| Title: | Latent Class Item Response Theory (LC-IRT) Models under Within-Item Multidimensionality |
|---|---|
| Description: | Framework for the Item Response Theory analysis of dichotomous and ordinal polytomous outcomes under the assumption of within-item multidimensionality and discreteness of the latent traits. The fitting algorithms allow for missing responses and for different item parametrizations and are based on the Expectation-Maximization paradigm. Individual covariates affecting the class weights may be included in the new version together with possibility of constraints on all model parameters. |
| Authors: | Francesco Bartolucci, Silvia Bacci - University of Perugia (IT) |
| Maintainer: | Francesco Bartolucci <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 2.1.1 |
| Built: | 2024-12-04 07:06:27 UTC |
| Source: | CRAN |
This package provides a flexible framework for the estimation of discrete two-tier Item Response Theory (IRT) models for the analysis of dichotomous and ordinal polytomous item responses. The class of models at issue is based on the assumption that one or more items are shared by (at most) two latent traits (i.e., within-item multidimensionality) and on the discreteness of latent traits (abilities). Every level of the abilities identify a latent class of subjects. The fitting algorithms are based on the Expectation-Maximization (EM) paradigm and allow for missing responses and for different item parametrizations. The package also allows for the inclusion of individual covariates affecting the class weights together with possibility of constraints on all model parameters.
| Package: | MultiLCIRT |
| Type: | Package |
| Version: | 2.1.1 |
| Date: | 2019-09-30 |
| License: | GPL (>= 2) |
Function est_multi_poly_within performs the parameter estimation of the same model considered in the R package MultiLCIRT when one or more items are shared by two latent traits (within-item multidimensionality); in addition, fixed values and constraints on support points and item parameters are allowed.
Francesco Bartolucci, Silvia Bacci - University of Perugia (IT)
Maintainer: Francesco Bartolucci <[email protected]>
Adams, R., Wilson, M., and Wang, W. (1997), The multidimensional random coefficients multinomial logit, Applied Psychological Measurement, 21, 1-24.
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.
Bartolucci, F., Bacci, S., and Gnaldi, M. (2015), Statistical Analysis of Questionnaires: A Unified Approach Based on R and Stata, Chapman and Hall/CRC press.
Bartolucci, F., Bacci, S., and Gnaldi, M. (2014), MultiLCIRT: An R package for multidimensional latent class item response models, Computational Statistics and Data Analysis, 71, 971-985.
Bock, R.D., Gibbons, R., and Muraki, E. (1988), Full-information item factor analysis, Applied Psychological Measurement, 12, 261-280.
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.
Gibbons, R. D., Darrell, R. B., Hedeker, D., Weiss, D. J., Segawa, E., Bhaumik, D. K., and Stover, A. (2007), Full-information item bifactor analysis of graded response data, Applied Psychological Measurement, 31, 4-19.
Gibbons, R. D. and Hedeker, D. R. (1992), Full-information item bi-factor analysis, Psychometrika, 57, 423-436.
## Not run: # Estimation of a two-tier LC-IRT model data(SF12_nomiss) S = SF12_nomiss[,1:12] X = SF12_nomiss[,13] # Define matrices to allocate each item on the latent variables multi1=rbind(1:6, 7:12) multi2=rbind(4:8, c(2:3, 10:12)) # Graded response model with two primary latent variables, each of them # having two dimensions (free discrimination and difficulty parameters; # two latent classes for both the latent variables; one covariate): tol = 10^-6 # decrease the 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,tol=tol, disp=TRUE,out_se=FALSE) # Display output summary(out1) ## End(Not run)## Not run: # Estimation of a two-tier LC-IRT model data(SF12_nomiss) S = SF12_nomiss[,1:12] X = SF12_nomiss[,13] # Define matrices to allocate each item on the latent variables multi1=rbind(1:6, 7:12) multi2=rbind(4:8, c(2:3, 10:12)) # Graded response model with two primary latent variables, each of them # having two dimensions (free discrimination and difficulty parameters; # two latent classes for both the latent variables; one covariate): tol = 10^-6 # decrease the 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,tol=tol, disp=TRUE,out_se=FALSE) # Display output summary(out1) ## End(Not run)
Function that given two matrices builds the corresponding block diagonal matrix.
