Polytomous Logit-Normit (Graded Logistic) Model Estimation

Description

Performs bivariate composite likelihood and full information maximum likelihood estimation for polytomous logit-normit (graded logistic) item response theory (IRT) models.

Details

This package currently contains several functions performing estimation of unidimensional (single latent trait) polytomous logit-normit models (also known graded logistic) using bivariate composite likelihood and full information maximum likelihood estimation.

Acknowledgment

Some code from the ltm package (version 0.9-7) was modified for counting the frequency of response patterns.

Author(s)

Carl Falk cffalk@gmail.com, Harry Joe

Maintainer: Carl F. Falk cffalk@gmail.com

References

Bartholomew, D., Knott, M., and Moustaki, I. (2011). Latent Variable Models and Factor Analysis: A Unified Approach, 3rd Edition. Wiley.

Maydeu-Olivares, A., and Joe, H. (2005). Limited and full information estimation and goodness-of-fit testing in 2^n contingency tables: A unified framework. Journal of the American Statistical Association, 100, 1009-1020.

Maydeu-Olivares, A., and Joe, H. (2006). Limited information and goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713-732.

Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 5-42.

5-item Test Data Set

Description

A simulated data set with a 5 items, each with 3 ordinal categories labeled as 0,1,2. Each row corresponds to a response pattern with the sixth column containing the frequency of each response pattern.

Usage

data(item5fr)

9 Item Test Data Set

Description

A simulated data set with a 9 items, each with 5 ordinal categories labeled as 0,1,2,3,4. Each row corresponds to a single individual's response on the 9 items.

Usage

data(item9cat5)

Full information maximum likelihood and bivariate composite likelihood estimation for polytomous logit-normit (graded logistic) model

Description

Full information maximum likelihood and bivariate composite likelihood estimation for polytomous logit-normit and Rasch models, via Newton Raphson iterations.

Usage

nrmlepln(x, ncat, nitem=NULL, alphas=NULL, betas=NULL, abound=c(-10,10),
         bbound=c(-1,10), nq=48, mxiter=200, m2=TRUE, iprint=FALSE)
nrmlerasch(x, ncat, nitem=NULL, alphas=NULL, abound=c(-10,10),
         bbound=c(-1,10), nq=48, mxiter=200, m2=TRUE, iprint=FALSE)
nrbcpln(x, ncat, nitem=NULL, alphas=NULL, betas=NULL, abound=c(-10,10),
         bbound=c(-1,10), nq=48, mxiter=200, se=TRUE, iprint=FALSE)

Arguments

Details

Estimation of graded logistic models is performed under the following parameterization:

Pr(y_i = k_i| \eta) = \left\{ \begin{array}{ll} 1-\Psi (\alpha_{i,k} + \beta_i \eta) & \mbox{if } k_i = 0\\ \Psi (\alpha_{i,k} + \beta_i \eta) - \Psi (\alpha_{i,k+1} + \beta_i \eta) & \mbox{if } 0 < k_i < m-1\\ \Psi (\alpha_{i,k+1} + \beta_i \eta) & \mbox{if } k_i = m-1 \end{array} \right.

Where the items are y_i, i = 1, \dots, n, and response categories are k=0, \dots, m-1. \eta is the latent trait, \Psi is the logistic distribution function, \alpha is an intercept (cutpoint) parameter, and \beta is a slope parameter. When the number of categories for the items is 2, this reduces to the 2PL parameterization:

Pr(y_i = 1| \eta) = \Psi (\alpha_1 + \beta_i \eta)

Both nrmlepln and nrbcpln perform estimation under these parameterizations, via Newton Raphson iterations, using full information maximum likelihood (nrmlepln) and bivariate composite likelihood (nrbcpln). See Maydeu-Olivares and Joe (2005, 2006) for more information on bivariate composite likelihood estimation (see also Varin, Reid, and Firth, 2011). Under nrmlerasch a common \beta parameter is estimated for all items.

Value

A list containing the following slots.

Slots

alphas

A vector of parameter estimates for alphas. Length is nitem\times(ncat-1). Estimates are in order by item, e.g., all alphas for item 1, followed by all alphas for item 2, and so on.

betas

A vector of parameter estimates for betas. Length is nitem.

nllk

Negative (composite) log-likelihood for polytomous logit-normit (or Rasch) model.

conv

Integer indicating whether estimation converged. Currently only returned for composite likelihood estimation.

sealphas

A vector of standard errors for the alpha estimates.

sebetas

A vector of standard errors for the beta estimates.

invhes

Inverse Hessian matrix for the MLE estimates.

vcov

Asymptotic covariance matrix for the composite likelihood estimates.

teststat

Value of M_2.

df

Degrees of freedom for M_2.

pval

P-value for M_2.

Author(s)

Carl F. Falk cffalk@gmail.com, Harry Joe

References

Bartholomew, D., Knott, M., and Moustaki, I. (2011). Latent Variable Models and Factor Analysis: A Unified Approach, 3rd Edition. Wiley.

