Description Usage Arguments Details Value Slots Author(s) References See Also Examples
Full information maximum likelihood and bivariate composite likelihood estimation for polytomous logit-normit and Rasch models, via Newton Raphson iterations.
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)
|
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. |
m2 |
Logical. If |
iprint |
Logical. Enables debugging / diagnostic information from C code that conducts estimation. |
se |
Logical. If |
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 reduces 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 parameter is estimated for all items.
A list containing the following slots.
alphas
A vector of parameter estimates for alphas. Length is
nitem
X (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.
Carl F. Falk cffalk@gmail.com, Harry Joe
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.
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 | ### 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)
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