Medical Research, 24(4), 462-487. doi:10.1177/0962280214521348
Erler, N. S., Rizopoulos, D., Rosmalen, J. V., Jaddoe, V. W., Franco, O. H., & Lesaffre, E. M. (2016). Dealing with missing covariates in epidemiologic studies: A comparison between multiple imputation and a full Bayesian approach. Statistics in Medicine, 35(17), 2955-2974. doi:10.1002/sim.6944
Ibrahim, J. G. (1990). Incomplete data in generalized linear models. Journal of the American Statistical Association, 85(411), 765-769. doi:10.1080/01621459.1990.10474938
Ibrahim, J. G., Chen, M. H., Lipsitz, S. R., & Herring, A. H. (2005). Missing-data methods for generalized linear models: A comparative review. Journal of the American Statistical Association, 100(469), 332-346. doi:10.1198/016214504000001844
Jamshidian, M., & Jennrich, R. I. (2000). Standard errors for EM estimation. Journal of the Royal Statistical Society (Series B), 62(2), 257-270. doi:10.1111/1467-9868.00230
Keller, B. T., & Enders, C. K. (2018). Blimp user's manual. Los Angeles, CA.
http://www.appliedmissingdata.com/multilevel-imputation.html
Lee, M. C., & Mitra, R. (2016). Multiply imputing missing values in data sets with mixed measurement scales using a sequence of generalised linear models. Computational Statistics & Data Analysis, 95(24), 24-38. doi:10.1016/j.csda.2015.08.004
Luedtke, O., Robitzsch, A., & West, S. (2020a). Analysis of interactions and nonlinear effects with missing data: A factored regression modeling approach using maximum likelihood estimation. Multivariate Behavioral Research, 55(3), 361-381. doi:10.1080/00273171.2019.1640104
Luedtke, O., Robitzsch, A., & West, S. (2020b). Regression models involving nonlinear effects with missing data: A sequential modeling approach using Bayesian estimation. Psychological Methods, 25(2), 157-181. doi:10.1037/met0000233
Zhang, Q., & Wang, L. (2017). Moderation analysis with missing data in the predictors. Psychological Methods, 22(4), 649-666. doi:10.1037/met0000104
See also the icdGLM package for estimation of generalized linear models with incomplete discrete covariates.
The imputation of covariates in substantive models with interactions or nonlinear terms can be also conducted with the JointAI package which is a wrapper to the JAGS software (see Erler et al., 2016). This package is also based on a sequential modelling approach.
The jomo package also accommodates substantive models (jomo::jomo.lm)
based on a joint modeling framework.
Substantive model compatible imputation based on fully conditional specification can be found in the smcfcs package (see Bartlett et al., 2015; Bartlett & Morris, 2015) or the Blimp stand-alone software (Keller & Enders, 2018).
## Not run: ############################################################################# # EXAMPLE 1: Simulated example linear regression with interaction effects ############################################################################# # The interaction model stats::lm( Y ~ X + Z + X:Z) is of substantive interest. # There can be missing values in all variables. data(data.mb01) dat <- data.mb01$missing #****************************************** # Model 1: ML approach #--- specify models # define integration nodes xnodes <- seq(-4,4,len=11) # nodes for X ynodes <- seq(-10,10,len=13) # nodes for Y. These ynodes are not really necessary for this dataset because # Y has no missing values. # define model for dependent variable Y dep <- list("model"="linreg", "formula"=Y ~ X*Z, "nodes"=ynodes ) # model P(X|Z) ind_X <- list( "model"="linreg", "formula"=X ~ Z, "nodes"=xnodes ) # all models for covariates ind <- list( "X"=ind_X ) #--- estimate model with numerical integration mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind ) summary(mod1) # extract some informations coef(mod1) vcov(mod1) logLik(mod1) #****************************************** # Model 2: Fully Bayesian approach / Multiple Imputation #--- define models dep <- list("model"="linreg", "formula"=Y ~ X*Z ) ind_X <- list( "model"="linreg", "formula"=X ~ Z ) ind_Z <- list( "model"="linreg", "formula"=Z ~ 1 ) ind <- list( "X"=ind_X, Z=ind_Z) #--- estimate model mod2 <- mdmb::frm_fb(dat, dep, ind, burnin=200, iter=1000) summary(mod2) #* plot parameters plot(mod2) #--- create list of multiply imputed datasets datlist <- mdmb::frm2datlist(mod2) # convert into object of class mids imp2 <- miceadds::datlist2mids(datlist) # estimate linear model on multiply imputed datasets mod2c <- with(imp2, stats::lm( Y ~ X*Z ) ) summary( mice::pool(mod2c) ) #****************************************** # Model 3: Multiple imputation in jomo package library(jomo) # impute with substantive model containing interaction effects formula <- Y ~ X*Z imp <- jomo::jomo.lm( formula=formula, data=dat, nburn=100, nbetween=100) # convert to object of class mids datlist <- miceadds::jomo2mids( imp ) # estimate linear model mod3 <- with(datlist, lm( Y ~ X*Z ) ) summary( mice::pool(mod3) ) ############################################################################# # EXAMPLE 2: Simulated example logistic regression with interaction effects ############################################################################# # Interaction model within a logistic regression Y ~ X + Z + X:Z # Y and Z are dichotomous variables. # attach data data(data.mb02) dat <- data.mb02$missing #****************************************** # Model 1: ML approach #--- specify model # define nodes xnodes <- seq(-5,5,len=15) # X - normally distributed variable ynodes <- c(0,1) # Y and Z dichotomous variable # model P(Y|X,Z) for dependent variable dep <- list("model"="logistic", "formula"=Y ~ X*Z, "nodes"=ynodes ) # model P(X|Z) ind_X <- list( "model"="linreg", "formula"=X ~ Z, "nodes"=xnodes ) # model P(Z) ind_Z <- list( "model"="logistic", "formula"=Z ~ 1, "nodes"=ynodes ) ind <- list( "Z"=ind_Z, "X"=ind_X ) #--- estimate model with numerical integration mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind ) summary(mod1) #****************************************** # Model 2: Fully Bayesian approach #--- specify model dep <- list("model"="logistic", "formula"=Y ~ X*Z ) ind_X <- list( "model"="linreg", "formula"=X ~ Z ) ind_Z <- list( "model"="logistic", "formula"=Z ~ 1 ) ind <- list( "Z"=ind_Z, "X"=ind_X ) #--- Bayesian estimation mod2 <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, burnin=500, iter=1000 ) summary(mod2) ############################################################################# # EXAMPLE 3: Confirmatory factor analysis ############################################################################# # A latent variable can be considered as missing data and the 'frm_em' function # is used to estimate the latent variable model. #--- simulate data N <- 1000 set.seed(91834) # latent variable theta <- stats::rnorm(N) # simulate items y1 <- 1.5 + 1*theta + stats::rnorm(N, sd=.7 ) y2 <- 1.9 + .7*theta + stats::rnorm(N, sd=1 ) y3 <- .9 + .7*theta + stats::rnorm(N, sd=.2 ) dat <- data.frame(y1,y2,y3) dat$theta <- NA #****************************************** # Model 1: ML approach #--- define model nodes <- seq(-4,4,len=21) ind_y1 <- list("model"="linreg", "formula"=y1 ~ offset(1*theta), "nodes"=nodes ) ind_y2 <- list( "model"="linreg", "formula"=y2 ~ theta, "nodes"=nodes, "coef_inits"=c(NA,1) ) ind_y3 <- list( "model"="linreg", "formula"=y3 ~ theta, "nodes"=nodes, "coef_inits"=c(1,1) ) dep <- list( "model"="linreg", "formula"=theta ~ 0, "nodes"=nodes ) ind <- list( "y1"=ind_y1, "y2"=ind_y2, "y3"=ind_y3) #*** estimate model with mdmb::frm_em mod1 <- mdmb::frm_em(dat, dep, ind) summary(mod1) #*** estimate model in lavaan library(lavaan) lavmodel <- " theta=~ 1*y1 + y2 + y3 theta ~~ theta " mod1b <- lavaan::cfa( model=lavmodel, data=dat ) summary(mod1b) # compare likelihood logLik(mod1) logLik(mod1b) ############################################################################# # EXAMPLE 4: Rasch model ############################################################################# #--- simulate data set.seed(91834) N <- 500 # latent variable theta0 <- theta <- stats::rnorm(N) # number of items I <- 7 dat <- sirt::sim.raschtype( theta, b=seq(-1.5,1.5,len=I) ) colnames(dat) <- paste0("I",1:I) dat$theta <- NA #****************************************** # Model 1: ML approach #--- define model nodes <- seq(-4,4,len=13) dep <- list("model"="linreg", "formula"=theta ~ 0, "nodes"=nodes ) ind <- list() for (ii in 1:I){ ind_ii <- list( "model"="logistic", formula= stats::as.formula( paste0("I",ii, " ~ offset(1*theta)") ) ) ind[[ii]] <- ind_ii } names(ind) <- colnames(dat)[-(I+1)] #--- estimate Rasch model with mdmb::frm_em mod1 <- mdmb::frm_em(dat, dep, ind ) summary(mod1) #--- estmate Rasch model with sirt package library(sirt) mod2 <- sirt::rasch.mml2( dat[,-(I+1)], theta.k=nodes, use.freqpatt=FALSE) summary(mod2) #** compare estimated parameters round(cbind(coef(mod1), c( mod2$sd.trait, -mod2$item$thresh[ seq(I,1)] ) ), 3) ############################################################################# # EXAMPLE 5: Regression model with measurement error in covariates ############################################################################# #--- simulate data set.seed(768) N <- 1000 # true score theta <- stats::rnorm(N) # heterogeneous error variance var_err <- stats::runif(N, .5, 1) # simulate observed score x <- theta + stats::rnorm(N, sd=sqrt(var_err) ) # simulate outcome y <- .3 + .7 * theta + stats::rnorm( N, sd=.8 ) dat0 <- dat <- data.frame( y=y, x=x, theta=theta ) #*** estimate model with true scores (which are unobserved in real datasets) mod0 <- stats::lm( y ~ theta, data=dat0 ) summary(mod0) #****************************************** # Model 1: Model-based approach #--- specify model dat$theta <- NA nodes <- seq(-4,4,len=15) dep <- list( "model"="linreg", "formula"=y ~ theta, "nodes"=nodes, "coef_inits"=c(NA, .4 ) ) ind <- list() ind[["theta"]] <- list( "model"="linreg", "formula"=theta ~ 1, "nodes"=nodes ) ind[["x"]] <- list( "model"="linreg", "formula"=x ~ 0 + offset(theta), "nodes"=nodes ) # assumption of heterogeneous known error variance ind[["x"]]$sigma_fixed <- sqrt( var_err ) #--- estimate regression model mod1 <- mdmb::frm_em(dat, dep, ind ) summary(mod1) #****************************************** # Model 2: Fully Bayesian estimation #--- specify model dep <- list( "model"="linreg", "formula"=y ~ theta ) ind <- list() ind[["theta"]] <- list( "model"="linreg", "formula"=theta ~ 1 ) ind[["x"]] <- list( "model"="linreg", "formula"=x ~ 0 + offset(theta) ) # assumption of heterogeneous known error variance ind[["x"]]$sigma_fixed <- sqrt( var_err ) data_init <- dat data_init$theta <- dat$x # estimate model mod2 <- mdmb::frm_fb(dat, dep, ind, burnin=200, iter=1000, data_init=data_init) summary(mod2) plot(mod2) ############################################################################# # EXAMPLE 6: Non-normally distributed covariates: # Positive values with Box-Cox transformation ############################################################################# # simulate data with chi-squared distributed covariate from # regression model Y ~ X set.seed(876) n <- 1500 df <- 2 x <- stats::rchisq( n, df=df ) x <- x / sqrt( 2*df) y <- 0 + 1*x R2 <- .25 # explained variance y <- y + stats::rnorm(n, sd=sqrt( (1-R2)/R2 * 1) ) dat0 <- dat <- data.frame( y=y, x=x ) # simulate missing responses prop_miss <- .5 cor_miss <- .7 resp_tend <- cor_miss*(dat$y-mean(y) )/ stats::sd(y) + stats::rnorm(n, sd=sqrt( 1 - cor_miss^2) ) dat[ resp_tend < stats::qnorm( prop_miss ), "x" ] <- NA summary(dat) #-- complete data mod0 <- stats::lm( y ~ x, data=dat0 ) summary(mod0) #-- listwise deletion mod1 <- stats::lm( y ~ x, data=dat ) summary(mod1) # normal distribution assumption for frm # define models dep <- list("model"="linreg", "formula"=y ~ x ) # normally distributed data ind_x1 <- list( "model"="linreg", "formula"=x ~ 1 ) # Box-Cox normal distribution ind_x2 <- list( "model"="bctreg", "formula"=x ~ 1, nodes=c( seq(0.05, 3, len=31), seq( 3.5, 9, by=.5 ) ) ) ind1 <- list( "x"=ind_x1 ) ind2 <- list( "x"=ind_x2 ) #--- incorrect normal distribution assumption mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind1 ) summary(mod1) #--- model chi-square distribution of predictor with Box-Cox transformation mod2 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind2 ) summary(mod2) ############################################################################# # EXAMPLE 7: Latent interaction model ############################################################################# # A latent interaction model Y ~ FX + FZ is of interest. Y is directly observed, # FX and FZ are both indirectly observed by three items #--- simulate data N <- 1000 set.seed(987) # latent variable FX <- stats::rnorm(N) FZ <- stats::rnorm(N) # simulate items x1 <- 1.5 + 1*FX + stats::rnorm(N, sd=.7 ) x2 <- 1.9 + .7*FX + stats::rnorm(N, sd=1 ) x3 <- .9 + .7*FX + stats::rnorm(N, sd=.2 ) z1 <- 1.5 + 1*FZ + stats::rnorm(N, sd=.7 ) z2 <- 1.9 + .7*FZ + stats::rnorm(N, sd=1 ) z3 <- .9 + .7*FZ + stats::rnorm(N, sd=.2 ) dat <- data.frame(x1,x2,x3,z1,z2,z3) dat$FX <- NA dat$FZ <- NA dat$y <- 2 + .5*FX + .3*FZ + .4*FX*FZ + rnorm( N, sd=1 ) # estimate interaction model with ML #--- define model nodes <- seq(-4,4,len=11) ind_x1 <- list("model"="linreg", "formula"=x1 ~ offset(1*FX), "nodes"=nodes ) ind_x2 <- list( "model"="linreg", "formula"=x2 ~ FX, "nodes"=nodes, "coef_inits"=c(NA,1) ) ind_x3 <- list( "model"="linreg", "formula"=x3 ~ FX, "nodes"=nodes, "coef_inits"=c(1,1) ) ind_FX <- list( "model"="linreg", "formula"=FX ~ 0, "nodes"=nodes ) ind_z1 <- list("model"="linreg", "formula"=z1 ~ offset(1*FZ), "nodes"=nodes ) ind_z2 <- list( "model"="linreg", "formula"=z2 ~ FZ, "nodes"=nodes, "coef_inits"=c(NA,1) ) ind_z3 <- list( "model"="linreg", "formula"=z3 ~ FZ, "nodes"=nodes, "coef_inits"=c(1,1) ) ind_FZ <- list( "model"="linreg", "formula"=FZ ~ 0 + FX, "nodes"=nodes ) ind <- list( "x1"=ind_x1, "x2"=ind_x2, "x3"=ind_x3, "FX"=ind_FX, "z1"=ind_z1, "z2"=ind_z2, "z3"=ind_z3, "FX"=ind_FZ ) dep <- list( "model"="linreg", formula=y ~ FX+FZ+FX*FZ, "coef_inits"=c(1,.2,.2,0) ) #*** estimate model with mdmb::frm_em mod1 <- mdmb::frm_em(dat, dep, ind) summary(mod1) ############################################################################# # EXAMPLE 8: Non-ignorable data in Y ############################################################################# # regression Y ~ X in which Y is missing depending Y itself library(mvtnorm) cor_XY <- .4 # correlation between X and Y prop_miss <- .5 # missing proportion cor_missY <- .7 # correlation with missing propensity N <- 3000 # sample size #----- simulate data set.seed(790) Sigma <- matrix( c(1, cor_XY, cor_XY, 1), 2, 2 ) mu <- c(0,0) dat <- mvtnorm::rmvnorm( N, mean=mu, sigma=Sigma ) colnames(dat) <- c("X","Y") dat <- as.data.frame(dat) #-- generate missing responses on Y depending on Y itself y1 <- dat$Y miss_tend <- cor_missY * y1 + rnorm(N, sd=sqrt( 1 - cor_missY^2) ) dat$Y[ miss_tend < quantile( miss_tend, prop_miss ) ] <- NA #--- ML estimation under assumption of ignorability nodes <- seq(-5,5,len=15) dep <- list("model"="linreg", "formula"=Y ~ X, "nodes"=nodes ) ind_X <- list( "model"="linreg", "formula"=X ~ 1, "nodes"=nodes ) ind <- list( "X"=ind_X ) mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind) summary(mod1) #--- ML estimation under assumption with specifying a model for non-ignorability # for response indicator resp_Y dat$resp_Y <- 1* ( 1 - is.na(dat$Y) ) dep <- list("model"="linreg", "formula"=Y ~ X, "nodes"=nodes ) ind_X <- list( "model"="linreg", "formula"=X ~ 1, "nodes"=nodes ) ind_respY <- list( "model"="logistic", "formula"=resp_Y ~ Y, "nodes"=nodes ) ind <- list( "X"=ind_X, "resp_Y"=ind_respY ) mod2 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind) summary(mod2) ############################################################################# # EXAMPLE 9: Ordinal variables: Graded response model ############################################################################# #--- simulate data N <- 2000 set.seed(91834) # latent variable theta <- stats::rnorm(N) # simulate items y1 <- 1*theta + stats::rnorm(N) y2 <- .7*theta + stats::rnorm(N) y3 <- .7*theta + stats::rnorm(N) # discretize variables y1 <- as.numeric( cut( y1, breaks=c(-Inf, -.5, 0.4, 1, Inf ) ) ) - 1 y2 <- as.numeric( cut( y2, breaks=c(-Inf, 0.3, 1, Inf ) ) ) - 1 y3 <- as.numeric( cut( y3, breaks=c(-Inf, .2, Inf ) ) ) - 1 # define dataset dat <- data.frame(y1,y2,y3) dat$theta <- NA #****************************************** # Model 1: Fully Bayesian estimation #--- define model ind_y1 <- list( "model"="oprobit", "formula"=y1 ~ offset(1*theta) ) ind_y2 <- list( "model"="oprobit", "formula"=y2 ~ theta ) ind_y3 <- list( "model"="oprobit", "formula"=y3 ~ theta ) dep <- list( "model"="linreg", "formula"=theta ~ 0 ) ind <- list( "y1"=ind_y1, "y2"=ind_y2, "y3"=ind_y3) # initial data data_init <- dat data_init$theta <- as.numeric( scale(dat$y1) ) + stats::rnorm(N, sd=.4 ) #-- estimate model iter <- 3000; burnin <- 1000 mod1 <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, data_init=data_init, iter=iter, burnin=burnin) summary(mod1) plot(mod1) ############################################################################# # EXAMPLE 10: Imputation for missig predictors in models with interaction # effects in multilevel regression models ############################################################################# library(miceadds) data(data.mb04, package="mdmb") dat <- data.mb04 #*** model specification mcmc_iter <- 4 # number of MCMC iterations for model parameter sampling model_formula <- y ~ cwc(x, idcluster) + gm(x, idcluster) + w + w*cwc(x, idcluster) + w*gm(x, idcluster) + ( 1 + cwc(x, idcluster) | idcluster) dep <- list("model"="mlreg", "formula"=model_formula, R_args=list(iter=mcmc_iter, outcome="normal") ) ind_x <- list( "model"="mlreg", "formula"=x ~ w + (1|idcluster), R_args=list(iter=mcmc_iter), sampling_level="idcluster" ) # group means of x are involved in the outcome model. Therefore, Metropolis-Hastings # sampling of missing values in x should be conducted at the level of clusters, # i.e. specifying sampling_level ind <- list("x"=ind_x) # --- estimate model mod1 <- mdmb::frm_fb(dat, dep, ind, aggregation=TRUE) # argument aggregation is necessary because group means are involved in regression formulas #------------- #*** imputation of a continuous level-2 variable w # create artificially some missings on w dat[ dat$idcluster %%3==0, "w" ] <- NA # define level-2 model with argument variable_level ind_w <- list( "model"="linreg", "formula"=w ~ 1, "variable_level"="idcluster" ) ind <- list( x=ind_x, w=ind_w) #* conduct imputations mod2 <- mdmb::frm_fb(dat, dep, ind, aggregation=TRUE) summary(mod2) #--- Model 1 with user-defined prior distributions for covariance matrices model_formula <- y ~ cwc(x, idcluster) + gm(x, idcluster) + w + w*cwc(x, idcluster) + w*gm(x, idcluster) + ( 1 + cwc(x, idcluster) | idcluster) # define scale degrees of freedom (nu) and scale matrix (S) for inverse Wishart distribution psi_nu0_list <- list( -3 ) psi_S0_list <- list( diag(0,2) ) dep <- list("model"="mlreg", "formula"=model_formula, R_args=list(iter=mcmc_iter, outcome="normal", psi_nu0_list=psi_nu0_list, psi_S0_list=psi_S0_list ) ) # define nu and S parameters for covariate model psi_nu0_list <- list( .4 ) psi_S0_list <- list( matrix(.2, nrow=1, ncol=1) ) ind_x <- list( "model"="mlreg", "formula"=x ~ w + (1|idcluster), R_args=list(iter=mcmc_iter, psi_nu0_list=psi_nu0_list, psi_S0_list=psi_S0_list), sampling_level="idcluster" ) ind <- list("x"=ind_x) # --- estimate model mod3 <- mdmb::frm_fb(dat, dep, ind, aggregation=TRUE) ############################################################################# # EXAMPLE 11: Bounded variable combined with Yeo-Johnson transformation ############################################################################# #*** simulate data set.seed(876) n <- 1500 x <- mdmb::ryjt_scaled( n, location=-.2, shape=.8, lambda=.9, probit=TRUE) R2 <- .25 # explained variance y <- 1*x + stats::rnorm(n, sd=sqrt( (1-R2)/R2 * stats::var(x)) ) dat0 <- dat <- data.frame( y=y, x=x ) # simulate missing responses prop_miss <- .5 cor_miss <- .7 resp_tend <- cor_miss*(dat$y-mean(y) )/ stats::sd(y) + stats::rnorm(n, sd=sqrt( 1 - cor_miss^2) ) dat[ resp_tend < stats::qnorm(prop_miss), "x" ] <- NA summary(dat) #*** define models dep <- list("model"="linreg", "formula"=y ~ x ) # distribution according to Yeo-Johnson transformation ind_x1 <- list( "model"="yjtreg", "formula"=x ~ 1 ) # distribution according to Probit Yeo-Johnson transformation ind_x2 <- list( "model"="yjtreg", "formula"=x ~ 1, R_args=list("probit"=TRUE ) ) ind1 <- list( "x"=ind_x1 ) ind2 <- list( "x"=ind_x2 ) #--- complete data mod0 <- stats::lm( y~x, data=dat0) summary(mod0) #--- Yeo-Johnson normal distribution (for unbounded variables) mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind1 ) summary(mod1) #--- Probit Yeo-Johnson normal distribution (for bounded variable on (0,1)) mod2 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind2) summary(mod2) #--- same model, but MCMC estimation mod3 <- mdmb::frm_fb(dat, dep, ind=ind2, burnin=2000, iter=5000) summary(mod3) plot(mod3) ############################################################################# # EXAMPLE 12: Yeo-Johnson transformation with estimated degrees of freedom ############################################################################# #*** simulate data set.seed(876) n <- 1500 x <- mdmb::ryjt_scaled( n, location=-.2, shape=.8, lambda=.9, df=10 ) R2 <- .25 # explained variance y <- 1*x + stats::rnorm(n, sd=sqrt( (1-R2)/R2 * stats::var(x)) ) dat0 <- dat <- data.frame( y=y, x=x ) # simulate missing responses prop_miss <- .5 cor_miss <- .7 resp_tend <- cor_miss*(dat$y-mean(y) )/ stats::sd(y) + stats::rnorm(n, sd=sqrt( 1-cor_miss^2) ) dat[ resp_tend < stats::qnorm(prop_miss), "x" ] <- NA summary(dat) #*** define models dep <- list("model"="linreg", "formula"=y ~ x ) # specify distribution with estimated degrees of freedom ind_x <- list( "model"="yjtreg", "formula"=x ~ 1, R_args=list(est_df=TRUE ) ) ind <- list( "x"=ind_x ) #--- Yeo-Johnson t distribution mod1 <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, iter=3000, burnin=1000 ) summary(mod1) ## End(Not run)## Not run: ############################################################################# # EXAMPLE 1: Simulated example linear regression with interaction effects ############################################################################# # The interaction model stats::lm( Y ~ X + Z + X:Z) is of substantive interest. # There can be missing values in all variables. data(data.mb01) dat <- data.mb01$missing #****************************************** # Model 1: ML approach #--- specify models # define integration nodes xnodes <- seq(-4,4,len=11) # nodes for X ynodes <- seq(-10,10,len=13) # nodes for Y. These ynodes are not really necessary for this dataset because # Y has no missing values. # define model for dependent variable Y dep <- list("model"="linreg", "formula"=Y ~ X*Z, "nodes"=ynodes ) # model P(X|Z) ind_X <- list( "model"="linreg", "formula"=X ~ Z, "nodes"=xnodes ) # all models for covariates ind <- list( "X"=ind_X ) #--- estimate model with numerical integration mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind ) summary(mod1) # extract some informations coef(mod1) vcov(mod1) logLik(mod1) #****************************************** # Model 2: Fully Bayesian approach / Multiple Imputation #--- define models dep <- list("model"="linreg", "formula"=Y ~ X*Z ) ind_X <- list( "model"="linreg", "formula"=X ~ Z ) ind_Z <- list( "model"="linreg", "formula"=Z ~ 1 ) ind <- list( "X"=ind_X, Z=ind_Z) #--- estimate model mod2 <- mdmb::frm_fb(dat, dep, ind, burnin=200, iter=1000) summary(mod2) #* plot parameters plot(mod2) #--- create list of multiply imputed datasets datlist <- mdmb::frm2datlist(mod2) # convert into object of class mids imp2 <- miceadds::datlist2mids(datlist) # estimate linear model on multiply imputed datasets mod2c <- with(imp2, stats::lm( Y ~ X*Z ) ) summary( mice::pool(mod2c) ) #****************************************** # Model 3: Multiple imputation in jomo package library(jomo) # impute with substantive model containing interaction effects formula <- Y ~ X*Z imp <- jomo::jomo.lm( formula=formula, data=dat, nburn=100, nbetween=100) # convert to object of class mids datlist <- miceadds::jomo2mids( imp ) # estimate linear model mod3 <- with(datlist, lm( Y ~ X*Z ) ) summary( mice::pool(mod3) ) ############################################################################# # EXAMPLE 2: Simulated example logistic regression with interaction effects ############################################################################# # Interaction model within a logistic regression Y ~ X + Z + X:Z # Y and Z are dichotomous variables. # attach data data(data.mb02) dat <- data.mb02$missing #****************************************** # Model 1: ML approach #--- specify model # define nodes xnodes <- seq(-5,5,len=15) # X - normally distributed variable ynodes <- c(0,1) # Y and Z dichotomous variable # model P(Y|X,Z) for dependent variable dep <- list("model"="logistic", "formula"=Y ~ X*Z, "nodes"=ynodes ) # model P(X|Z) ind_X <- list( "model"="linreg", "formula"=X ~ Z, "nodes"=xnodes ) # model P(Z) ind_Z <- list( "model"="logistic", "formula"=Z ~ 1, "nodes"=ynodes ) ind <- list( "Z"=ind_Z, "X"=ind_X ) #--- estimate model with numerical integration mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind ) summary(mod1) #****************************************** # Model 2: Fully Bayesian approach #--- specify model dep <- list("model"="logistic", "formula"=Y ~ X*Z ) ind_X <- list( "model"="linreg", "formula"=X ~ Z ) ind_Z <- list( "model"="logistic", "formula"=Z ~ 1 ) ind <- list( "Z"=ind_Z, "X"=ind_X ) #--- Bayesian estimation mod2 <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, burnin=500, iter=1000 ) summary(mod2) ############################################################################# # EXAMPLE 3: Confirmatory factor analysis ############################################################################# # A latent variable can be considered as missing data and the 'frm_em' function # is used to estimate the latent variable model. #--- simulate data N <- 1000 set.seed(91834) # latent variable theta <- stats::rnorm(N) # simulate items y1 <- 1.5 + 1*theta + stats::rnorm(N, sd=.7 ) y2 <- 1.9 + .7*theta + stats::rnorm(N, sd=1 ) y3 <- .9 + .7*theta + stats::rnorm(N, sd=.2 ) dat <- data.frame(y1,y2,y3) dat$theta <- NA #****************************************** # Model 1: ML approach #--- define model nodes <- seq(-4,4,len=21) ind_y1 <- list("model"="linreg", "formula"=y1 ~ offset(1*theta), "nodes"=nodes ) ind_y2 <- list( "model"="linreg", "formula"=y2 ~ theta, "nodes"=nodes, "coef_inits"=c(NA,1) ) ind_y3 <- list( "model"="linreg", "formula"=y3 ~ theta, "nodes"=nodes, "coef_inits"=c(1,1) ) dep <- list( "model"="linreg", "formula"=theta ~ 0, "nodes"=nodes ) ind <- list( "y1"=ind_y1, "y2"=ind_y2, "y3"=ind_y3) #*** estimate model with mdmb::frm_em mod1 <- mdmb::frm_em(dat, dep, ind) summary(mod1) #*** estimate model in lavaan library(lavaan) lavmodel <- " theta=~ 1*y1 + y2 + y3 theta ~~ theta " mod1b <- lavaan::cfa( model=lavmodel, data=dat ) summary(mod1b) # compare likelihood logLik(mod1) logLik(mod1b) ############################################################################# # EXAMPLE 4: Rasch model ############################################################################# #--- simulate data set.seed(91834) N <- 500 # latent variable theta0 <- theta <- stats::rnorm(N) # number of items I <- 7 dat <- sirt::sim.raschtype( theta, b=seq(-1.5,1.5,len=I) ) colnames(dat) <- paste0("I",1:I) dat$theta <- NA #****************************************** # Model 1: ML approach #--- define model nodes <- seq(-4,4,len=13) dep <- list("model"="linreg", "formula"=theta ~ 0, "nodes"=nodes ) ind <- list() for (ii in 1:I){ ind_ii <- list( "model"="logistic", formula= stats::as.formula( paste0("I",ii, " ~ offset(1*theta)") ) ) ind[[ii]] <- ind_ii } names(ind) <- colnames(dat)[-(I+1)] #--- estimate Rasch model with mdmb::frm_em mod1 <- mdmb::frm_em(dat, dep, ind ) summary(mod1) #--- estmate Rasch model with sirt package library(sirt) mod2 <- sirt::rasch.mml2( dat[,-(I+1)], theta.k=nodes, use.freqpatt=FALSE) summary(mod2) #** compare estimated parameters round(cbind(coef(mod1), c( mod2$sd.trait, -mod2$item$thresh[ seq(I,1)] ) ), 3) ############################################################################# # EXAMPLE 5: Regression model with measurement error in covariates ############################################################################# #--- simulate data set.seed(768) N <- 1000 # true score theta <- stats::rnorm(N) # heterogeneous error variance var_err <- stats::runif(N, .5, 1) # simulate observed score x <- theta + stats::rnorm(N, sd=sqrt(var_err) ) # simulate outcome y <- .3 + .7 * theta + stats::rnorm( N, sd=.8 ) dat0 <- dat <- data.frame( y=y, x=x, theta=theta ) #*** estimate model with true scores (which are unobserved in real datasets) mod0 <- stats::lm( y ~ theta, data=dat0 ) summary(mod0) #****************************************** # Model 1: Model-based approach #--- specify model dat$theta <- NA nodes <- seq(-4,4,len=15) dep <- list( "model"="linreg", "formula"=y ~ theta, "nodes"=nodes, "coef_inits"=c(NA, .4 ) ) ind <- list() ind[["theta"]] <- list( "model"="linreg", "formula"=theta ~ 1, "nodes"=nodes ) ind[["x"]] <- list( "model"="linreg", "formula"=x ~ 0 + offset(theta), "nodes"=nodes ) # assumption of heterogeneous known error variance ind[["x"]]$sigma_fixed <- sqrt( var_err ) #--- estimate regression model mod1 <- mdmb::frm_em(dat, dep, ind ) summary(mod1) #****************************************** # Model 2: Fully Bayesian estimation #--- specify model dep <- list( "model"="linreg", "formula"=y ~ theta ) ind <- list() ind[["theta"]] <- list( "model"="linreg", "formula"=theta ~ 1 ) ind[["x"]] <- list( "model"="linreg", "formula"=x ~ 0 + offset(theta) ) # assumption of heterogeneous known error variance ind[["x"]]$sigma_fixed <- sqrt( var_err ) data_init <- dat data_init$theta <- dat$x # estimate model mod2 <- mdmb::frm_fb(dat, dep, ind, burnin=200, iter=1000, data_init=data_init) summary(mod2) plot(mod2) ############################################################################# # EXAMPLE 6: Non-normally distributed covariates: # Positive values with Box-Cox transformation ############################################################################# # simulate data with chi-squared distributed covariate from # regression model Y ~ X set.seed(876) n <- 1500 df <- 2 x <- stats::rchisq( n, df=df ) x <- x / sqrt( 2*df) y <- 0 + 1*x R2 <- .25 # explained variance y <- y + stats::rnorm(n, sd=sqrt( (1-R2)/R2 * 1) ) dat0 <- dat <- data.frame( y=y, x=x ) # simulate missing responses prop_miss <- .5 cor_miss <- .7 resp_tend <- cor_miss*(dat$y-mean(y) )/ stats::sd(y) + stats::rnorm(n, sd=sqrt( 1 - cor_miss^2) ) dat[ resp_tend < stats::qnorm( prop_miss ), "x" ] <- NA summary(dat) #-- complete data mod0 <- stats::lm( y ~ x, data=dat0 ) summary(mod0) #-- listwise deletion mod1 <- stats::lm( y ~ x, data=dat ) summary(mod1) # normal distribution assumption for frm # define models dep <- list("model"="linreg", "formula"=y ~ x ) # normally distributed data ind_x1 <- list( "model"="linreg", "formula"=x ~ 1 ) # Box-Cox normal distribution ind_x2 <- list( "model"="bctreg", "formula"=x ~ 1, nodes=c( seq(0.05, 3, len=31), seq( 3.5, 9, by=.5 ) ) ) ind1 <- list( "x"=ind_x1 ) ind2 <- list( "x"=ind_x2 ) #--- incorrect normal distribution assumption mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind1 ) summary(mod1) #--- model chi-square distribution of predictor with Box-Cox transformation mod2 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind2 ) summary(mod2) ############################################################################# # EXAMPLE 7: Latent interaction model ############################################################################# # A latent interaction model Y ~ FX + FZ is of interest. Y is directly observed, # FX and FZ are both indirectly observed by three items #--- simulate data N <- 1000 set.seed(987) # latent variable FX <- stats::rnorm(N) FZ <- stats::rnorm(N) # simulate items x1 <- 1.5 + 1*FX + stats::rnorm(N, sd=.7 ) x2 <- 1.9 + .7*FX + stats::rnorm(N, sd=1 ) x3 <- .9 + .7*FX + stats::rnorm(N, sd=.2 ) z1 <- 1.5 + 1*FZ + stats::rnorm(N, sd=.7 ) z2 <- 1.9 + .7*FZ + stats::rnorm(N, sd=1 ) z3 <- .9 + .7*FZ + stats::rnorm(N, sd=.2 ) dat <- data.frame(x1,x2,x3,z1,z2,z3) dat$FX <- NA dat$FZ <- NA dat$y <- 2 + .5*FX + .3*FZ + .4*FX*FZ + rnorm( N, sd=1 ) # estimate interaction model with ML #--- define model nodes <- seq(-4,4,len=11) ind_x1 <- list("model"="linreg", "formula"=x1 ~ offset(1*FX), "nodes"=nodes ) ind_x2 <- list( "model"="linreg", "formula"=x2 ~ FX, "nodes"=nodes, "coef_inits"=c(NA,1) ) ind_x3 <- list( "model"="linreg", "formula"=x3 ~ FX, "nodes"=nodes, "coef_inits"=c(1,1) ) ind_FX <- list( "model"="linreg", "formula"=FX ~ 0, "nodes"=nodes ) ind_z1 <- list("model"="linreg", "formula"=z1 ~ offset(1*FZ), "nodes"=nodes ) ind_z2 <- list( "model"="linreg", "formula"=z2 ~ FZ, "nodes"=nodes, "coef_inits"=c(NA,1) ) ind_z3 <- list( "model"="linreg", "formula"=z3 ~ FZ, "nodes"=nodes, "coef_inits"=c(1,1) ) ind_FZ <- list( "model"="linreg", "formula"=FZ ~ 0 + FX, "nodes"=nodes ) ind <- list( "x1"=ind_x1, "x2"=ind_x2, "x3"=ind_x3, "FX"=ind_FX, "z1"=ind_z1, "z2"=ind_z2, "z3"=ind_z3, "FX"=ind_FZ ) dep <- list( "model"="linreg", formula=y ~ FX+FZ+FX*FZ, "coef_inits"=c(1,.2,.2,0) ) #*** estimate model with mdmb::frm_em mod1 <- mdmb::frm_em(dat, dep, ind) summary(mod1) ############################################################################# # EXAMPLE 8: Non-ignorable data in Y ############################################################################# # regression Y ~ X in which Y is missing depending Y itself library(mvtnorm) cor_XY <- .4 # correlation between X and Y prop_miss <- .5 # missing proportion cor_missY <- .7 # correlation with missing propensity N <- 3000 # sample size #----- simulate data set.seed(790) Sigma <- matrix( c(1, cor_XY, cor_XY, 1), 2, 2 ) mu <- c(0,0) dat <- mvtnorm::rmvnorm( N, mean=mu, sigma=Sigma ) colnames(dat) <- c("X","Y") dat <- as.data.frame(dat) #-- generate missing responses on Y depending on Y itself y1 <- dat$Y miss_tend <- cor_missY * y1 + rnorm(N, sd=sqrt( 1 - cor_missY^2) ) dat$Y[ miss_tend < quantile( miss_tend, prop_miss ) ] <- NA #--- ML estimation under assumption of ignorability nodes <- seq(-5,5,len=15) dep <- list("model"="linreg", "formula"=Y ~ X, "nodes"=nodes ) ind_X <- list( "model"="linreg", "formula"=X ~ 1, "nodes"=nodes ) ind <- list( "X"=ind_X ) mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind) summary(mod1) #--- ML estimation under assumption with specifying a model for non-ignorability # for response indicator resp_Y dat$resp_Y <- 1* ( 1 - is.na(dat$Y) ) dep <- list("model"="linreg", "formula"=Y ~ X, "nodes"=nodes ) ind_X <- list( "model"="linreg", "formula"=X ~ 1, "nodes"=nodes ) ind_respY <- list( "model"="logistic", "formula"=resp_Y ~ Y, "nodes"=nodes ) ind <- list( "X"=ind_X, "resp_Y"=ind_respY ) mod2 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind) summary(mod2) ############################################################################# # EXAMPLE 9: Ordinal variables: Graded response model ############################################################################# #--- simulate data N <- 2000 set.seed(91834) # latent variable theta <- stats::rnorm(N) # simulate items y1 <- 1*theta + stats::rnorm(N) y2 <- .7*theta + stats::rnorm(N) y3 <- .7*theta + stats::rnorm(N) # discretize variables y1 <- as.numeric( cut( y1, breaks=c(-Inf, -.5, 0.4, 1, Inf ) ) ) - 1 y2 <- as.numeric( cut( y2, breaks=c(-Inf, 0.3, 1, Inf ) ) ) - 1 y3 <- as.numeric( cut( y3, breaks=c(-Inf, .2, Inf ) ) ) - 1 # define dataset dat <- data.frame(y1,y2,y3) dat$theta <- NA #****************************************** # Model 1: Fully Bayesian estimation #--- define model ind_y1 <- list( "model"="oprobit", "formula"=y1 ~ offset(1*theta) ) ind_y2 <- list( "model"="oprobit", "formula"=y2 ~ theta ) ind_y3 <- list( "model"="oprobit", "formula"=y3 ~ theta ) dep <- list( "model"="linreg", "formula"=theta ~ 0 ) ind <- list( "y1"=ind_y1, "y2"=ind_y2, "y3"=ind_y3) # initial data data_init <- dat data_init$theta <- as.numeric( scale(dat$y1) ) + stats::rnorm(N, sd=.4 ) #-- estimate model iter <- 3000; burnin <- 1000 mod1 <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, data_init=data_init, iter=iter, burnin=burnin) summary(mod1) plot(mod1) ############################################################################# # EXAMPLE 10: Imputation for missig predictors in models with interaction # effects in multilevel regression models ############################################################################# library(miceadds) data(data.mb04, package="mdmb") dat <- data.mb04 #*** model specification mcmc_iter <- 4 # number of MCMC iterations for model parameter sampling model_formula <- y ~ cwc(x, idcluster) + gm(x, idcluster) + w + w*cwc(x, idcluster) + w*gm(x, idcluster) + ( 1 + cwc(x, idcluster) | idcluster) dep <- list("model"="mlreg", "formula"=model_formula, R_args=list(iter=mcmc_iter, outcome="normal") ) ind_x <- list( "model"="mlreg", "formula"=x ~ w + (1|idcluster), R_args=list(iter=mcmc_iter), sampling_level="idcluster" ) # group means of x are involved in the outcome model. Therefore, Metropolis-Hastings # sampling of missing values in x should be conducted at the level of clusters, # i.e. specifying sampling_level ind <- list("x"=ind_x) # --- estimate model mod1 <- mdmb::frm_fb(dat, dep, ind, aggregation=TRUE) # argument aggregation is necessary because group means are involved in regression formulas #------------- #*** imputation of a continuous level-2 variable w # create artificially some missings on w dat[ dat$idcluster %%3==0, "w" ] <- NA # define level-2 model with argument variable_level ind_w <- list( "model"="linreg", "formula"=w ~ 1, "variable_level"="idcluster" ) ind <- list( x=ind_x, w=ind_w) #* conduct imputations mod2 <- mdmb::frm_fb(dat, dep, ind, aggregation=TRUE) summary(mod2) #--- Model 1 with user-defined prior distributions for covariance matrices model_formula <- y ~ cwc(x, idcluster) + gm(x, idcluster) + w + w*cwc(x, idcluster) + w*gm(x, idcluster) + ( 1 + cwc(x, idcluster) | idcluster) # define scale degrees of freedom (nu) and scale matrix (S) for inverse Wishart distribution psi_nu0_list <- list( -3 ) psi_S0_list <- list( diag(0,2) ) dep <- list("model"="mlreg", "formula"=model_formula, R_args=list(iter=mcmc_iter, outcome="normal", psi_nu0_list=psi_nu0_list, psi_S0_list=psi_S0_list ) ) # define nu and S parameters for covariate model psi_nu0_list <- list( .4 ) psi_S0_list <- list( matrix(.2, nrow=1, ncol=1) ) ind_x <- list( "model"="mlreg", "formula"=x ~ w + (1|idcluster), R_args=list(iter=mcmc_iter, psi_nu0_list=psi_nu0_list, psi_S0_list=psi_S0_list), sampling_level="idcluster" ) ind <- list("x"=ind_x) # --- estimate model mod3 <- mdmb::frm_fb(dat, dep, ind, aggregation=TRUE) ############################################################################# # EXAMPLE 