# 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

 `1` ```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 `fmodel1pp`, `fmodel2pp`, and `fmodel3pp` for related models, and `fmodel4pl` for the logistic variant of this model.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ```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 ```