Full information maximum likelihood and bivariate composite likelihood estimation for polytomous logitnormit (graded logistic) model
Description
Full information maximum likelihood and bivariate composite likelihood estimation for polytomous logitnormit and Rasch models, via Newton Raphson iterations.
Usage
1 2 3 4 5 6  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

ncat 
Number of ordinal categories for each item, coded as
0,...,( 
nitem 
Number of items. If omitted, it is assumed that 
alphas 
A vector of length 
betas 
A vector of length 
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 
m2 
Logical. If 
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) = { 1Ψ (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 < m1, Ψ (α_i,k+1 + β_i*η) if k_i = m1}.
Where the items are y_i, i = 1, …, n, and response categories are k=0, …, m1. 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 MaydeuOlivares 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

betas 
A vector of paraemter estimates for betas. Length is 
nllk 
Negative (composite) loglikelihood for polytomous logitnormit (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 
Pvalue 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.
MaydeuOlivares, A., and Joe, H. (2005). Limited and full information estimation and goodnessoffit testing in 2^n contingency tables: A unified framework. Journal of the American Statistical Association, 100, 10091020.
MaydeuOlivares, A., and Joe, H. (2006). Limited information and goodnessoffit testing in multidimensional contingency tables. Psychometrika, 71, 713732.
Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 542.
See Also
startalphas
startbetas
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32  ### 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)

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