MLCIRTwithin (version 1.1)

est_multi_poly_within: Estimate latent class item response theory (LC-IRT) models for dichotomous and polytomous responses under within-item multidimensionality

Description

The function performs maximum likelihood estimation of the parameters of the IRT models assuming a discrete distribution for the ability and within-item multidimensionality. Every ability level corresponds to a latent class of subjects in the reference population. Maximum likelihood estimation is based on Expectation- Maximization algorithm.

Usage

est_multi_poly_within(S, yv = rep(1,ns), k1, k2, X = NULL, start = c("deterministic","random","external"),  link = c("global", "local"),disc = FALSE, difl = FALSE, multi1, multi2, piv1 = NULL, piv2 = NULL, Phi = NULL, ga1c = NULL, ga2c = NULL, De1 = NULL, De2 = NULL, fort = FALSE, tol = 10^-10, disp = FALSE, output = FALSE, out_se = FALSE, glob = FALSE)

Arguments

S

matrix of all response sequences observed at least once in the sample and listed row-by-row (use NA for missing response)

yv

vector of the frequencies of every response configuration in S

k1

number of ability levels (or latent classes) for the 1st latent variable

k2

number of ability levels (or latent classes) for the 2nd latent variable

X

matrix of covariates that affects the weights

start

method of initialization of the algorithm

link

type of link function ("global" for global logits, "local" for local logits); with global logits a graded response model results; with local logits a partial credit model results (with dichotomous responses, global logits is the same as using local logits resulting in the Rasch or the 2PL model depending on the value assigned to disc)

disc

indicator of constraints on the discriminating indices (FALSE = all equal to one, TRUE = free)

difl

indicator of constraints on the difficulty levels (FALSE = free, TRUE = rating scale parametrization)

multi1

matrix with a number of rows equal to the number of dimensions and elements in each row equal to the indices of the items measuring the dimension corresponding to that row for the 1st latent variable

multi2

matrix with a number of rows equal to the number of dimensions and elements in each row equal to the indices of the items measuring the dimension corresponding to that row for the 2nd latent variable

piv1

initial value of the vector of weights of the latent classes (if start="external") for the 1st latent variable

piv2

initial value of the vector of weights of the latent classes (if start="external") for the 2nd latent variable

Phi

initial value of the matrix of the conditional response probabilities (if start="external")

ga1c

initial value of the complete vector of discriminating indices (if start="external") for the 1st latent variable

ga2c

initial value of the complete vector of discriminating indices (if start="external") for the 2nd latent variable

De1

initial value of regression coefficients for the covariates (if start="external") affecting the 1st latent variable

De2

initial value of regression coefficients for the covariates (if start="external") affecting the 2st latent variable

fort

to use Fortran routines when possible

tol

tolerance level for checking convergence of the algorithm as relative difference between consecutive log-likelihoods

disp

to display the likelihood evolution step by step

output

to return additional outputs (Piv, Pp, lkv, Xlabel, XX1dis, XX2dis)

out_se

to return standard errors

glob

to use global logits in the covariates

Value

piv1

estimated vector of weights of the latent classes (average of the weights in case of model with covariates) for the 1st latent variable

piv2

estimated vector of weights of the latent classes (average of the weights in case of model with covariates) for the 2nd latent variable

Th1

estimated matrix of ability levels for each dimension and latent class for the 1st latent variable

Th2

estimated matrix of ability levels for each dimension and latent class for the 2nd latent variable

Bec

estimated vector of difficulty levels for every item (split in two vectors if difl=TRUE)

ga1c

estimated vector of discriminating indices for every item (with all elements equal to 1 with Rasch parametrization) for the 1st latent variable

ga2c

estimated vector of discriminating indices for every item (with all elements equal to 1 with Rasch parametrization) for the 2nd latent variable

fv1

vector indicating the reference item chosen for each latent dimension for the 1st latent variable

fv2

vector indicating the reference item chosen for each latent dimension for the 2nd latent variable

Phi

array of the conditional response probabilities for every item and each of the k1*k2 latent classes

De1

matrix of regression coefficients for the multinomial (or global if glob=TRUE) logit model on the class weights for the 1st latent variable

De2

matrix of regression coefficients for the multinomial (or global if glob=TRUE) logit model on the class weights for the 2nd latent variable

Piv

matrix of the weights for every covariate pattern configuration (if output=TRUE)

