fmodelnrm: Latent Trait Posterior of the Nominal Response Model

In ltbayes: Simulation-Based Bayesian Inference for Latent Traits of Item Response Models

Description

fmodelnrm evaluates the (unnormalized) posterior density of the latent trait of the nominal response model with given prior distribution, and computes the probabilities for each item and response category given the latent trait.

Usage

fmodelnrm(zeta, y, apar, bpar, prior = dnorm, ...)

Arguments

zeta	Latent trait value.
y	Matrix of size s by m of response patterns such that the posterior is computed by conditioning on the event that the response pattern is one of the s response patterns. For conditioning on a single response pattern s = 1 and so the matrix is 1 by m. Elements of y should be integers from 0 to r-1 where r is the number of response categories.
apar	Matrix of size m by r of "slope" parameters.
bpar	Matrix of size m by r of "intercept" parameters.
prior	Function that evaluates the prior distribution of the latent trait. The default is the standard normal distribution.
...	Additional arguments to be passed to the prior distribution.

Details

The nominal response model is parameterized here as

P(Y_{ij} = y|ζ_i) \propto \exp(α_{jy}ζ_i + β_{jy})

where Y_{ij} = 0, 1,…,r-1 and α_{jk} and β_{jk} are the "slope" (apar) and "intercept" (bpar) parameters, respectively. The nominal response model is also sometimes called the nominal categories model and was first proposed by Bock (1972).

Value

post	The log of the unnormalized posterior distribution evaluated at zeta.
prob	Matrix of size m by 2 array of item response probabilities.

Note

When estimating the item parameters, constraints on α_{jk} and β_{jk} are necessary for identification, such as α_{j0} = 0 and β_{j0} = 0, but these are not reflected here since a variety of constraints can be used.

Author(s)

Timothy R. Johnson

References

Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29-51.

Examples

samp <- 5000 # samples from posterior distribution
burn <- 1000 # burn-in samples to discard

alph <- matrix(c(-1, 0, 1), 5, 3, byrow = TRUE)
beta <- matrix(0, 5, 3)

post <- postsamp(fmodelnrm, c(0,1,2,1,0), 
	apar = alph, bpar = beta, control = list(nbatch = samp + burn))

post <- data.frame(sample = 1:samp, 
	zeta = post$batch[(burn + 1):(samp + burn)])
	
with(post, plot(sample, zeta), type = "l")  # trace plot of sampled realizations
with(post, plot(density(zeta, adjust = 2))) # density estimate of posterior distribution

with(posttrace(fmodelnrm, c(0,1,2,1,0), apar = alph, bpar = beta),
	plot(zeta, post, type = "l")) # profile of log-posterior density

information(fmodelnrm, c(0,1,2,1,0), apar = alph, bpar = beta) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodelnrm, c(0,1,2,1,0), apar = alph, bpar = beta) # posterior mode

with(post, quantile(zeta, probs = c(0.025, 0.975))) # posterior credibility interval
profileci(fmodelnrm, c(0,1,2,1,0), 
   apar = alph, bpar = beta) # profile likelihood confidence interval

ltbayes documentation built on May 2, 2019, 12:40 p.m.