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Last data update: 2014.03.03

R: Latent Trait Posterior of the One-Parameter Binary Probit...

fmodel1pp

R Documentation

Latent Trait Posterior of the One-Parameter Binary Probit Model

Description

fmodel1pp evaluates the (unnormalized) posterior density of the latent trait of a one-parameter binary probit item response model with given prior distribution, and computes the probabilities for each item and response category given the latent trait.

Usage

fmodel1pp(zeta, y, bpar, prior = dnorm, ...)

Arguments

`zeta`	Latent trait value.
`y`	Vector of length m for a single response pattern, or matrix of size s by m of a set of s item response patterns. In the latter case the posterior is computed by conditioning on the event that the response pattern is one of the s response patterns. Elements of `y` should be 0 or 1.
`bpar`	Vector of m "difficulty" parameters.
`prior`	Function that evaluates the prior distribution of the latent trait. The default is a standard normal distribution.
`...`	Additional arguments to be passed to `prior`.

Details

The item response model is parameterized as

P(Y_{ij} = 1|ζ_i) = Φ(ζ_i - β_j),

where where β_j is the difficulty parameter (bpar), ζ_i is the latent trait (zeta), and Φ is the distribution function of a standard normal distribution.

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

This function is designed to be called by other functions in the ltbayes package, but could be useful on its own. This function calls fmodel4pp since it is a special case.

Author(s)

Timothy R. Johnson

Examples

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

beta <- -2:2        # difficulty parameters

post <- postsamp(fmodel1pp, c(1,1,0,0,0), 
	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(fmodel1pp, c(1,1,0,0,0), 
	bpar = beta),	plot(zeta, post, type = "l")) # profile of log-posterior density

information(fmodel1pp, c(1,1,0,0,0), bpar = beta) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodel1pp, c(1,1,0,0,0), bpar = beta) # posterior mode

with(post, quantile(zeta, probs = c(0.025, 0.975))) # posterior credibility interval
profileci(fmodel1pp, c(1,1,0,0,0), bpar = beta) # profile likelihood confidence interval