fmodelrsm: Latent Trait Posterior of the Rating Scale Model

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

Description

fmodelpcm evaluates the (unnormalized) posterior density of the latent trait of a rating scale item response model with a given prior distribution, and computes the probability for each item and response category given the latent trait.

Usage

fmodelrsm(zeta, y, 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 integers from 0 to r-1 where r is the number of response categories.
cpar	Vector of length r-1 of category "threshold" parameters.
dpar	Vector of length m of item "difficulty" parameters.
prior	Function that evaluates the prior distribution of the latent trait. The default is the standard normal distribution.
...	Additional arguments to be passed to prior.

Details

The rating scale model is parameterized here as a special case of the partial credit model where

P(Y_{ij} = y|ζ_i) \propto \exp(yζ_i - ∑_{k=0}^yβ_{jk})

for y = 0, 1,…, r-1 where β_{jk} = δ_j + γ_k and γ_0 = 0. The parameters δ_j and γ_k are the 'difficulty' (dpar) and 'threshold' (cpar) parameters, respectively. This model was proposed by Andersen (1977) and Andrich (1978a, 1978b).

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

The number of response categories (r) is inferred from the number of columns in dpar, not from the maximum value in y.

Because the rating scale model can be viewed as a special case of the partial credit model, the latter can be used with fmodelpcm to specify more general models.

Author(s)

Timothy R. Johnson

References

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42, 69-81.

Andrich, D. (1978a). A rating formulation for ordered response categories. Psychometrika, 43, 561-573.

Andrich, D. (1978b). Application of a psychometric rating scale model to ordered categories which are scored with successive integers. Applied Psychological Measurement, 2, 581-594.

Examples

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

gamm <- c(0, 0)
delt <- rep(0, 5)

post <- postsamp(fmodelrsm, c(0,1,2,1,0), 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(fmodelrsm, c(0,1,2,1,0), cpar = gamm, dpar = delt),
	plot(zeta, post, type = "l")) # profile of log-posterior density

information(fmodelrsm, c(0,1,2,1,0), cpar = gamm, dpar = delt) # Fisher information

with(post, mean(zeta)) # posterior mean
postmode(fmodelrsm, c(0,1,2,1,0), cpar = gamm, dpar = delt) # posterior mode

with(post, quantile(zeta, probs = c(0.025, 0.975))) # posterior credibility interval
profileci(fmodelrsm, c(0,1,2,1,0),
	cpar = gamm, dpar = delt) # profile likelihood confidence interval

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