Maydeu-Olivares, A., and Joe, H. (2005). Limited and full information estimation and goodness-of-fit testing in 2^n contingency tables: A unified framework. Journal of the American Statistical Association, 100, 1009-1020.

Maydeu-Olivares, A., and Joe, H. (2006). Limited information and goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713-732.

Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 5-42.

See Also

startalphas startbetas

Examples

 ### Matrix of response patterns and frequencies
 data(item5fr)
 
 ## ML estimation
 nrmleplnout<-nrmlepln(item5fr, ncat=3, nitem=5)
 print(nrmleplnout)
 
 ## BCL estimation
 nrbcplnout<-nrbcpln(item5fr, ncat=3, nitem=5)
 print(nrbcplnout)
 
 ## ML Rasch estimation
 nrmleraschout<-nrmlerasch(item5fr, ncat=3, nitem=5)
 print(nrmleraschout)
 
 
 ### Raw data
 data(item9cat5)
 
 ## ML estimation
 nrmleplnout<-nrmlepln(item9cat5, ncat=5)
 print(nrmleplnout)
 
 ## BCL estimation
 nrbcplnout<-nrbcpln(item9cat5, ncat=5, se=FALSE)
 print(nrbcplnout)
 
 ## ML Rasch estimation
 nrmleraschout<-nrmlerasch(item9cat5, ncat=5)
 print(nrmleraschout)
 

Simulate data from polytomous logit-normit (graded logistic) model

Description

Simulate data from polytomous logit-normit (graded logistic) model

Usage

simulpln(n, nitem, ncat, alphas, betas)

Arguments

Details

Data from graded logistic models is generated under the following parameterization:

Pr(y_i = k_i| \eta) = \left\{ \begin{array}{ll} 1-\Psi (\alpha_{i,k} + \beta_i \eta) & \mbox{if } k_i = 0\\ \Psi (\alpha_{i,k} + \beta_i \eta) - \Psi (\alpha_{i,k+1} + \beta_i \eta) & \mbox{if } 0 < k_i < m-1\\ \Psi (\alpha_{i,k+1} + \beta_i \eta) & \mbox{if } k_i = m-1 \end{array} \right.

Where the items are y_i, i = 1, \dots, n, and response categories are k=0, \dots, m-1. \eta is the latent trait, \Psi is the logistic distribution function, \alpha is an intercept (cutpoint) parameter, and \beta is a slope parameter. When the number of categories for the items is 2, this reduces to the 2PL parameterization:

Pr(y_i = 1| \eta) = \Psi (\alpha_1 + \beta_i \eta)

Value

A data matrix in which each row represents a response pattern and the final column represents the frequency of each response pattern.

Author(s)

Carl F. Falk cffalk@gmail.com, Harry Joe

See Also

nrmlepln nrmlerasch nrbcpln

Examples

   n<-500;
   ncat<-3;
   nitem<-5
   alphas=c(0,-.5,  .2,-1,  .4,-.6,  .3,-.2,  .5,-.5)
   betas=c(1,1,1,.5,.5)
   
   set.seed(1234567)
   datfr<-simulpln(n,nitem,ncat,alphas,betas)
   nrmleplnout<-nrmlepln(datfr, ncat=ncat, nitem=nitem)
   nrmleplnout

Starting values for polytomous logit-normit model

Description

Computes starting values for estimation of polytomous logit-normit model.

Usage

startalphas(x, ncat, nitem = NULL)
startbetas(x, ncat, nitem = NULL)

Arguments

Details

startalphas computes starting values for the (decreasing) cutpoints for the items based on logit transformed probabilities, assuming independent items.

startbetas computes starting values for slopes under the polytomous logit-normit model, using a method based on values that are proportional to the average correlations of each item with all other items. Starting values are currently bounded between -.2 and 1.

Value

A vector of starting values, depending on which function was called.

Author(s)

Carl F. Falk cffalk@gmail.com, Harry Joe

See Also

nrmlepln nrmlerasch nrbcpln

Examples

### Raw data
data(item9cat5)

myAlphas<-startalphas(item9cat5, ncat=5)
print(myAlphas)

myBetas<-startbetas(item9cat5, ncat=5)
print(myBetas)

nrbcplnout<-nrbcpln(item9cat5, ncat=5, alphas=myAlphas, betas=myBetas, se=FALSE)
print(nrbcplnout)

## Matrix of response patterns and frequencies
data(item5fr)

myAlphas<-startalphas(item5fr, ncat=3, nitem=5)
print(myAlphas)

myBetas<-startbetas(item5fr, ncat=3, nitem=5)
print(myBetas)

nrbcplnout<-nrbcpln(item5fr, ncat=3, nitem=5, alphas=myAlphas, betas=myBetas, se=FALSE)
print(nrbcplnout)