11: Bounded variable combined with Yeo-Johnson transformation ############################################################################# #*** simulate data set.seed(876) n <- 1500 x <- mdmb::ryjt_scaled( n, location=-.2, shape=.8, lambda=.9, probit=TRUE) R2 <- .25 # explained variance y <- 1*x + stats::rnorm(n, sd=sqrt( (1-R2)/R2 * stats::var(x)) ) dat0 <- dat <- data.frame( y=y, x=x ) # simulate missing responses prop_miss <- .5 cor_miss <- .7 resp_tend <- cor_miss*(dat$y-mean(y) )/ stats::sd(y) + stats::rnorm(n, sd=sqrt( 1 - cor_miss^2) ) dat[ resp_tend < stats::qnorm(prop_miss), "x" ] <- NA summary(dat) #*** define models dep <- list("model"="linreg", "formula"=y ~ x ) # distribution according to Yeo-Johnson transformation ind_x1 <- list( "model"="yjtreg", "formula"=x ~ 1 ) # distribution according to Probit Yeo-Johnson transformation ind_x2 <- list( "model"="yjtreg", "formula"=x ~ 1, R_args=list("probit"=TRUE ) ) ind1 <- list( "x"=ind_x1 ) ind2 <- list( "x"=ind_x2 ) #--- complete data mod0 <- stats::lm( y~x, data=dat0) summary(mod0) #--- Yeo-Johnson normal distribution (for unbounded variables) mod1 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind1 ) summary(mod1) #--- Probit Yeo-Johnson normal distribution (for bounded variable on (0,1)) mod2 <- mdmb::frm_em(dat=dat, dep=dep, ind=ind2) summary(mod2) #--- same model, but MCMC estimation mod3 <- mdmb::frm_fb(dat, dep, ind=ind2, burnin=2000, iter=5000) summary(mod3) plot(mod3) ############################################################################# # EXAMPLE 12: Yeo-Johnson transformation with estimated degrees of freedom ############################################################################# #*** simulate data set.seed(876) n <- 1500 x <- mdmb::ryjt_scaled( n, location=-.2, shape=.8, lambda=.9, df=10 ) R2 <- .25 # explained variance y <- 1*x + stats::rnorm(n, sd=sqrt( (1-R2)/R2 * stats::var(x)) ) dat0 <- dat <- data.frame( y=y, x=x ) # simulate missing responses prop_miss <- .5 cor_miss <- .7 resp_tend <- cor_miss*(dat$y-mean(y) )/ stats::sd(y) + stats::rnorm(n, sd=sqrt( 1-cor_miss^2) ) dat[ resp_tend < stats::qnorm(prop_miss), "x" ] <- NA summary(dat) #*** define models dep <- list("model"="linreg", "formula"=y ~ x ) # specify distribution with estimated degrees of freedom ind_x <- list( "model"="yjtreg", "formula"=x ~ 1, R_args=list(est_df=TRUE ) ) ind <- list( "x"=ind_x ) #--- Yeo-Johnson t distribution mod1 <- mdmb::frm_fb(dat=dat, dep=dep, ind=ind, iter=3000, burnin=1000 ) summary(mod1) ## End(Not run)
Several regression functions which allow for sampling weights and prior distributions.
The function yjt_regression performs a linear regression in which the
response variable is transformed according to the Yeo-Johnson transformation
(Yeo & Johnson, 2000; see yjt_dist) and the residuals are
distributed following the scaled distribution. The degrees of freedom
of the distribution can be fixed or estimated (est_df=TRUE).
The function bct_regression has same functionality like the
Yeo-Johnson transformation but employs a Box-Cox transformation
of the outcome variable.
The Yeo-Johnson transformation can be extended by a probit transformation
(probit=TRUE) to cover the case of bounded variables on .
The function logistic_regression performs logistic regression
for dichotomous data.
The function oprobit_regression performs ordinal probit regression
for ordinal polytomous data.
#---- linear regression with Yeo-Johnson transformed scaled t distribution yjt_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL, df=Inf, lambda_fixed=NULL, probit=FALSE, est_df=FALSE, df_min=0.5, df_max=100, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL) ## S3 method for class 'yjt_regression' coef(object, ...) ## S3 method for class 'yjt_regression' logLik(object, ...) ## S3 method for class 'yjt_regression' predict(object, newdata=NULL, trafo=TRUE, ...) ## S3 method for class 'yjt_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'yjt_regression' vcov(object, ...) #---- linear regression with Box-Cox transformed scaled t distribution bct_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL, df=Inf, lambda_fixed=NULL, est_df=FALSE, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL) ## S3 method for class 'bct_regression' coef(object, ...) ## S3 method for class 'bct_regression' logLik(object, ...) ## S3 method for class 'bct_regression' predict(object, newdata=NULL, trafo=TRUE, ...) ## S3 method for class 'bct_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'bct_regression' vcov(object, ...) #---- logistic regression logistic_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL) ## S3 method for class 'logistic_regression' coef(object, ...) ## S3 method for class 'logistic_regression' logLik(object, ...) ## S3 method for class 'logistic_regression' predict(object, newdata=NULL, ...) ## S3 method for class 'logistic_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'logistic_regression' vcov(object, ...) #---- ordinal probit regression oprobit_regression(formula, data, weights=NULL, beta_init=NULL, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL, control_optim_fct=NULL) ## S3 method for class 'oprobit_regression' coef(object, ...) ## S3 method for class 'oprobit_regression' logLik(object, ...) ## S3 method for class 'oprobit_regression' predict(object, newdata=NULL, ...) ## S3 method for class 'oprobit_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'oprobit_regression' vcov(object, ...)#---- linear regression with Yeo-Johnson transformed scaled t distribution yjt_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL, df=Inf, lambda_fixed=NULL, probit=FALSE, est_df=FALSE, df_min=0.5, df_max=100, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL) ## S3 method for class 'yjt_regression' coef(object, ...) ## S3 method for class 'yjt_regression' logLik(object, ...) ## S3 method for class 'yjt_regression' predict(object, newdata=NULL, trafo=TRUE, ...) ## S3 method for class 'yjt_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'yjt_regression' vcov(object, ...) #---- linear regression with Box-Cox transformed scaled t distribution bct_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL, df=Inf, lambda_fixed=NULL, est_df=FALSE, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL) ## S3 method for class 'bct_regression' coef(object, ...) ## S3 method for class 'bct_regression' logLik(object, ...) ## S3 method for class 'bct_regression' predict(object, newdata=NULL, trafo=TRUE, ...) ## S3 method for class 'bct_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'bct_regression' vcov(object, ...) #---- logistic regression logistic_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL) ## S3 method for class 'logistic_regression' coef(object, ...) ## S3 method for class 'logistic_regression' logLik(object, ...) ## S3 method for class 'logistic_regression' predict(object, newdata=NULL, ...) ## S3 method for class 'logistic_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'logistic_regression' vcov(object, ...) #---- ordinal probit regression oprobit_regression(formula, data, weights=NULL, beta_init=NULL, use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL, control_optim_fct=NULL) ## S3 method for class 'oprobit_regression' coef(object, ...) ## S3 method for class 'oprobit_regression' logLik(object, ...) ## S3 method for class 'oprobit_regression' predict(object, newdata=NULL, ...) ## S3 method for class 'oprobit_regression' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'oprobit_regression' vcov(object, ...)
formula |
Formula |
data |
Data frame. The dependent variable must be coded as 0 and 1. |
weights |
Optional vector of sampling weights |
beta_init |
Optional vector of initial regression coefficients |
beta_prior |
Optional list containing priors of all parameters (see Examples for definition of this list). |
df |
Fixed degrees of freedom for scaled |
lambda_fixed |
Optional fixed value for |
probit |
Logical whether probit transformation should be employed for
bounded outcome in |
est_df |
Logical indicating whether degrees of freedom in
|
df_min |
Minimum value for estimated degrees of freedom |
df_max |
Maximum value for estimated degrees of freedom |
use_grad |
Computation method for gradients in |
h |
Numerical differentiation parameter. |
optimizer |
Type of optimizer to be chosen. Options are
|
maxiter |
Maximum number of iterations |
control |
Optional arguments to be passed to optimization function
( |
control_optim_fct |
Optional control argument for gradient in optimization |
object |
Object of class |
newdata |
Design matrix for |
trafo |
Logical indicating whether fitted values should be on the
transformed metric ( |
digits |
Number of digits for rounding |
file |
File name if the |
... |
Further arguments to be passed. |
List containing values
coefficients |
Estimated regression coefficients |
vcov |
Estimated covariance matrix |
partable |
Parameter table |
y |
Vector of values of dependent variable |
Xdes |
Design matrix |
weights |
Sampling weights |
fitted.values |
Fitted values in metric of probabilities |
linear.predictor |
Fitted values in metric of logits |
loglike |
Log likelihood value |
logprior |
Log prior value |
logpost |
Log posterior value |
deviance |
Deviance |
loglike_case |
Case-wise likelihood |
ic |
Information criteria |
R2 |
Pseudo R-square value according to McKelvey and Zavoina |
Alexander Robitzsch
McKelvey, R., & Zavoina, W. (1975). A statistical model for the analysis of ordinal level dependent variables. Journal of Mathematical Sociology, 4(1), 103-120. doi:10.1080/0022250X.1975.9989847
Yeo, I.-K., & Johnson, R. (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87(4), 954-959. doi:10.1093/biomet/87.4.954
See yjt_dist or car::yjPower
for functions for the Yeo-Johnson transformation.
