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

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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

x

A data matrix. Data can be in one of two formats: 1) raw data where the number of rows corresponds to an individual's response and each column represents an item, and 2) a matrix of dimensions nrecX(nitem+1) where each row corresponds to a response pattern and the last column is the frequency of that response pattern. A data matrix of the second type requires input for nitem and nrec.

ncat

Number of ordinal categories for each item, coded as 0,...,(ncat-1). Currently supported are items that have the same number of categories.

nitem

Number of items. If omitted, it is assumed that x contains a data matrix of the first type (raw data) and the number of columns in x will be selected as the number of items.

alphas

A vector of length nitemX(ncat-1) corresponding to starting values for the (decreasing) cutpoints for the items. If omitted, these will be computed from the function startalphas.

betas

A vector of length nitem corresponding to starting values for the beta vectors of slopes. If omitted, these will be computed from the function startbetas. For the polytomous logit-normit, there is one slope for each item; for the Rasch model, there is a common slope beta for all of the items.

abound

Vector of length 2 that sets upper and lower bounds on parameter estimation for alphas. Currently experimental; changing defaults it not recommended. Estimation problems are more likely solved by changing starting values.

bbound

Vector of length 2 that sets upper and lower bounds on parameter estimation for betas. Currently experimental; changing defaults it not recommended. Estimation problems are more likely solved by changing starting values.

nq

Number of quadrature points to use during estimation. This argument is currently experimental. It is recommended to use the default of 48.

mxiter

Maximum number of iterations for estimation.

se

Logical. If TRUE, calculates standard errors for the bivariate composite likelihood method.

m2

Logical. If TRUE, computes goodness-of-fit statistics from Maydeu-Olivares and Joe (2005, 2006; i.e., M_2).

iprint

Logical. Enables debugging / diagnostic information from C code that conducts estimation.

Details

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

Pr(y_i = k_i| η) = { 1-Ψ (α_i,k + β_i*η) if k_i = 0, Ψ (α_i,k + β_i*η) - Ψ (α_i,k+1 + β_i*η) if 0 < k_i < m-1, Ψ (α_i,k+1 + β_i*η) if k_i = m-1}.

Where the items are y_i, i = 1, …, n, and response categories are k=0, …, m-1. η is the latent trait, Ψ is the logistic distribution function, α is an intercept (cutpoint) parameter, and β is a slope parameter. When the number of categories for the items is 2, this reduceds to the 2PL parameterization:

Pr(y_i = 1| η) = Ψ (α_1 + β_i η)

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 β paramter is estimated for all items.

Value

alphas

A vector of parameter estimates for alphas. Length is nitemX(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 paraemter 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 fredom 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

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### 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)

## Not run: 
### 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)

## End(Not run)