blkdiag(A, B)blkdiag(A, B)
A |
first matrix to be included |
B |
second matrix to be included |
C |
resulting block diagonal matrix |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_between, estimated abilities, item parameters, and regression coefficients are displayed
## S3 method for class 'est_multi_poly_between' coef(object, ...)## S3 method for class 'est_multi_poly_between' coef(object, ...)
object |
output from est_multi_poly_between |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_within, estimated abilities, item parameters, and regression coefficients are displayed for the 1st and the 2nd latent variable
## S3 method for class 'est_multi_poly_within' coef(object, ...)## S3 method for class 'est_multi_poly_within' coef(object, ...)
object |
output from est_multi_poly_within |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_between, the inferior and superior limits of confidence intervals at a given level are displayed for abilities, item parameters, and regression coefficients
## S3 method for class 'est_multi_poly_between' confint(object, parm, level=0.95, ...)## S3 method for class 'est_multi_poly_between' confint(object, parm, level=0.95, ...)
object |
output from est_multi_poly_between |
parm |
empity object |
level |
confidence level |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_within, the inferior and superior limits of confidence intervals at a given level are displayed for abilities, item parameters, and regression coefficients for the 1st and the 2nd latent variable
## S3 method for class 'est_multi_poly_within' confint(object, parm, level=0.95, ...)## S3 method for class 'est_multi_poly_within' confint(object, parm, level=0.95, ...)
object |
output from est_multi_poly_within |
parm |
empity object |
level |
confidence level |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
It estimates marginal regression models to datasets consisting of a categorical response and one or more covariates by a Fisher-scoring algorithm; this is an internal function that also works with response variables having a different number of response categories.
est_multi_glob_genZ(Y, X, model = c("m","l","g"), ind = 1:nrow(Y), de = NULL, Z = NULL, z = NULL, Dis = NULL, dis = NULL, disp=FALSE, only_sc = FALSE, Int = NULL, der_single = FALSE, maxit = 10)est_multi_glob_genZ(Y, X, model = c("m","l","g"), ind = 1:nrow(Y), de = NULL, Z = NULL, z = NULL, Dis = NULL, dis = NULL, disp=FALSE, only_sc = FALSE, Int = NULL, der_single = FALSE, maxit = 10)
Y |
matrix of response configurations |
X |
array of all distinct covariate configurations |
model |
type of logit (m = multinomial, l = local, g = global) |
ind |
vector to link responses to covariates |
de |
initial vector of regression coefficients |
Z |
design matrix |
z |
intercept associated with the design matrix |
Dis |
matrix for inequality constraints on de |
dis |
vector for inequality constraints on de |
disp |
to display partial output |
only_sc |
to exit giving only the score |
Int |
matrix of the fixed intercepts |
der_single |
to require single derivatives |
maxit |
maximum number of iterations |
be |
estimated vector of regression coefficients |
lk |
log-likelihood at convergence |
Pdis |
matrix of the probabilities for each distinct covariate configuration |
P |
matrix of the probabilities for each covariate configuration |
sc |
score for the vector of regression coefficients |
FI |
Fisher information matrix |
de |
estimated vector of (free) regression coefficients |
scde |
score for the vector of (free) regression coefficients |
FIde |
Fisher information matrix for the vector of (free) regression coefficients |
Sc |
matrix of individual scores for the vector of regression coefficients (if der_single=TRUE) |
Scde |
matrix of individual scores for the vector of (free) regression coefficients (if der_single=TRUE) |
Francesco Bartolucci - University of Perugia (IT)
Colombi, R. and Forcina, A. (2001), Marginal regression models for the analysis of positive association of ordinal response variables, Biometrika, 88, 1007-1019.