See stats::lm and
stats::glm for linear and logistic
regression models.
############################################################################# # EXAMPLE 1: Simulated example logistic regression ############################################################################# #--- simulate dataset set.seed(986) N <- 500 x <- stats::rnorm(N) y <- 1*( stats::runif(N) < stats::plogis( -0.8 + 1.2 * x ) ) data <- data.frame( x=x, y=y ) #--- estimate logistic regression with mdmb::logistic_regression mod1 <- mdmb::logistic_regression( y ~ x, data=data ) summary(mod1) ## Not run: #--- estimate logistic regression with stats::glm mod1b <- stats::glm( y ~ x, data=data, family="binomial") summary(mod1b) #--- estimate logistic regression with prior distributions b0 <- list( "dnorm", list(mean=0, sd=100) ) # first parameter b1 <- list( "dcauchy", list(location=0, scale=2.5) ) # second parameter beta_priors <- list( b0, b1 ) # order in list defines priors for parameters #* estimation mod2 <- mdmb::logistic_regression( y ~ x, data=data, beta_prior=beta_priors ) summary(mod2) ############################################################################# # EXAMPLE 2: Yeo-Johnson transformed scaled t regression ############################################################################# #*** create simulated data set.seed(9865) n <- 1000 x <- stats::rnorm(n) y <- .5 + 1*x + .7*stats::rt(n, df=8 ) y <- mdmb::yj_antitrafo( y, lambda=.5 ) dat <- data.frame( y=y, x=x ) # display data graphics::hist(y) #--- Model 1: fit regression model with transformed normal distribution (df=Inf) mod1 <- mdmb::yjt_regression( y ~ x, data=dat ) summary(mod1) #--- Model 2: fit regression model with transformed scaled t distribution (df=10) mod2 <- mdmb::yjt_regression( y ~ x, data=dat, df=10) summary(mod2) #--- Model 3: fit regression model with transformed normal distribution (df=Inf) # and fixed transformation parameter lambda of .5 mod3 <- mdmb::yjt_regression( y ~ x, data=dat, lambda_fixed=.5) summary(mod3) #--- Model 4: fit regression model with transformed normal distribution (df=Inf) # and fixed transformation parameter lambda of 1 # -> This model corresponds to least squares regression mod4 <- mdmb::yjt_regression( y ~ x, data=dat, lambda_fixed=1) summary(mod4) # fit with lm function mod4b <- stats::lm( y ~ x, data=dat ) summary(mod4b) #--- Model 5: fit regression model with estimated degrees of freedom mod5 <- mdmb::yjt_regression( y ~ x, data=dat, est_df=TRUE) summary(mod5) #** compare log-likelihood values logLik(mod1) logLik(mod2) logLik(mod3) logLik(mod4) logLik(mod4b) logLik(mod5) ############################################################################# # EXAMPLE 3: Regression with Box-Cox and Yeo-Johnson transformations ############################################################################# #*** simulate data set.seed(985) n <- 1000 x <- stats::rnorm(n) y <- .5 + 1*x + stats::rnorm(n, sd=.7 ) y <- mdmb::bc_antitrafo( y, lambda=.5 ) dat <- data.frame( y=y, x=x ) #--- Model 1: fit regression model with Box-Cox transformation mod1 <- mdmb::bct_regression( y ~ x, data=dat ) summary(mod1) #--- Model 2: fit regression model with Yeo-Johnson transformation mod2 <- mdmb::yjt_regression( y ~ x, data=dat ) summary(mod2) #--- compare fit logLik(mod1) logLik(mod2) ############################################################################# # EXAMPLE 4: Ordinal probit regression ############################################################################# #--- simulate data set.seed(987) N <- 1500 x <- stats::rnorm(N) z <- stats::rnorm(N) # regression coefficients b0 <- -.5 ; b1 <- .6 ; b2 <- .1 # vector of thresholds thresh <- c(-1, -.3, 1) yast <- b0 + b1 * x + b2*z + stats::rnorm(N) y <- as.numeric( cut( yast, c(-Inf,thresh,Inf) ) ) - 1 dat <- data.frame( x=x, y=y, z=z ) #--- probit regression mod <- mdmb::oprobit_regression( formula=y ~ x + z + I(x*z), data=dat) summary(mod) ## End(Not run)############################################################################# # EXAMPLE 1: Simulated example logistic regression ############################################################################# #--- simulate dataset set.seed(986) N <- 500 x <- stats::rnorm(N) y <- 1*( stats::runif(N) < stats::plogis( -0.8 + 1.2 * x ) ) data <- data.frame( x=x, y=y ) #--- estimate logistic regression with mdmb::logistic_regression mod1 <- mdmb::logistic_regression( y ~ x, data=data ) summary(mod1) ## Not run: #--- estimate logistic regression with stats::glm mod1b <- stats::glm( y ~ x, data=data, family="binomial") summary(mod1b) #--- estimate logistic regression with prior distributions b0 <- list( "dnorm", list(mean=0, sd=100) ) # first parameter b1 <- list( "dcauchy", list(location=0, scale=2.5) ) # second parameter beta_priors <- list( b0, b1 ) # order in list defines priors for parameters #* estimation mod2 <- mdmb::logistic_regression( y ~ x, data=data, beta_prior=beta_priors ) summary(mod2) ############################################################################# # EXAMPLE 2: Yeo-Johnson transformed scaled t regression ############################################################################# #*** create simulated data set.seed(9865) n <- 1000 x <- stats::rnorm(n) y <- .5 + 1*x + .7*stats::rt(n, df=8 ) y <- mdmb::yj_antitrafo( y, lambda=.5 ) dat <- data.frame( y=y, x=x ) # display data graphics::hist(y) #--- Model 1: fit regression model with transformed normal distribution (df=Inf) mod1 <- mdmb::yjt_regression( y ~ x, data=dat ) summary(mod1) #--- Model 2: fit regression model with transformed scaled t distribution (df=10) mod2 <- mdmb::yjt_regression( y ~ x, data=dat, df=10) summary(mod2) #--- Model 3: fit regression model with transformed normal distribution (df=Inf) # and fixed transformation parameter lambda of .5 mod3 <- mdmb::yjt_regression( y ~ x, data=dat, lambda_fixed=.5) summary(mod3) #--- Model 4: fit regression model with transformed normal distribution (df=Inf) # and fixed transformation parameter lambda of 1 # -> This model corresponds to least squares regression mod4 <- mdmb::yjt_regression( y ~ x, data=dat, lambda_fixed=1) summary(mod4) # fit with lm function mod4b <- stats::lm( y ~ x, data=dat ) summary(mod4b) #--- Model 5: fit regression model with estimated degrees of freedom mod5 <- mdmb::yjt_regression( y ~ x, data=dat, est_df=TRUE) summary(mod5) #** compare log-likelihood values logLik(mod1) logLik(mod2) logLik(mod3) logLik(mod4) logLik(mod4b) logLik(mod5) ############################################################################# # EXAMPLE 3: Regression with Box-Cox and Yeo-Johnson transformations ############################################################################# #*** simulate data set.seed(985) n <- 1000 x <- stats::rnorm(n) y <- .5 + 1*x + stats::rnorm(n, sd=.7 ) y <- mdmb::bc_antitrafo( y, lambda=.5 ) dat <- data.frame( y=y, x=x ) #--- Model 1: fit regression model with Box-Cox transformation mod1 <- mdmb::bct_regression( y ~ x, data=dat ) summary(mod1) #--- Model 2: fit regression model with Yeo-Johnson transformation mod2 <- mdmb::yjt_regression( y ~ x, data=dat ) summary(mod2) #--- compare fit logLik(mod1) logLik(mod2) ############################################################################# # EXAMPLE 4: Ordinal probit regression ############################################################################# #--- simulate data set.seed(987) N <- 1500 x <- stats::rnorm(N) z <- stats::rnorm(N) # regression coefficients b0 <- -.5 ; b1 <- .6 ; b2 <- .1 # vector of thresholds thresh <- c(-1, -.3, 1) yast <- b0 + b1 * x + b2*z + stats::rnorm(N) y <- as.numeric( cut( yast, c(-Inf,thresh,Inf) ) ) - 1 dat <- data.frame( x=x, y=y, z=z ) #--- probit regression mod <- mdmb::oprobit_regression( formula=y ~ x + z + I(x*z), data=dat) summary(mod) ## End(Not run)
Extracts offset values by applying a formula with an offset term to a data frame.
offset_values_extract(formula, data)offset_values_extract(formula, data)
formula |
An R Formula |
data |
Data frame |
Vector containing offset values
stats::offset,
stats::model.offset
############################################################################# # EXAMPLE 1: Toy example for extraction of offset values ############################################################################# data(data.ma01, package="miceadds") dat <- data.ma01 dat1 <- mdmb::offset_values_extract( formula=~ migrant + offset(books), data=dat ) dat1[1:5] ## [1] 6 6 5 2 6############################################################################# # EXAMPLE 1: Toy example for extraction of offset values ############################################################################# data(data.ma01, package="miceadds") dat <- data.ma01 dat1 <- mdmb::offset_values_extract( formula=~ migrant + offset(books), data=dat ) dat1[1:5] ## [1] 6 6 5 2 6
Fits and evaluates the ordinal probit model.
#---- ordinal probit model doprobit(x, thresh, max_val=99) fit_oprobit(x, par_init=NULL, weights=NULL) ## S3 method for class 'fit_oprobit' coef(object, ...) ## S3 method for class 'fit_oprobit' logLik(object, ...) ## S3 method for class 'fit_oprobit' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_oprobit' vcov(object, ...)#---- ordinal probit model doprobit(x, thresh, max_val=99) fit_oprobit(x, par_init=NULL, weights=NULL) ## S3 method for class 'fit_oprobit' coef(object, ...) ## S3 method for class 'fit_oprobit' logLik(object, ...) ## S3 method for class 'fit_oprobit' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_oprobit' vcov(object, ...)
x |
Numeric vector |
thresh |
Vector of thresholds |
max_val |
Maximum value for computing thresholds |
par_init |
Optional vector of initial parameters |
weights |
Optional vector of sampling weights |
object |
Object of class |
digits |
Number of digits used for rounding in |
file |
File name for the |
... |
Further arguments to be passed |
Vector or an object of fitted distribution depending on the called function
See oprobit_regression for fitting a regression model in which
the response variable follows an ordinal probit model.