Pp

matrix of the posterior probabilities for each response configuration and latent class (if output=TRUE)

lk

log-likelihood at convergence of the EM algorithm

np

number of free parameters

aic

Akaike Information Criterion index

bic

Bayesian Information Criterion index

ent

Entropy index to measure the separation of classes

lkv

Vector to trace the log-likelihood evolution across iterations (if output=TRUE)

seDe1

Standard errors for vector De1 (if out_se=TRUE) for the 1st latent variable

seDe2

Standard errors for vector De2 (if out_se=TRUE) for the 2nd latent variable

seTh1

Standard errors for vector Th1 (if out_se=TRUE) for the 1st latent variable

seTh2

Standard errors for vector Th2 (if out_se=TRUE) for the 2nd latent variable

seBec

Standard errors for vector Bec of difficulty parameters (if out_se=TRUE)

sega1

Standard errors for vector ga1c of discrimination indices (if out_se=TRUE) for the 1st latent variable

sega2

Standard errors for vector ga2c of discrimination indices (if out_se=TRUE) for the 2nd latent variable

Vn

Estimated variance-covariance matrix for all parameter estimates (if out_se=TRUE)

Xlabel

Structure of the design matrix, for internal use (if output=TRUE)

XX1dis

Design matrix for the covariates affecting the 1st latent variable (if output=TRUE)

XX2dis

Design matrix for the covariates affecting the 2st latent variable (if output=TRUE)

Details

In order to ensure the model identifiability, the following conditions must hold. First, the maximum number of items shared by the two latent variables is equal to the total number of items minus one, that is, the union of rows of multi1 must differ from the union of rows of multi2. Second, suitable constraints on the item parameters are required: one discriminanting index must be equal to 1 and one difficulty parameter must be equal to 0 for each dimension. The constrained items may be chosen in an arbitrary way: by default the algorithm select the first element of each row of multi1 and multi2. As a consequence, the user must pay attention to specify matrices multi1 and multi2 so that different items are constrained for each dimension.

References

Bacci, S. and Bartolucci, F. (2015), A multidimensional finite mixture SEM for non-ignorable missing responses to test items, Structural Equation Modeling, 22, 352-365.

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

Bacci, S., Bartolucci, F. and Gnaldi, M. (2014), A class of Multidimensional Latent Class IRT models for ordinal polytomous item responses, Communications in Statistics - Theory and Methods, 43, 787-800.

Examples

## Not run: 
# # Fit the model under different within-item multidimensional structures
# # for SF12_nomiss data
# data(SF12_nomiss)
# S = SF12_nomiss[,1:12]
# X = SF12_nomiss[,13]
# 
# # Graded response model with two latent variables sharing six items (free
# # discrimination and difficulty parameters; two latent classes for each
# # latent variable; one covariate):
# multi1 = c(1:5, 8:12)
# multi2 = c(6:12, 1)
# tol = 10^-6  # decrease tolerance to obtain more reliable results
# out1 = est_multi_poly_within(S=S,k1=2,k2=2,X=X,link="global",disc=TRUE,
#                              multi1=multi1,multi2=multi2,disp=TRUE,
#                              out_se=TRUE,tol=tol) 
#                              
# # Partial credit model with two latent variables sharing eleven items
# # (free discrimination and difficulty parameters; two latent classes for
# # the 1st latent variable and three latent classes for the 2nd latent
# # variable; one covariate):
# multi1 = 1:12
# multi2 = 2:12
# out2 = est_multi_poly_within(S=S,k1=2,k2=3,X=X,link="local",disc=TRUE,
#                              multi1=multi1,multi2=multi2,disp=TRUE,tol=tol)
#                                
# # Display output:
# out2$lk
# out2$Th1 
# out2$piv1
# out2$Th2
# out2$piv2
# out2$De1
# out2$De2
# ## End(Not run)

## Not run: 
# ## Fit the model under different situations for RLMS data
# # Example of use of the function to account for non-ignorable missing
# # item responses 
# data(RLMS)
# X = RLMS[,1:4]
# Y = RLMS[,6:9]
# YR = cbind(Y,1*(!is.na(Y)))
# multi1 = 1:4
# multi2 = 5:8
# tol = 10^-6  # decrease tolerance to obtain more reliable results
# 
# # MAR model
# out0 = est_multi_poly_within(YR,k1=3,k2=2,X=X,link="global",
#                  disc=TRUE,multi1=multi1,multi2=multi2,disp=TRUE,
#                  out_se=TRUE,glob=TRUE,tol=tol) 
#                  
# # NMAR model
# multi1 = 1:8
# out1 = est_multi_poly_within(YR,k1=3,k2=2,X=X,link="global",
#                  disc=TRUE,multi1=multi1,multi2=multi2,disp=TRUE,
#                  out_se=TRUE,glob=TRUE,tol=tol)
#                    
# # testing effect of the latent trait on missingness
# c(out0$bic,out1$bic)
# (test1 = out1$ga1c[-1]/out1$sega1c[-1])
# ## End(Not run)