Glonek, G. F. V. and McCullagh, P. (1995), Multivariate logistic models, Journal of the Royal Statistical Society, Series B, 57, 533-546.
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.
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)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)
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 |
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 |
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) |
Francesco Bartolucci, Silvia Bacci - University of Perugia (IT)
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.
## 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)## 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)
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.
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)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)
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 |
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 |
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.
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) |
Francesco Bartolucci, Silvia Bacci - University of Perugia (IT)
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.
## 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$De2 ## End(Not run) ## Not run: ## Fit the model under different situations for RLMS data # Example of use of the function to account for non-ignorable missing # item responses data(RLMS) X = RLMS[,1:4] Y = RLMS[,6:9] YR = cbind(Y,1*(!is.na(Y))) multi1 = 1:4 multi2 = 5:8 tol = 10^-6 # decrease tolerance to obtain more reliable results # MAR model out0 = est_multi_poly_within(YR,k1=3,k2=2,X=X,link="global", disc=TRUE,multi1=multi1,multi2=multi2,disp=TRUE, out_se=TRUE,glob=TRUE,tol=tol) # NMAR model multi1 = 1:8 out1 = est_multi_poly_within(YR,k1=3,k2=2,X=X,link="global", disc=TRUE,multi1=multi1,multi2=multi2,disp=TRUE, out_se=TRUE,glob=TRUE,tol=tol) # testing effect of the latent trait on missingness c(out0$bic,out1$bic) (test1 = out1$ga1c[-1]/out1$sega1c[-1]) ## End(Not run) ## Not run: ## Fit the model under different external constraints on abilities and/or item parameters data(SF12_nomiss) S = SF12_nomiss[,1:12] X = SF12_nomiss[,13] multi1m = rbind(1:5, 8:12) # two dimensions for the 1st latent variable multi2m = rbind(6:9, c(10:12, 1)) # two dimensions for the 2nd latent variable k1 = 2 k2 = 2 # Fixed ability levels; all item parameters can be free Zth1 = matrix(0,nrow(multi1m)*k1,0) zth1 = c(rep(-1, times=nrow(multi1m)), rep(1, times=nrow(multi1m))) Zth2 = matrix(0,nrow(multi2m)*k2,0) zth2 = c(rep(-1, times=nrow(multi2m)), rep(1, times=nrow(multi2m))) # item difficulties: 10*4 + 2*2 = 44 (10 items with 5 categories plus 2 items with 3 categories) Zbe = diag(44) # item discriminating parameters = 10 items loading on the 1st latent variable plus 8 items loading # on the 2nd latent variable Zga1 = diag(10); Zga2 = diag(8) zga1 = rep(0,nrow(Zga1)); zga1[1] = 1 zga2 = rep(0,nrow(Zga2)); zga2[1] = 1 out1c = est_multi_poly_within(S=S,k1=k1,k2=k2,X=X,link="global",disc=TRUE,multi1=multi1m, multi2=multi2m,disp=TRUE,out_se=TRUE,Zth1=Zth1,zth1=zth1,Zth2=Zth2, zth2=zth2,Zbe=Zbe,Zga1=Zga1,zga1=zga1,Zga2=Zga2,zga2=zga2) summary(out1c) out1c$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$ga1t ## End(Not run)## Not run: # Fit the model under different within-item multidimensional structures # for SF12_nomiss data data(SF12_nomiss) S = SF12_nomiss[,1:12] X = SF12_nomiss[,13] # Graded response model with two latent variables sharing six items (free # discrimination and difficulty parameters; two latent classes for each # latent variable; one covariate): multi1 = c(1:5, 8:12) multi2 = c(6:12, 1) tol = 10^-6 # decrease tolerance to obtain more reliable results out1 = est_multi_poly_within(S=S,k1=2,k2=2,X=X,link="global",disc=TRUE, multi1=multi1,multi2=multi2,disp=TRUE, out_se=TRUE,tol=tol) # Partial credit model with two latent variables sharing eleven items # (free discrimination and difficulty parameters; two latent classes for # the 1st latent variable and three latent classes for the 2nd latent # variable; one covariate): multi1 = 1:12 multi2 = 2:12 out2 = est_multi_poly_within(S=S,k1=2,k2=3,X=X,link="local",disc=TRUE, multi1=multi1,multi2=multi2,disp=TRUE,tol=tol) # Display output: summary(out2) out2$lk out2$Th1 out2$Th1s out2$piv1 out2$Th2 out2$Th2s out2$piv2 out2$De1 out2$De2 ## 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) ## 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 ## End(Not run)
Function used within est_multi_poly_between to compute observed log-likelihood and score.