############################################################################# # EXAMPLE 1: Fit an ordinal probit distribution ############################################################################# #-- simulate data set.seed(987) N <- 1500 # define thresholds thresh <- c(0,.3, .7, 1.6) # latent continuous data yast <- stats::rnorm(N) # discretized ordinal data x <- as.numeric( cut( yast, c(-Inf,thresh,Inf) ) ) - 1 #-- fit ordinal probit distribution mod <- mdmb::fit_oprobit(x=x) summary(mod) logLik(mod) vcov(mod)############################################################################# # EXAMPLE 1: Fit an ordinal probit distribution ############################################################################# #-- simulate data set.seed(987) N <- 1500 # define thresholds thresh <- c(0,.3, .7, 1.6) # latent continuous data yast <- stats::rnorm(N) # discretized ordinal data x <- as.numeric( cut( yast, c(-Inf,thresh,Inf) ) ) - 1 #-- fit ordinal probit distribution mod <- mdmb::fit_oprobit(x=x) summary(mod) logLik(mod) vcov(mod)
Removes rows with some missing entries in a data frame for variables
appearing in the R formula formula. This operation is also
known as listwise deletion.
remove_NA_data_frame(formula, data, weights=NULL)remove_NA_data_frame(formula, data, weights=NULL)
formula |
An R formula |
data |
Data frame |
weights |
Optional vector of sample weights |
Data frame with some rows removed according to missing data
############################################################################# # EXAMPLE 1: Removing rows in a data frame (listwise deletion) ############################################################################# data(data.ma01, package="miceadds") dat <- data.ma01 #*** remove rows with some missings according to a formula dat1 <- mdmb::remove_NA_data_frame( formula=read ~ migrant + books, data=dat) #*** remove rows according to two formulas formula1 <- read ~ migrant + books formula2 <- paredu ~ migrant + female # create formula consisting of formula1 and formula2 formula3 <- paste( "~", deparse(formula1[[2]]), "+", deparse(formula1[[3]]), " + ", deparse(formula2[[2]]), "+", deparse(formula2[[3]]) ) dat2 <- mdmb::remove_NA_data_frame( formula=as.formula(formula3), data=dat) dim(dat) dim(dat1$data) dim(dat2$data) ## > dim(dat) ## [1] 4073 11 ## > dim(dat1$data) ## [1] 3371 11 ## > dim(dat2$data) ## [1] 3090 11############################################################################# # EXAMPLE 1: Removing rows in a data frame (listwise deletion) ############################################################################# data(data.ma01, package="miceadds") dat <- data.ma01 #*** remove rows with some missings according to a formula dat1 <- mdmb::remove_NA_data_frame( formula=read ~ migrant + books, data=dat) #*** remove rows according to two formulas formula1 <- read ~ migrant + books formula2 <- paredu ~ migrant + female # create formula consisting of formula1 and formula2 formula3 <- paste( "~", deparse(formula1[[2]]), "+", deparse(formula1[[3]]), " + ", deparse(formula2[[2]]), "+", deparse(formula2[[3]]) ) dat2 <- mdmb::remove_NA_data_frame( formula=as.formula(formula3), data=dat) dim(dat) dim(dat1$data) dim(dat2$data) ## > dim(dat) ## [1] 4073 11 ## > dim(dat1$data) ## [1] 3371 11 ## > dim(dat2$data) ## [1] 3090 11
Collection of functions for the Yeo-Johnson transformation
(Yeo & Johnson, 2000) and the corresponding distribution family of scaled
distribution with and without Yeo-Johnson transformation
(see Details). The Yeo-Johnson transformation can also be applied for bounded variables
on which uses a probit transformation (see Details; argument probit).
The Box-Cox transformation (bc; Sakia, 1992)
can be applied for variables with positive values.
# Yeo-Johnson transformation and its inverse transformation yj_trafo(y, lambda, use_rcpp=TRUE, probit=FALSE) yj_antitrafo(y, lambda, probit=FALSE) #---- scaled t distribution with Yeo-Johnson transformation dyjt_scaled(x, location=0, shape=1, lambda=1, df=Inf, log=FALSE, probit=FALSE) ryjt_scaled(n, location=0, shape=1, lambda=1, df=Inf, probit=FALSE) fit_yjt_scaled(x, df=Inf, par_init=NULL, lambda_fixed=NULL, weights=NULL, probit=FALSE) ## S3 method for class 'fit_yjt_scaled' coef(object, ...) ## S3 method for class 'fit_yjt_scaled' logLik(object, ...) ## S3 method for class 'fit_yjt_scaled' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_yjt_scaled' vcov(object, ...) # Box-Cox transformation and its inverse transformation bc_trafo(y, lambda) bc_antitrafo(y, lambda) #---- scaled t distribution with Box-Cox transformation dbct_scaled(x, location=0, shape=1, lambda=1, df=Inf, log=FALSE, check_zero=TRUE) rbct_scaled(n, location=0, shape=1, lambda=1, df=Inf) fit_bct_scaled(x, df=Inf, par_init=NULL, lambda_fixed=NULL, weights=NULL) ## S3 method for class 'fit_bct_scaled' coef(object, ...) ## S3 method for class 'fit_bct_scaled' logLik(object, ...) ## S3 method for class 'fit_bct_scaled' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_bct_scaled' vcov(object, ...) #---- scaled t distribution dt_scaled(x, location=0, shape=1, df=Inf, log=FALSE) rt_scaled(n, location=0, shape=1, df=Inf) fit_t_scaled(x, df=Inf, par_init=NULL, weights=NULL) ## S3 method for class 'fit_t_scaled' coef(object, ...) ## S3 method for class 'fit_t_scaled' logLik(object, ...) ## S3 method for class 'fit_t_scaled' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_t_scaled' vcov(object, ...)# Yeo-Johnson transformation and its inverse transformation yj_trafo(y, lambda, use_rcpp=TRUE, probit=FALSE) yj_antitrafo(y, lambda, probit=FALSE) #---- scaled t distribution with Yeo-Johnson transformation dyjt_scaled(x, location=0, shape=1, lambda=1, df=Inf, log=FALSE, probit=FALSE) ryjt_scaled(n, location=0, shape=1, lambda=1, df=Inf, probit=FALSE) fit_yjt_scaled(x, df=Inf, par_init=NULL, lambda_fixed=NULL, weights=NULL, probit=FALSE) ## S3 method for class 'fit_yjt_scaled' coef(object, ...) ## S3 method for class 'fit_yjt_scaled' logLik(object, ...) ## S3 method for class 'fit_yjt_scaled' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_yjt_scaled' vcov(object, ...) # Box-Cox transformation and its inverse transformation bc_trafo(y, lambda) bc_antitrafo(y, lambda) #---- scaled t distribution with Box-Cox transformation dbct_scaled(x, location=0, shape=1, lambda=1, df=Inf, log=FALSE, check_zero=TRUE) rbct_scaled(n, location=0, shape=1, lambda=1, df=Inf) fit_bct_scaled(x, df=Inf, par_init=NULL, lambda_fixed=NULL, weights=NULL) ## S3 method for class 'fit_bct_scaled' coef(object, ...) ## S3 method for class 'fit_bct_scaled' logLik(object, ...) ## S3 method for class 'fit_bct_scaled' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_bct_scaled' vcov(object, ...) #---- scaled t distribution dt_scaled(x, location=0, shape=1, df=Inf, log=FALSE) rt_scaled(n, location=0, shape=1, df=Inf) fit_t_scaled(x, df=Inf, par_init=NULL, weights=NULL) ## S3 method for class 'fit_t_scaled' coef(object, ...) ## S3 method for class 'fit_t_scaled' logLik(object, ...) ## S3 method for class 'fit_t_scaled' summary(object, digits=4, file=NULL, ...) ## S3 method for class 'fit_t_scaled' vcov(object, ...)
y |
Numeric vector |
lambda |
Transformation parameter |
use_rcpp |
Logical indicating whether Rcpp package should be used |
probit |
Logical indicating whether probit transformation should be
applied for bounded variables on |
x |
Numeric vector |
location |
Location parameter of (transformed) scaled |
shape |
Shape parameter of (transformed) scaled |
df |
Degrees of freedom of (transformed) scaled |
log |
Logical indicating whether logarithm of the density should be computed |
check_zero |
Logical indicating whether check for inadmissible values should be conducted |
n |
Number of observations to be simulated |
par_init |
Optional vector of initial parameters |
lambda_fixed |
Optional value for fixed |
weights |
Optional vector of sampling weights |
object |
Object of class |
digits |
Number of digits used for rounding in |
file |
File name for the |
... |
Further arguments to be passed |
Let be the Yeo-Johnson transformation. A random variable
is distribution as Scaled with Yeo-Johnson transformation with location
, scale and transformation parameter
iff and is distributed
with df degrees of freedom.
For a bounded variable on , the probit transformation
is applied such that with a
distributed variable .
For a Yeo-Johnson normally distributed variable, a normally distributed variable results in
case of . For a Box-Cox normally distributed variable, a normally
distributed variable results for .
Vector or an object of fitted distribution depending on the called function
Sakia, S. M. (1992). The Box-Cox transformation technique: A review. The Statistician, 41(2), 169-178. doi:10.2307/2348250
Yeo, I.-K., & Johnson, R. (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87(4), 954-959. doi:10.1093/biomet/87.4.954
See yjt_regression for fitting a regression model in which
the response variable is distributed according to the scaled
distribution with Yeo-Johnson transformation.
See car::yjPower for fitting the Yeo-Johnson
transformation in the car package. See car::bcPower for the
Box-Cox transformation.
The scaled distribution can be also found in
metRology::dt.scaled (metRology package).
See stats::dt for the distribution.
See the fitdistrplus package or the general
stats4::mle function
for fitting several distributions in R.
