ltbayes (version 0.3)

fmodel4pl: Latent Trait Posterior of the Four-Parameter Binary Logistic Model

Description

fmodel4pl evaluates the (unnormalized) posterior density of the latent trait of a four-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

fmodel4pl(zeta, y, apar, bpar, cpar, dpar, 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.

cpar

Vector of m lower asymptote parameters.

dpar

Vector of m upper asymptote 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 $$P(Y_{ij} = 1|\zeta_i) = \gamma_j + (\delta_j - \gamma_j) / (1 + \exp(-\alpha_j(\zeta_i - \beta_j))),$$ where $\alpha_j$ is the discrimination parameter (apar), $\beta_j$ is the difficulty parameter (bpar), $0 \le \gamma_j < 1$ is the lower asymptote parameter (cpar), $0 < \delta_j \le 1$ is the upper asymptote parameter (dpar), and $\zeta_i$ is the latent trait (zeta). The four-parameter binary logistic model is discussed by Barton and Lord (1981).

References

Barton, M. A. & Lord, R. M. (1981). An upper asymptote for the three-parameter logistic item-response model. New Jersey: Educational Testing Service.

See Also

See fmodel1pl, fmodel2pl, and fmodel3pl for related models, and fmodel4pp 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
gamm <- rep(0.1, 5) # lower asymptote parameters
delt <- rep(0.9, 5)	# upper asymptote parameters

post <- postsamp(fmodel4pl, c(1,1,0,0,0), 
	apar = alph, bpar = beta, cpar = gamm, dpar = delt,
	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(fmodel4pl, c(1,1,0,0,0), 
	apar = alph, bpar = beta, cpar = gamm, dpar = delt),
	plot(zeta, post, type = "l")) # profile of log-posterior density

information(fmodel4pl, c(1,1,0,0,0), 
	apar = alph, bpar = beta, cpar = gamm, dpar = delt) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodel4pl, c(1,1,0,0,0), 
	apar = alph, bpar = beta, cpar = gamm, dpar = delt) # posterior mode

with(post, quantile(zeta, probs = c(0.025, 0.975))) # posterior credibility interval
profileci(fmodel4pl, c(1,1,0,0,0), apar = alph, 
	bpar = beta, cpar = gamm, dpar = delt) # profile likelihood confidence interval