lk_obs_score_between(part_comp, lde, lpart, lgat, S, R, yv, k, rm, lv, J, fv, disc, glob, refitem, miss, ltype, XXdis, Xlabel, ZZ0,fort, Zpar, zpar, Zga, zga, items)lk_obs_score_between(part_comp, lde, lpart, lgat, S, R, yv, k, rm, lv, J, fv, disc, glob, refitem, miss, ltype, XXdis, Xlabel, ZZ0,fort, Zpar, zpar, Zga, zga, items)
part_comp |
complete vector of parameters |
lde |
length of de |
lpart |
length of part |
lgat |
length of gat |
S |
matrix of responses |
R |
matrix of observed responses indicator |
yv |
vector of frequencies |
k |
number of latent classes for the latent variable |
rm |
number of dimensions for the latent variable |
lv |
number of response categories for each item |
J |
number of items |
fv |
indicator of constrained parameters |
disc |
presence of discrimination parameters |
glob |
indicator of global parametrization for the covariates |
refitem |
vector of reference items |
miss |
indicator of presence of missing responses |
ltype |
type of logit |
XXdis |
array of covariates for the latent variable |
Xlabel |
indicator for covariate configuration |
ZZ0 |
design matrix |
fort |
to use Fortran |
Zpar |
array for the specification of constraints on the support points of the latent variable and for the item difficulty parameters |
zpar |
vector for the specification of constraints on the support points of the latent variable and for 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 |
items |
items affected by the latent variable |
lk |
log-likelihood function |
sc |
score vector |
Francesco Bartolucci - University of Perugia (IT)
Function used within est_multi_poly_within to compute observed log-likelihood and score.
lk_obs_score_within(part_comp, lde1, lde2, lpart, lga1t, lga2t, S, R, yv, k1, k2, rm1, rm2, lv, J, fv, disc, glob, refitem, miss, ltype, XX1dis, XX2dis, Xlabel, ZZ0, fort, Zpar, zpar, Zga1, zga1, Zga2, zga2, items1, items2)lk_obs_score_within(part_comp, lde1, lde2, lpart, lga1t, lga2t, S, R, yv, k1, k2, rm1, rm2, lv, J, fv, disc, glob, refitem, miss, ltype, XX1dis, XX2dis, Xlabel, ZZ0, fort, Zpar, zpar, Zga1, zga1, Zga2, zga2, items1, items2)
part_comp |
complete vector of parameters |
lde1 |
length of de1 |
lde2 |
length of de2 |
lpart |
length of part |
lga1t |
length of ga1t |
lga2t |
length of ga2t |
S |
matrix of responses |
R |
matrix of observed responses indicator |
yv |
vector of frequencies |
k1 |
number of latent classes for the 1st latent variable |
k2 |
number of latent classes for the 2nd latent variable |
rm1 |
number of dimensions for the 1st latent variable |
rm2 |
number of dimensions for the 2nd latent variable |
lv |
number of response categories for each item |
J |
number of items |
fv |
indicator of constrained parameters |
disc |
presence of discrimination parameters |
glob |
indicator of global parametrization for the covariates |
refitem |
vector of reference items |
miss |
indicator of presence of missing responses |
ltype |
type of logit |
XX1dis |
array of covariates for the 1st latent variable |
XX2dis |
array of covariates for the 2nd latent variable |
Xlabel |
indicator for covariate configuration |
ZZ0 |
design matrix |
fort |
to use Fortran |
Zpar |
array for the specification of constraints on the support points of the 1st and the 2nd latent variable and for the item difficulty parameters |
zpar |
vector for the specification of constraints on the support points of the 1st and the 2nd latent variable and for 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 |
items1 |
items affected by the 1st latent variable |
items2 |
items affected by the 2nd latent variable |
lk |
log-likelihood function |
sc |
score vector |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_between, the log-likelihood at convergence is displayed
## S3 method for class 'est_multi_poly_between' logLik(object, ...)## S3 method for class 'est_multi_poly_between' logLik(object, ...)
