fmodel4pp: Latent Trait Posterior of the Four-Parameter Binary Probit...

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

Description

fmodel4pp evaluates the (unnormalized) posterior density of the latent trait of a four-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

fmodel4pp(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. Elements of y should be 0 or 1.
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.

Details

The item response model is parameterized as

P(Y_{ij} = 1|ζ_i) = γ_j + (δ_j - γ_j)Φ(α_j(ζ_i - β_j)),

where α_j is the discrimination parameter (apar), β_j is the difficulty parameter (bpar), 0 ≤ γ_j < 1 is the lower asymptote parameter (cpar), 0 < δ_j ≤ 1 is the upper asymptote parameter (dpar), ζ_i is the latent trait (zeta), and Φ is the distribution function of the standard normal distribution. This model is a variant of the logistic model discussed by Barton and Lord (1981).

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.

Author(s)

Timothy R. Johnson

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 fmodel1pp, fmodel2pp, and fmodel3pp for related models, and fmodel4pl for the logistic 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(fmodel4pp, 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(fmodel4pp, 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(fmodel4pp, c(1,1,0,0,0), 
	apar = alph, bpar = beta, cpar = gamm, dpar = delt) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodel4pp, 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(fmodel4pp, c(1,1,0,0,0), apar = alph, 
	bpar = beta, cpar = gamm, dpar = delt) # profile likelihood confidence interval

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