R: Estimate multidimensional and multilevel LC IRT model for...
est_multi_poly_clust
R Documentation
Estimate multidimensional and multilevel LC IRT model for dichotomous and polytomous responses
Description
The function performs maximum likelihood estimation of the parameters of the IRT models
assuming a discrete distribution for the ability and a discrete distribution for the latent variable
at cluster level. 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_clust(S, kU, kV, W = NULL, X = NULL, clust,
start = 0, link = 0, disc = 0, difl = 0,
multi = 1:J, piv = NULL, Phi = NULL,
gac = NULL, DeU = NULL, DeV = NULL,
fort = FALSE, tol = 10^-10, disp = FALSE,
output = 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)
kU
number of support points (or latent classes at cluster level)
kV
number of ability levels (or latent classes at individual level)
W
matrix of covariates that affects the weights at cluster level
X
matrix of covariates that affects the weights at individual level
clust
vector of cluster indicator for each unit
start
method of initialization of the algorithm (0 = deterministic, 1 = random, 2 = arguments given as
input)
link
type of link function (0 = no link function, 1 = global logits, 2 = local logits);
with no link function the Latent Class model results; with global logits the Graded Response model
results; with local logits the Partial Credit 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 (0 = all equal to one, 1 = free)
difl
indicator of constraints on the difficulty levels (0 = free, 1 = rating scale parameterization)
multi
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
piv
initial value of the vector of weights of the latent classes (if start=2)
Phi
initial value of the matrix of the conditional response probabilities (if start=2)
gac
initial value of the complete vector of discriminating indices (if start=2)
DeU
initial value of regression coefficients for the covariates in W (if start=2)
DeV
initial value of regression coefficients for the covariates in X (if start=2)
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 (Phi,Pp,Piv)
Value
piv
estimated vector of weights of the latent classes (average of the weights in case of model with covariates)
Th
estimated matrix of ability levels for each dimension and latent class
Bec
estimated vector of difficulty levels for every item (split in two vectors if difl=1)
gac
estimated vector of discriminating indices for every item (with all elements equal to 1
with Rasch parametrization)
fv
vector indicating the reference item chosen for each latent dimension
Phi
array of the conditional response probabilities for every item and latent class
De
matrix of regression coefficients for the multinomial logit model on the class weights
Piv
matrix of the weights for every response configuration (if output=TRUE)
Pp
matrix of the posterior probabilities for each response configuration and latent class
(if output=TRUE)
lk
log-likelhood at convergence of the EM algorithm
np
number of free parameters
aic
Akaike Information Criterion index
bic
Bayesian Information Criterion index
ent
Etropy index to measure the separation of classes
lkv
Vector to trace the log-likelihood evolution across iterations (if output=TRUE)
seDe
Standard errors for De (if output=TRUE)
separ
Standard errors for vector of parameters containing Th and Be (if output=TRUE)
sega
Standard errors for vector of discrimination indices (if output=TRUE)
Vn
Estimated variance-covariance matrix for all parameter estimates (if output=TRUE)
Author(s)
Francesco Bartolucci, Silvia Bacci, Michela Gnaldi - University of Perugia (IT)
References
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, Communication in Statistics - Theory and Methods, 43, 787-800.
Examples
## Not run:
# generate covariate at cluster level
nclust = 200
W = matrix(round(rnorm(nclust)*2,0)/2,nclust,1)
la = exp(W)/(1+exp(W))
U = 1+1*(runif(nclust)<la)
clust = NULL
for(h in 1:nclust){
nh = round(runif(1,5,20))
clust = c(clust,h*rep(1,nh))
}
n = length(clust)
# generate covariates
DeV = rbind(c(1.75,1.5),c(-0.25,-1.5),c(-0.5,-1),c(0.5,1))
X = matrix(round(rnorm(2*n)*2,0)/2,n,2)
Piv = cbind(0,cbind(U[clust]==1,U[clust]==2,X)%*%DeV)
Piv = exp(Piv)*(1/rowSums(exp(Piv)))
V = rep(0,n)
for(i in 1:n) V[i] = which(rmultinom(1,1,Piv[i,])==1)
# generate responses
la = c(0.2,0.5,0.8)
Y = matrix(0,n,10)
for(i in 1:n) Y[i,] = runif(10)<la[V[i]]
# fit the model with k1=3 and k2=2 classes
out1 = est_multi_poly_clust(Y,kU=2,kV=3,W=W,X=X,clust=clust)
out2 = est_multi_poly_clust(Y,kU=2,kV=3,W=W,X=X,clust=clust,disp=TRUE,
output=TRUE)
out3 = est_multi_poly_clust(Y,kU=2,kV=3,W=W,X=X,clust=clust,disp=TRUE,
output=TRUE,start=2,Phi=out2$Phi,gac=out2$gac,
DeU=out2$DeU,DeV=out2$DeV)
# Rasch
out4 = est_multi_poly_clust(Y,kU=2,kV=3,W=W,X=X,clust=clust,link=1,
disp=TRUE,output=TRUE)
out5 = est_multi_poly_clust(Y,kU=2,kV=3,W=W,X=X,clust=clust,link=1,
disc=1,disp=TRUE,output=TRUE)
## End(Not run)
providing 
.