object |
output from est_multi_poly_between |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_within, the log-likelihood at convergence is displayed
## S3 method for class 'est_multi_poly_within' logLik(object, ...)## S3 method for class 'est_multi_poly_within' logLik(object, ...)
object |
output from est_multi_poly_within |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_between, the call of it is written
## S3 method for class 'est_multi_poly_between' print(x, ...)## S3 method for class 'est_multi_poly_between' print(x, ...)
x |
output from est_multi_poly_between |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_within, the call of it is written
## S3 method for class 'est_multi_poly_within' print(x, ...)## S3 method for class 'est_multi_poly_within' print(x, ...)
x |
output from est_multi_poly_within |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
It provides matrix of probabilities under different parametrizations and for the case of response variables having a different number of categories.
prob_multi_glob_gen(X, model, be, ind=(1:dim(X)[3]))prob_multi_glob_gen(X, model, be, ind=(1:dim(X)[3]))
X |
array of all distinct covariate configurations |
model |
type of logit (g = global, l = local, m = multinomial) |
be |
initial vector of regression coefficients |
ind |
vector to link responses to covariates |
Pdis |
matrix of distinct probability vectors |
P |
matrix of the probabilities for each covariate configuration |
Francesco Bartolucci - University of Perugia (IT)
Colombi, R. and Forcina, A. (2001), Marginal regression models for the analysis of positive association of ordinal response variables, Biometrika, 88, 1007-1019.
Glonek, G. F. V. and McCullagh, P. (1995), Multivariate logistic models, Journal of the Royal Statistical Society, Series B, 57, 533-546.
This dataset contains the data about job satisfaction described in: Bartolucci, F., Bacci, S., and Gnaldi, M. (2015), Statistical Analysis of Questionnaires: A Unified Approach Based on R and Stata, Chapman and Hall/CRC press
data(RLMS)data(RLMS)
A data frame with 1485 observations about four polytomous items with covariates:
maritalmarital status of the respondent
educationeducational level of the respondent
gendergender of the respondent
ageage of the respondent
workwork status of the respondent
Y11st item response
Y22nd item response
Y33rd item response
Y44th item response
Bartolucci, F., Bacci, S., and Gnaldi, M. (2015), Statistical Analysis of Questionnaires: A Unified Approach Based on R and Stata, Chapman and Hall/CRC press
data(RLMS) ## maybe str(RLMS) str(RLMS)data(RLMS) ## maybe str(RLMS) str(RLMS)
It searches for the global maximum of the log-likelihood of between-item muldimensional models given a vector of possible number of classes to try for.
