est_multi_poly_between {MLCIRTwithin}

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

Description

The function performs maximum likelihood estimation of the parameters of the IRT models assuming a discrete distribution for the ability and between-item multidimensionality. Every ability level corresponds to a latent class of subjects in the reference population. The class of models is based on a between-item multidimensional formulation with each item loading on a dimension of a given latent variable. Maximum likelihood estimation is based on Expectation- Maximization algorithm.

Usage

est_multi_poly_between(S, yv = rep(1, ns), k, X = NULL, start = c("deterministic",
                       "random","external"), link = c("global","local"), disc = FALSE,
                       difl = FALSE, multi = 1:J, Phi = NULL, gat = NULL, De = NULL,
                       fort = FALSE, tol = 10^-10, maxitc = 10^4, disp = FALSE,
                       output = FALSE, out_se = FALSE, glob = FALSE, Zth=NULL,zth=NULL,
                       Zbe=NULL, zbe=NULL,Zga=NULL,zga=NULL)

Arguments

S

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

yv

vector of the frequencies of every response configuration in S

k

number of ability levels (or latent classes) for the 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); difl = TRUE is only admitted in the presence of items with the same number of categories

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 for the latent variable

Phi

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

gat

initial value of the vector of free discriminating indices (if start="external")

De

initial value of regression coefficients for the covariates (if start="external")

fort

to use Fortran routines when possible

tol

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

maxitc

maximum number of iterations of the algorithm

disp

to display the likelihood evolution step by step

output

to return additional outputs (Piv,Pp,lkv, Xlabel, XXdis)

out_se

to return standard errors

glob

to use global logits in the covariates

Zth

matrix for the specification of constraints on the support points

zth

vector for the specification of constraints on the support points

Zbe

matrix for the specification of constraints on the item difficulty parameters

zbe

vector for the specification of constraints on the item difficulty parameters

Zga

matrix for the specification of constraints on the item discriminating indices

zga

vector for the specification of constraints on the item discriminating indices

Value

piv

estimated vector of weights of the latent classes (average of the weights in case of model with covariates)

fv

vector indicating the reference item chosen for each latent dimension of the latent variable

tht

estimated matrix of free ability levels for each dimension

Th

complete matrix of free and constrained ability levels for each dimension and latent class of the latent variable

bet

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

Bec

complete vector of free and constrained difficulty levels for every item (split in two vectors if difl=TRUE)

gat

estimated vector of free discriminating indices for every item (with all elements equal to 1 if disc=FALSE)

gac

complete vector of free and constrained discriminating indices for every item (with all elements equal to 1 if disc=FALSE)

De

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

Phi

array of the conditional response probabilities for every item and each of the k latent classes

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

entropy index to measure the separation of classes

pivs

estimated vector of (ordered) weights of the latent classes (average of the weights in case of model with covariates)

Ths

standardized ability levels

Becs

standardized values of item difficulty parameters

gacs

standardized values of item discriminating indices

call

call of function

Pp

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

lkv

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

Xlabel

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

XXdis

design matrix for the covariates affecting the latent variable (if output=TRUE)

Piv

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

setht

standard errors for vector tht (if out_se=TRUE)

seTh

standard errors for vector Th (if out_se=TRUE)

sebet

standard errors for vector bet (if out_se=TRUE)

seBec

standard errors for vector Bec (if out_se=TRUE)

segat

standard errors for vector gat (if out_se=TRUE)

segac

standard errors for vector gac (if out_se=TRUE)

seDe

standard errors for vector De (if out_se=TRUE)

Vnt

estimated variance-covariance matrix for free parameter estimates (if out_se=TRUE)

Vn

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

Author(s)

Francesco Bartolucci, Silvia Bacci - 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, Communications in Statistics - Theory and Methods, 43, 787-800.

Examples

## Not run: 
# Fit a Graded response model with two dimensions (free discrimination
# and difficulty parameters; three latent classes):
data(SF12_nomiss)
S = SF12_nomiss[,1:12]
X = SF12_nomiss[,13]
multi0 = rbind(c(1:5, 8), c(6:7,9:12))
k=3

out1 =  est_multi_poly_between(S=S,k=k,X=X,link="global",disc=TRUE,
                               multi=multi0,fort=TRUE,disp=TRUE,out_se=TRUE) 

# Display output:
summary(out1)
out1$lk
out1$Th
out1$piv
out1$De

## End(Not run)

## Not run: 
## Fit the model under different external constraints on abilities and/or item parameters
# Fixed ability levels; all item parameters can be free  
S1 = pmin(as.matrix(S),2) # all items have the same number of categories
Zth = matrix(0,nrow(multi0)*k,0)
zth = c(rep(-1, times=nrow(multi0)), rep(0, times=nrow(multi0)),  rep(1, times=nrow(multi0)))
Zbe = diag(ncol(S1)*2)  # free item difficulties: 12*2  = 24 (12 items with 3 categories)
Zga = diag(ncol(S1));  # free item discriminating parameters = 12 items loading on U
outc1 = est_multi_poly_between(S=S1,k=k,X=X,link="global",disc=TRUE, multi=multi0,disp=TRUE,
                               out_se=TRUE,Zth=Zth,zth=zth,Zbe=Zbe,Zga=Zga) 
outc1$Th
outc1$tht
outc1$Bec                                                          

# Add equality constraints on item parameters                         
# Same difficulties for pairs of items 1-7, 2-8, 3-9, 4-10, 5-11, 6-12; 
# same discriminating indices for items 2 and 3;
# free ability levels
Zbe = (matrix(1,2,1)%x%diag(12))[,-1]
Zga = as.matrix(rep(0, times=12)); Zga[2,1] = 1; Zga[3,1] = 1; 
Zga1p1 = matrix(0, nrow=3, ncol=9); Zga1p2 = diag(9); Zga1p = rbind(Zga1p1, Zga1p2)
Zga = cbind(Zga, Zga1p)
# discriminating index of item 1 constrained to 1 for the model identifiability
zga = rep(0,nrow(Zga)); zga[1] = 1 
outc2 = est_multi_poly_between(S=S1,k=k,X=X,link="global",disc=TRUE, 
                             multi = multi0,disp=TRUE,tol=10^-4,
                             out_se=TRUE,Zbe=Zbe, Zga=Zga, zga=zga)
outc2$tht
outc2$Th
outc2$Ths
outc2$Bec
outc2$Becs
outc2$gac 
outc2$gacs

## End(Not run)