MLCIRTwithin (version 1.0)

est_multi_poly_within: Estimate LC IRT model for dichotomous and polytomous responses under within multidimensionality

Description

The function performs maximum likelihood estimation of the parameters of the IRT models assuming a discrete distribution for the ability and within 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 = 0, link = 1,
        disc = 0, difl = 0, multi1 = 1:J, multi2 = 1:J, piv1 = NULL, piv2 = NULL, 
        Phi = NULL, gac = NULL, De = 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 (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 Credi

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)

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=2) for the 1st latent variable

piv2

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

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)

De

initial value of regression coefficients for the covariates (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)

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 2nd 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=1)
• 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
• 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)
• seDe1
  Standard errors for De (if output=TRUE) for the 1st latent variable
• seDe2
  Standard errors for De (if output=TRUE) for the 2nd latent variable
• separ
  Standard errors for vector of parameters containing Th and Be (if out_se=TRUE)
• sega1
  Standard errors for vector of discrimination indices (if out_se=TRUE) for the 1st latent variable
• sega2
  Standard errors for vector of discrimination indices (if out_se=TRUE) for the 2nd latent variable
• Vn
  Estimated variance-covariance matrix for all parameter estimates (if output=TRUE)

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

## Fit the model under different situations for NAEP data
data(naep)
multi1 = 1:12
multi2 = 2:12
# Rasch model and 2PL with two dimensions (within)
out10 = est_multi_poly_within(naep,k1=3,k2=3,multi1=multi1,multi2=multi2,link=1,disp=TRUE,
                  out_se=TRUE) 
out11 = est_multi_poly_within(naep,k1=3,k2=3,multi1=multi1,multi2=multi2,link=1,disc=1,
                  disp=TRUE,out_se=TRUE)

# fit the model under different situations for RLMS data
# example of use of the function to account for non-ignorable missing responses
data(RLMS)
X = RLMS[,1:4]
Y = RLMS[,6:9]
YR = cbind(Y,1*(!is.na(Y)))
multi1 = 1:4
multi2 = 5:8
# MAR model
out0 = est_multi_poly_within(YR,X=X,k1=3,k2=2,multi1=multi1,multi2=multi2,link=1,difl=1,
                 disc=1,glob=1,disp=TRUE,out_se=TRUE) 
# NMAR model
multi1 = 1:8
out1 = est_multi_poly_within(YR,X=X,k1=3,k2=2,multi1=multi1,multi2=multi2,link=1,difl=1,
                 disc=1,glob=1,disp=TRUE,out_se=TRUE) 
# testing effect of the latent trait on missingness
c(out0$bic,out1$bic)
test1 = out1$ga1c[-1]/out1$sega1