search.model_between(S, yv = rep(1, ns), kv, X = NULL, link = c("global","local"), disc = FALSE, difl = FALSE, multi = 1:J, fort = FALSE, tol1 = 10^-6, tol2 = 10^-10, glob = FALSE, disp = FALSE, output = FALSE, out_se = FALSE, nrep = 2, Zth=NULL,zth=NULL, Zbe=NULL, zbe=NULL,Zga=NULL,zga=NULL)search.model_between(S, yv = rep(1, ns), kv, X = NULL, link = c("global","local"), disc = FALSE, difl = FALSE, multi = 1:J, fort = FALSE, tol1 = 10^-6, tol2 = 10^-10, glob = FALSE, disp = FALSE, output = FALSE, out_se = FALSE, nrep = 2, Zth=NULL,zth=NULL, Zbe=NULL, zbe=NULL,Zga=NULL,zga=NULL)
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 |
kv |
vector of the possible numbers of latent classes |
X |
matrix of covariates affecting the weights |
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) |
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 |
fort |
to use Fortran routines when possible |
tol1 |
tolerance level for checking convergence of the algorithm as relative difference between consecutive log-likelihoods (initial check based on random starting values) |
tol2 |
tolerance level for checking convergence of the algorithm as relative difference between consecutive log-likelihoods (final converngece) |
glob |
to use global logits in the covariates |
disp |
to display the likelihood evolution step by step |
output |
to return additional outputs (Piv,Pp,lkv) |
out_se |
to return standard errors |
nrep |
number of repetitions of each random initialization |
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 |
out.single |
output of each single model for each |
aicv |
Akaike Information Criterion index for each |
bicv |
Bayesian Information Criterion index for each |
entv |
Entropy index for each |
necv |
NEC index for each |
lkv |
log-likelihood at convergence of the EM algorithm for each |
errv |
trace of any errors occurred during the estimation process for each |
Francesco Bartolucci, Silvia Bacci - University of Perugia (IT)
Bartolucci, F., Bacci, S. and Gnaldi, M. (2014), MultiLCIRT: An R package for multidimensional latent class item response models, Computational Statistics & Data Analysis, 71, 971-985.
## Not run: # Fit a Graded response model with two latent variables (free discrimination # and difficulty parameters; two 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)) out1 = search.model_between(S=S,kv=1:3,X=X,link="global",disc=TRUE, multi=multi0,fort=TRUE,disp=TRUE,out_se=TRUE) # Display output out1$lkv out1$bicv # Display output with 2 classes: out1$out.single[[2]] out1$out.single[[2]]$lktrace out1$out.single[[2]]$Th out1$out.single[[2]]$piv out1$out.single[[2]]$gac out1$out.single[[2]]$Bec ## End(Not run)## Not run: # Fit a Graded response model with two latent variables (free discrimination # and difficulty parameters; two latent classes): data(SF12_nomiss) S = SF12_nomiss[,1:12] X = SF12_nomiss[,13] multi0 = rbind(c(1:5, 8), c(6:7,9:12)) out1 = search.model_between(S=S,kv=1:3,X=X,link="global",disc=TRUE, multi=multi0,fort=TRUE,disp=TRUE,out_se=TRUE) # Display output out1$lkv out1$bicv # Display output with 2 classes: out1$out.single[[2]] out1$out.single[[2]]$lktrace out1$out.single[[2]]$Th out1$out.single[[2]]$piv out1$out.single[[2]]$gac out1$out.single[[2]]$Bec ## End(Not run)
It searches for the global maximum of the log-likelihood of within-item muldimensional models given a vector of possible number of classes to try for.
