nrmlepln: Full information maximum likelihood and bivariate composite...
In pln: Polytomous Logit-Normit (Graded Logistic) Model Estimation

nrmlepln

R Documentation

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

`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 `nrec\times`(`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 `nitem\times`(`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.
`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.
`se`	Logical. If `TRUE`, calculates standard errors for the bivariate composite likelihood method.

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.

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)

pln documentation built on Oct. 17, 2024, 1:06 a.m.