MLCIRTwithin (version 2.1.1)

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 two-tier 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. The class of models is based on a particular within-item multidimensional formulation with each item loading on at most two uncorrelated latent variables. Maximum likelihood estimation is based on the 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, Phi = NULL,
                      ga1t = NULL, ga2t = NULL, De1 = NULL, De2 = NULL, fort = FALSE,
                      tol = 10^-10, maxitc = 10^4, disp = FALSE, output = FALSE,
                      out_se = FALSE, glob = FALSE, Zth1 = NULL, zth1 = NULL, Zth2=NULL,
                      zth2=NULL, Zbe=NULL, zbe=NULL, Zga1=NULL, zga1=NULL, Zga2=NULL,
                      zga2=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

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); difl = TRUE is only admitted in the presence of items with the same number of categories

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

Phi

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

ga1t

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

ga2t

initial value of the vector of free 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 2nd 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

maxitc

maximum number of iterations of the algorithm

disp

to display the likelihood evolution step by step

output

to return additional outputs (Piv1, Piv2, Pp1, Pp2, lkv, Xlabel, XX1dis, XX2dis)

out_se

to return standard errors

glob

to use global logits in the covariates

Zth1

matrix for the specification of constraints on the support points for the 1st latent variable

zth1

vector for the specification of constraints on the support points for the 1st latent variable

Zth2

matrix for the specification of constraints on the support points for the 2nd latent variable

zth2

vector for the specification of constraints on the support points for the 2nd latent variable

Zbe

matrix for the specification of constraints on the item difficulty parameters

zbe

vector for the specification of constraints on the item difficulty parameters

Zga1

matrix for the specification of constraints on the item discriminating indices for the 1st latent variable

zga1

vector for the specification of constraints on the item discriminating indices for the 1st latent variable

Zga2

matrix for the specification of constraints on the item discriminating indices for the 2nd latent variable

zga2

vector for the specification of constraints on the item discriminating indices for the 2nd latent variable

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

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

th1t

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

th2t

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

Th1

complete matrix of free and constrained ability levels for each dimension and latent class for the 1st latent variable

Th2

complete matrix of free and constrained ability levels for each dimension and latent class for the 2nd 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)

ga1t

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

ga2t

estimated vector of free discriminating indices for every item (with all elements equal to 1 if disc=FALSE) for the 2nd latent variable

ga1c

complete vector of free and constrained discriminating indices for every item for the 1st latent variable (with all elements equal to 1 if disc=FALSE and NA for items that do not load on the 1st latent variable)

ga2c

complete vector of free and constrained discriminating indices for every item for the 2nd latent variable (with all elements equal to 1 if disc=FALSE and NA for items that do not load on the 2nd latent variable)

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

Phi

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

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

piv1s

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

piv2s

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

Th1s

standardized ability levels for the 1st latent variable, ordered according to the first dimension

Th2s

standardized ability levels for the 2nd latent variable, ordered according to the first dimension

Becs

standardized values of item difficulty parameters

ga1cs

standardized values of item discriminating indices for the 1st latent variable

ga2cs

standardized values of item discriminating indices for the 2nd latent variable

call

call of function

Pp1

matrix of the posterior probabilities for each response configuration and latent class for the 1st latent variable (if output=TRUE)

Pp2

matrix of the posterior probabilities for each response configuration and latent class for the 2nd latent variable (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)

XX1dis

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

XX2dis

design matrix for the covariates affecting the 2nd latent variable (if output=TRUE)

Piv1

matrix of the weights for every covariate pattern configuration for the 1st latent variable (if output=TRUE)

Piv2

matrix of the weights for every covariate pattern configuration for the 2nd latent variable (if output=TRUE)

seth1t

standard errors for vector th1t (if out_se=TRUE)

seth2t

standard errors for vector th2t (if out_se=TRUE)

seTh1

standard errors for vector Th1 (if out_se=TRUE)

seTh2

standard errors for vector Th2 (if out_se=TRUE)

sebet

standard errors for vector bet (if out_se=TRUE)

seBec

standard errors for vector Bec (if out_se=TRUE)

sega1t

standard errors for vector ga1t (if out_se=TRUE)

sega2t

standard errors for vector ga2t (if out_se=TRUE)

sega1c

standard errors for vector ga1c (if out_se=TRUE)

sega2c

standard errors for vector ga2c (if out_se=TRUE)

seDe1

standard errors for vector De1 (if out_se=TRUE)

seDe2

standard errors for vector De2 (if out_se=TRUE)

Vnt

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

Vn

complete variance-covariance matrix for all parameters (if out_se=TRUE)

Details

In order to ensure the model identifiability, the following conditions must hold. First, 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 selects 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. Second, 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. These conditions may be skipped specifying in a suitable way the entries of Zth1, zth1, Zth2, zth2, Zbe, zbe, Zga1, zga1, Zga2, and zga2, according to the following equations:

Th1 = Zth1 %*% th1t + zth1

Th2 = Zth2 %*% th2t + zth2

Bec = Zbe %*% bet + zbe

ga1c = Zga1 %*% ga1t + zga1

ga2c = Zga2 %*% ga2t + zga2,

where Th1, Th2, Bec, ga1c, ga2c denote the complete matrices/vectors of support points (Th1, Th2), item difficulties (Bec), and item discriminating indices (ga1c, ga2c), whereas th1t, th2t, bet, ga1t, ga2t are the corresponding matrices/vectors of free (i.e., unconstrained) parameters.