search.model_within(S, yv = rep(1, ns), kv1, kv2, X = NULL, link = c("global","local"), disc = FALSE, difl = FALSE, multi1, multi2, fort = FALSE, tol1 = 10^-6, tol2 = 10^-10, glob = FALSE, disp = FALSE, output = FALSE, out_se = FALSE, nrep = 2, Zth1 = NULL, zth1 = NULL, Zth2=NULL, zth2=NULL, Zbe=NULL, zbe=NULL, Zga1=NULL, zga1=NULL, Zga2=NULL, zga2=NULL)search.model_within(S, yv = rep(1, ns), kv1, kv2, X = NULL, link = c("global","local"), disc = FALSE, difl = FALSE, multi1, multi2, fort = FALSE, tol1 = 10^-6, tol2 = 10^-10, glob = FALSE, disp = FALSE, output = FALSE, out_se = FALSE, nrep = 2, Zth1 = NULL, zth1 = NULL, Zth2=NULL, zth2=NULL, Zbe=NULL, zbe=NULL, Zga1=NULL, zga1=NULL, Zga2=NULL, zga2=NULL)
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 |
kv1 |
vector of the possible numbers of ability levels (or latent classes) for the 1st latent variable |
kv2 |
vector of the possible numbers of ability levels (or latent classes) for the 2nd latent variable |
X |
matrix of covariates affecting the weights |
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 |
fort |
to use Fortran routines when possible |
tol1 |
tolerance level for checking convergence of the algorithm as relative difference between consecutive log-likelihoods (initial check based on random starting values) |
tol2 |
tolerance level for checking convergence of the algorithm as relative difference between consecutive log-likelihoods (final convergence) |
glob |
to use global logits in the covariates |
disp |
to display the likelihood evolution step by step |
output |
to return additional outputs (Piv,Pp,lkv) |
out_se |
to return standard errors |
nrep |
number of repetitions of each random initialization |
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 |
out.single |
output of each single model for each |
aicv |
Akaike Information Criterion index for each |
bicv |
Bayesian Information Criterion index for each |
entv |
Entropy index for each |
necv |
NEC index for each |
lkv |
log-likelihood at convergence of the EM algorithm for each |
errv |
trace of any errors occurred during the estimation process for each |
Francesco Bartolucci, Silvia Bacci - University of Perugia (IT)
Bartolucci, F., Bacci, S. and Gnaldi, M. (2014), MultiLCIRT: An R package for multidimensional latent class item response models, Computational Statistics & Data Analysis, 71, 971-985.
## 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] # Partial credit model with two latent variables sharing six items # (free difficulty parameters and constrained discriminating parameters; # 1 to 3 latent classes for the 1st latent variable and 1 to 2 classes for the 2nd latent variable; # one covariate): multi1 = c(1:5, 8:12) multi2 = c(6:12, 1) out1 = search.model_within(S=S,kv1=1:3,kv2=1:2,X=X,link="global",disc=FALSE, multi1=multi1,multi2=multi2,disp=TRUE, out_se=TRUE,tol1=10^-4, tol2=10^-7, nrep=1) # Main output out1$lkv out1$aicv out1$bicv # Model with 2 latent classes for each latent variable out1$out.single[[4]]$k1 out1$out.single[[4]]$k2 out1$out.single[[4]]$Th1 out1$out.single[[4]]$Th2 out1$out.single[[4]]$piv1 out1$out.single[[4]]$piv2 out1$out.single[[4]]$ga1c out1$out.single[[4]]$ga2c out1$out.single[[4]]$Bec ## End(Not run)## Not run: # Fit the model under different within-item multidimensional structures # for SF12_nomiss data data(SF12_nomiss) S = SF12_nomiss[,1:12] X = SF12_nomiss[,13] # Partial credit model with two latent variables sharing six items # (free difficulty parameters and constrained discriminating parameters; # 1 to 3 latent classes for the 1st latent variable and 1 to 2 classes for the 2nd latent variable; # one covariate): multi1 = c(1:5, 8:12) multi2 = c(6:12, 1) out1 = search.model_within(S=S,kv1=1:3,kv2=1:2,X=X,link="global",disc=FALSE, multi1=multi1,multi2=multi2,disp=TRUE, out_se=TRUE,tol1=10^-4, tol2=10^-7, nrep=1) # Main output out1$lkv out1$aicv out1$bicv # Model with 2 latent classes for each latent variable out1$out.single[[4]]$k1 out1$out.single[[4]]$k2 out1$out.single[[4]]$Th1 out1$out.single[[4]]$Th2 out1$out.single[[4]]$piv1 out1$out.single[[4]]$piv2 out1$out.single[[4]]$ga1c out1$out.single[[4]]$ga2c out1$out.single[[4]]$Bec ## End(Not run)
This data set contains the responses of 620 oncological patients to 12 ordinal polytomous items that measure the health-related quality of life, according to the Italian release of Short-Form 12 version 2 (SF-12v2); patient's age is also provided.
