ltbayes (version 0.3)

fmodel1pl: Latent Trait Posterior of the One-Parameter Binary Logistic Model

Description

fmodel1pl evaluates the (unnormalized) posterior density of the latent trait of a one-parameter binary logistic item response model with given prior distribution, and computes the probabilities for each item and response category given the latent trait.

Usage

fmodel1pl(zeta, y, bpar, prior = dnorm, ...)

Arguments

zeta

Scalar or vector of latent trait values.

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

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 $$P(Y_{ij} = 1|\zeta_i) = 1 / (1 + \exp(-(\zeta_i - \beta_j))),$$ where $\beta_j$ is the difficulty parameter (bpar) and $\zeta_i$ is the latent trait (zeta).

See Also

See fmodel2pl, fmodel3pl, and fmodel4pl for related models, and fmodel1pp for the probit variant of this model.

Examples

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

beta <- -2:2        # difficulty parameters

post <- postsamp(fmodel1pl, 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(fmodel1pl, c(1,1,0,0,0), 
	bpar = beta),	plot(zeta, post, type = "l")) # profile of log-posterior density

information(fmodel1pl, c(1,1,0,0,0), bpar = beta) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodel1pl, c(1,1,0,0,0), bpar = beta) # posterior mode

with(post, quantile(zeta, probs = c(0.025, 0.975))) # posterior credibility interval
profileci(fmodel1pl, c(1,1,0,0,0), bpar = beta) # profile likelihood confidence interval