Full information maximum likelihood and bivariate composite likelihood estimation for polytomous logit-normit and Rasch models, via Newton Raphson iterations.
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)A list containing the following slots.
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.
Number of ordinal categories for each item, coded as
0,...,(ncat-1). Currently supported are items that have the same number of categories.
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.
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.
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.
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.
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.
Number of quadrature points to use during estimation. This argument is currently experimental. It is recommended to use the default of 48.
Maximum number of iterations for estimation.
Logical. If TRUE, computes goodness-of-fit statistics
from Maydeu-Olivares and Joe (2005, 2006; i.e., \(M_2\)).
Logical. Enables debugging / diagnostic information from C code that conducts estimation.
Logical. If TRUE, calculates standard errors for the bivariate composite
likelihood method.
alphasA 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.
betasA vector of parameter estimates for betas. Length is nitem.
nllkNegative (composite) log-likelihood for polytomous logit-normit (or Rasch) model.
convInteger indicating whether estimation converged. Currently only returned for composite likelihood estimation.
sealphasA vector of standard errors for the alpha estimates.
sebetasA vector of standard errors for the beta estimates.
invhesInverse Hessian matrix for the MLE estimates.
vcovAsymptotic covariance matrix for the composite likelihood estimates.
teststatValue of \(M_2\).
dfDegrees of freedom for \(M_2\).
pvalP-value for \(M_2\).
Carl F. Falk cffalk@gmail.com, Harry Joe
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.
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.
startalphas
startbetas
### 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)
# \donttest{
### 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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