ltbayes (version 0.3)

fmodel2pp: Latent Trait Posterior of the Two-Parameter Binary Probit Model

Description

fmodel2pp evaluates the (unnormalized) posterior density of the latent trait of a two-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

fmodel2pp(zeta, y, apar, 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. Element

apar

Vector of m "discrimination" parameters.

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.

Value

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

Details

The item response model is parameterized as $$P(Y_{ij} = 1|\zeta_i) = \Phi(-\alpha_j(\zeta_i - \beta_j)),$$ where $\Phi$ is the distribution function of a standard normal distribution, $\alpha_j$ is the discrimination parameter (apar), $\beta_j$ is the difficulty parameter (bpar), and $\zeta_i$ is the latent trait (zeta).

See Also

See fmodel1pp, fmodel3pp, and fmodel4pp for related models, and fmodel2pp for a probit variant of this model.

Examples

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

alph <- rep(1, 5)   # discrimination parameters
beta <- -2:2        # difficulty parameters

post <- postsamp(fmodel2pp, c(1,1,0,0,0), 
	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(fmodel2pp, c(1,1,0,0,0), 
	apar = alph, bpar = beta),
	plot(zeta, post, type = "l")) # profile of log-posterior density

information(fmodel2pp, c(1,1,0,0,0), 
	apar = alph, bpar = beta) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodel2pp, c(1,1,0,0,0), 
	apar = alph, bpar = beta) # posterior mode

with(post, quantile(zeta, probs = c(0.025, 0.975))) # posterior credibility interval
profileci(fmodel2pp, c(1,1,0,0,0), apar = alph, 
	bpar = beta) # profile likelihood confidence interval