data(SF12)data(SF12)
A dataframe with 620 observations on 12 items and one covariate:
Y1general health
Y2limits in moderate activities
Y3limits in climbing several flights of stairs
Y4accomplished less than he/she would like, as a result of his/her physical health
Y5limited in the kind of work or daily activities, as a result of his/her physical health
Y6accomplished less than he/she would like, as a result of his/her emotional health
Y7did work less carefully than usual, as a result of his/her emotional health
Y8how much did pain interfere with normal work
Y9how much of the time have he/she felt calm and peaceful
Y10how much of the time did he/she have a lot of energy
Y11how much of the time have he/she felt downhearted and depressed
Y12how much of the time physical health or emotional health interfered with social activities
ageage of the respondent
All items have 5 response categories, with the exception of items Y2 and Y3 having 3 response categories: the minimum value 0 correspond to a low level of quality of life, whereas the maximum value corresponds to a high level of quality of life. A proportion of 0.205 patients (127 out of 620) has missing responses (NA) on one or more items.
Ware, J., Kosinski, M., Turner-Bowker, D. and Gandek, B. (2002), SF-12v2. How to score version 2 of the SF-12 health survey, QualityMetric Incorporated: Lincoln.
data(SF12) dim(SF12) ## maybe str(SF12) str(SF12)data(SF12) dim(SF12) ## maybe str(SF12) str(SF12)
This data set contains the responses of 493 oncological patients to 12 ordinal polytomous items that measure the health-related quality of life, according to the Italian release of Short-Form 12 version 2 (SF-12v2); patient's age is also provided.
data(SF12)data(SF12)
A dataframe with 493 observations on 12 items and one covariate:
Y1general health
Y2limits in moderate activities
Y3limits in climbing several flights of stairs
Y4accomplished less than he/she would like, as a result of his/her physical health
Y5limited in the kind of work or daily activities, as a result of his/her physical health
Y6accomplished less than he/she would like, as a result of his/her emotional health
Y7did work less carefully than usual, as a result of his/her emotional health
Y8how much did pain interfere with normal work
Y9how much of the time have he/she felt calm and peaceful
Y10how much of the time did he/she have a lot of energy
Y11how much of the time have he/she felt downhearted and depressed
Y12how much of the time physical health or emotional health interfered with social activities
ageage of the respondent
All items have 5 response categories, with the exception of items Y2 and Y3 having 3 response categories: the minimum value 0 correspond to a low level of quality of life, whereas the maximum value corresponds to a high level of quality of life. All records are complete.
Ware, J., Kosinski, M., Turner-Bowker, D. and Gandek, B. (2002), SF-12v2. How to score version 2 of the SF-12 health survey, QualityMetric Incorporated: Lincoln.
data(SF12_nomiss) dim(SF12_nomiss) ## maybe str(SF12_nomiss) str(SF12_nomiss)data(SF12_nomiss) dim(SF12_nomiss) ## maybe str(SF12_nomiss) str(SF12_nomiss)
Given the output from est_multi_poly_between, it is written in a readable form
## S3 method for class 'est_multi_poly_between' summary(object, ...)## S3 method for class 'est_multi_poly_between' summary(object, ...)
object |
output from est_multi_poly_between |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_within, it is written in a readable form
## S3 method for class 'est_multi_poly_within' summary(object, ...)## S3 method for class 'est_multi_poly_within' summary(object, ...)
object |
output from est_multi_poly_within |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_between, the estimated variance-and-covariance matrix is displayed
## S3 method for class 'est_multi_poly_between' vcov(object, ...)## S3 method for class 'est_multi_poly_between' vcov(object, ...)
object |
output from est_multi_poly_between |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)
Given the output from est_multi_poly_within, the estimated variance-and-covariance matrix is displayed
## S3 method for class 'est_multi_poly_within' vcov(object, ...)## S3 method for class 'est_multi_poly_within' vcov(object, ...)
object |
output from est_multi_poly_within |
... |
further arguments passed to or from other methods |
Francesco Bartolucci - University of Perugia (IT)