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.

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.

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

Bonifay, W. E. (2015), An illustration of the two-tier item factor analysis model, in S. P. Reise and D. A. Revicki (eds), Handbook of Item Response Theory Modeling, p. 207-225, Routledge.

Cai, L. (2010), A two-tier full-information item factor analysis model with applications, Psychometrika, 75, 581-612.

Cai, L., Yang, J. S., and Hansen, M. (2011), Generalized full-information item bifactor analysis, Psychological Methods, 16, 221-248.

Examples

# NOT RUN {
# }
# 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:
summary(out2)
out2$lk
out2$Th1 
out2$Th1s
out2$piv1
out2$Th2
out2$Th2s
out2$piv2
out2$De1
out2$De2
# }
# 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])
# }
# NOT RUN {
# }
# NOT RUN {
## Fit the model under different external constraints on abilities and/or item parameters
data(SF12_nomiss)
S = SF12_nomiss[,1:12]
X = SF12_nomiss[,13]
multi1m = rbind(1:5, 8:12) # two dimensions for the 1st latent variable 
multi2m = rbind(6:9, c(10:12, 1)) # two dimensions for the 2nd latent variable 
k1 = 2
k2 = 2

# Fixed ability levels; all item parameters can be free
Zth1 = matrix(0,nrow(multi1m)*k1,0)
zth1 = c(rep(-1, times=nrow(multi1m)),  rep(1, times=nrow(multi1m)))
Zth2 = matrix(0,nrow(multi2m)*k2,0)
zth2 = c(rep(-1, times=nrow(multi2m)),  rep(1, times=nrow(multi2m)))  
# item difficulties: 10*4 + 2*2 = 44 (10 items with 5 categories plus 2 items with 3 categories)
Zbe = diag(44)
# item discriminating parameters = 10 items loading on the 1st latent variable plus 8 items loading
# on the 2nd latent variable
Zga1 = diag(10); Zga2 = diag(8) 
zga1 = rep(0,nrow(Zga1)); zga1[1] = 1
zga2 = rep(0,nrow(Zga2)); zga2[1] = 1
out1c = est_multi_poly_within(S=S,k1=k1,k2=k2,X=X,link="global",disc=TRUE,multi1=multi1m,
                              multi2=multi2m,disp=TRUE,out_se=TRUE,Zth1=Zth1,zth1=zth1,Zth2=Zth2,
                              zth2=zth2,Zbe=Zbe,Zga1=Zga1,zga1=zga1,Zga2=Zga2,zga2=zga2)   
summary(out1c)
out1c$Bec                             

# Constraint difficulties of the first threshold to be equal for all items 
# and difficulties of the second threshold to be equal for all items; 
# free ability levels
multi1u = c(1:3, 6:10) # one dimension for the 1st latent variable 
multi2u = c(4:10, 1)  # one dimension for the 2nd latent variable
S1 = pmin(as.matrix(S[, -c(2,3)]),2)  # all items have the same number of categories
Zbe = as.matrix((matrix(1,10,1)%x%diag(2))[,-1])  
out2c = est_multi_poly_within(S=S1,k1=2,k2=2,X=X,link="global",disc=TRUE,
                             multi1=multi1u,multi2=multi2u,disp=TRUE,
                             out_se=TRUE,Zbe=Zbe)
out2c$Bec   

# Same difficulties for pairs of items 1-6, 2-7, 3-8, 4-9, 5-10; 
# free ability levels
Zbe = (matrix(1,2,1)%x%diag(10))[,-1]
out3c = est_multi_poly_within(S=S1,k1=2,k2=2,X=X,link="global",disc=TRUE,
                             multi1=multi1u,multi2=multi2u,disp=TRUE,
                             out_se=TRUE,Zbe=Zbe)
out3c$Bec  

# Add equality constraints on some discriminating indices for the 1st latent variable
Zbe = (matrix(1,2,1)%x%diag(10))[,-1]
Zga1 = diag(length(multi1u)); 
# discriminating index of item 1 constrained to 1 for the model identifiability
# discriminating index of item 3 equal to discriminating index of item 2
Zga1 = Zga1[, -c(1, 3)];
Zga1[3, 1] = 1 
zga1 = rep(0,nrow(Zga1)); zga1[1] = 1
out4c = est_multi_poly_within(S=S1,k1=2,k2=2,X=X,link="global",disc=TRUE,
                             multi1=multi1u,multi2=multi2u,disp=TRUE,tol=10^-4,
                             out_se=TRUE,Zbe=Zbe, Zga1=Zga1, zga1=zga1)   
out4c$Bec 
out4c$ga1c
out4c$ga1t                                                           
# }