############################################################################# # EXAMPLE 1: Transforming values according to Yeo-Johnson transformation ############################################################################# # vector of y values y <- seq(-3,3, len=100) # non-negative lambda values plot( y, mdmb::yj_trafo( y, lambda=1 ), type="l", ylim=8*c(-1,1), ylab=expression( g[lambda] (y) ) ) lines( y, mdmb::yj_trafo( y, lambda=2 ), lty=2 ) lines( y, mdmb::yj_trafo( y, lambda=.5 ), lty=3 ) lines( y, mdmb::yj_trafo( y, lambda=0 ), lty=4 ) # non-positive lambda values plot( y, mdmb::yj_trafo( y, lambda=-1 ), type="l", ylim=8*c(-1,1), ylab=expression(g[lambda] (y) ) ) lines( y, mdmb::yj_trafo( y, lambda=-2 ), lty=2 ) lines( y, mdmb::yj_trafo( y, lambda=-.5 ), lty=3 ) lines( y, mdmb::yj_trafo( y, lambda=0 ), lty=4 ) ## Not run: ############################################################################# # EXAMPLE 2: Density of scaled t distribution ############################################################################# # define location and scale parameter m0 <- 0.3 sig <- 1.5 #-- compare density of scaled t distribution with large degrees of freedom # with normal distribution y1 <- mdmb::dt_scaled( y, location=m0, shape=sig, df=100 ) y2 <- stats::dnorm( y, mean=m0, sd=sig ) max(abs(y1-y2)) ############################################################################# # EXAMPLE 3: Simulating and fitting the scaled t distribution ############################################################################# #-- simulate data with 10 degrees of freedom set.seed(987) df0 <- 10 # define degrees of freedom x <- mdmb::rt_scaled( n=1E4, location=m0, shape=sig, df=df0 ) #** fit data with df=10 degrees of freedom fit1 <- mdmb::fit_t_scaled(x=x, df=df0 ) #** compare with fit from normal distribution fit2 <- mdmb::fit_t_scaled(x=x, df=Inf ) # df=Inf is the default #-- some comparisons coef(fit1) summary(fit1) logLik(fit1) AIC(fit1) AIC(fit2) ############################################################################# # EXAMPLE 4: Simulation and fitting of scaled t distribution with # Yeo-Johnson transformation ############################################################################# # define parameters of transformed scaled t distribution m0 <- .5 sig <- 1.5 lam <- .5 # evaluate density x <- seq( -5, 5, len=100 ) y <- mdmb::dyjt_scaled( x, location=m0, shape=sig, lambda=lam ) graphics::plot( x, y, type="l") # transform original values mdmb::yj_trafo( y=x, lambda=lam ) #** simulate data set.seed(987) x <- mdmb::ryjt_scaled(n=3000, location=m0, shape=sig, lambda=lam ) graphics::hist(x, breaks=30) #*** Model 1: Fit data with lambda to be estimated fit1 <- mdmb::fit_yjt_scaled(x=x) summary(fit1) coef(fit1) #*** Model 2: Fit data with lambda fixed to simulated lambda fit2 <- mdmb::fit_yjt_scaled(x=x, lambda_fixed=lam) summary(fit2) coef(fit2) #*** Model 3: Fit data with lambda fixed to 1 fit3 <- mdmb::fit_yjt_scaled(x=x, lambda_fixed=1) #-- compare log-likelihood values logLik(fit1) logLik(fit2) logLik(fit3) ############################################################################# # EXAMPLE 5: Approximating the chi square distribution # with yjt and bct distribution ############################################################################# #-- simulate data set.seed(987) n <- 3000 df0 <- 5 x <- stats::rchisq( n=n, df=df0 ) #-- plot data graphics::hist(x, breaks=30) #-- fit data with yjt distribution fit1 <- mdmb::fit_yjt_scaled(x) summary(fit1) c1 <- coef(fit1) #-- fit data with bct distribution fit2 <- mdmb::fit_bct_scaled(x) summary(fit2) c2 <- coef(fit2) # compare log-likelihood values logLik(fit1) logLik(fit2) #-- plot chi square distribution and approximating yjt distribution y <- seq( .01, 3*df0, len=100 ) dy <- stats::dchisq( y, df=df0 ) graphics::plot( y, dy, type="l", ylim=c(0, max(dy) )*1.1 ) # approximation with scaled t distribution and Yeo-Johnson transformation graphics::lines( y, mdmb::dyjt_scaled(y, location=c1[1], shape=c1[2], lambda=c1[3]), lty=2) # approximation with scaled t distribution and Box-Cox transformation graphocs::lines( y, mdmb::dbct_scaled(y, location=c2[1], shape=c2[2], lambda=c2[3]), lty=3) # appoximating normal distribution graphics::lines( y, stats::dnorm( y, mean=df0, sd=sqrt(2*df0) ), lty=4) graphics::legend( .6*max(y), .9*max(dy), c("chi square", "yjt", "bct", "norm"), lty=1:4) ############################################################################# # EXAMPLE 6: Bounded variable on (0,1) with Probit Yeo-Johnson transformation ############################################################################# set.seed(876) n <- 1000 x <- stats::rnorm(n) y <- stats::pnorm( 1*x + stats::rnorm(n, sd=sqrt(.5) ) ) dat <- data.frame( y=y, x=x ) #*** fit Probit Yeo-Johnson distribution mod1 <- mdmb::fit_yjt_scaled(x=y, probit=TRUE) summary(mod1) #*** estimation using regression model mod2 <- mdmb::yjt_regression( y ~ x, data=dat, probit=TRUE ) summary(mod2) ## End(Not run)############################################################################# # EXAMPLE 1: Transforming values according to Yeo-Johnson transformation ############################################################################# # vector of y values y <- seq(-3,3, len=100) # non-negative lambda values plot( y, mdmb::yj_trafo( y, lambda=1 ), type="l", ylim=8*c(-1,1), ylab=expression( g[lambda] (y) ) ) lines( y, mdmb::yj_trafo( y, lambda=2 ), lty=2 ) lines( y, mdmb::yj_trafo( y, lambda=.5 ), lty=3 ) lines( y, mdmb::yj_trafo( y, lambda=0 ), lty=4 ) # non-positive lambda values plot( y, mdmb::yj_trafo( y, lambda=-1 ), type="l", ylim=8*c(-1,1), ylab=expression(g[lambda] (y) ) ) lines( y, mdmb::yj_trafo( y, lambda=-2 ), lty=2 ) lines( y, mdmb::yj_trafo( y, lambda=-.5 ), lty=3 ) lines( y, mdmb::yj_trafo( y, lambda=0 ), lty=4 ) ## Not run: ############################################################################# # EXAMPLE 2: Density of scaled t distribution ############################################################################# # define location and scale parameter m0 <- 0.3 sig <- 1.5 #-- compare density of scaled t distribution with large degrees of freedom # with normal distribution y1 <- mdmb::dt_scaled( y, location=m0, shape=sig, df=100 ) y2 <- stats::dnorm( y, mean=m0, sd=sig ) max(abs(y1-y2)) ############################################################################# # EXAMPLE 3: Simulating and fitting the scaled t distribution ############################################################################# #-- simulate data with 10 degrees of freedom set.seed(987) df0 <- 10 # define degrees of freedom x <- mdmb::rt_scaled( n=1E4, location=m0, shape=sig, df=df0 ) #** fit data with df=10 degrees of freedom fit1 <- mdmb::fit_t_scaled(x=x, df=df0 ) #** compare with fit from normal distribution fit2 <- mdmb::fit_t_scaled(x=x, df=Inf ) # df=Inf is the default #-- some comparisons coef(fit1) summary(fit1) logLik(fit1) AIC(fit1) AIC(fit2) ############################################################################# # EXAMPLE 4: Simulation and fitting of scaled t distribution with # Yeo-Johnson transformation ############################################################################# # define parameters of transformed scaled t distribution m0 <- .5 sig <- 1.5 lam <- .5 # evaluate density x <- seq( -5, 5, len=100 ) y <- mdmb::dyjt_scaled( x, location=m0, shape=sig, lambda=lam ) graphics::plot( x, y, type="l") # transform original values mdmb::yj_trafo( y=x, lambda=lam ) #** simulate data set.seed(987) x <- mdmb::ryjt_scaled(n=3000, location=m0, shape=sig, lambda=lam ) graphics::hist(x, breaks=30) #*** Model 1: Fit data with lambda to be estimated fit1 <- mdmb::fit_yjt_scaled(x=x) summary(fit1) coef(fit1) #*** Model 2: Fit data with lambda fixed to simulated lambda fit2 <- mdmb::fit_yjt_scaled(x=x, lambda_fixed=lam) summary(fit2) coef(fit2) #*** Model 3: Fit data with lambda fixed to 1 fit3 <- mdmb::fit_yjt_scaled(x=x, lambda_fixed=1) #-- compare log-likelihood values logLik(fit1) logLik(fit2) logLik(fit3) ############################################################################# # EXAMPLE 5: Approximating the chi square distribution # with yjt and bct distribution ############################################################################# #-- simulate data set.seed(987) n <- 3000 df0 <- 5 x <- stats::rchisq( n=n, df=df0 ) #-- plot data graphics::hist(x, breaks=30) #-- fit data with yjt distribution fit1 <- mdmb::fit_yjt_scaled(x) summary(fit1) c1 <- coef(fit1) #-- fit data with bct distribution fit2 <- mdmb::fit_bct_scaled(x) summary(fit2) c2 <- coef(fit2) # compare log-likelihood values logLik(fit1) logLik(fit2) #-- plot chi square distribution and approximating yjt distribution y <- seq( .01, 3*df0, len=100 ) dy <- stats::dchisq( y, df=df0 ) graphics::plot( y, dy, type="l", ylim=c(0, max(dy) )*1.1 ) # approximation with scaled t distribution and Yeo-Johnson transformation graphics::lines( y, mdmb::dyjt_scaled(y, location=c1[1], shape=c1[2], lambda=c1[3]), lty=2) # approximation with scaled t distribution and Box-Cox transformation graphocs::lines( y, mdmb::dbct_scaled(y, location=c2[1], shape=c2[2], lambda=c2[3]), lty=3) # appoximating normal distribution graphics::lines( y, stats::dnorm( y, mean=df0, sd=sqrt(2*df0) ), lty=4) graphics::legend( .6*max(y), .9*max(dy), c("chi square", "yjt", "bct", "norm"), lty=1:4) ############################################################################# # EXAMPLE 6: Bounded variable on (0,1) with Probit Yeo-Johnson transformation ############################################################################# set.seed(876) n <- 1000 x <- stats::rnorm(n) y <- stats::pnorm( 1*x + stats::rnorm(n, sd=sqrt(.5) ) ) dat <- data.frame( y=y, x=x ) #*** fit Probit Yeo-Johnson distribution mod1 <- mdmb::fit_yjt_scaled(x=y, probit=TRUE) summary(mod1) #*** estimation using regression model mod2 <- mdmb::yjt_regression( y ~ x, data=dat, probit=TRUE ) summary(mod2) ## End(Not run)