EDMS 779 Assignment

MCMC exercises using BUGS: Variations on the IRT example

The problems below builds on a problem we discussed in class, namely an IRT example provided by Prof. Roberts' (BUGS setup for an IRT example). Use BUGS to answer the questions, and send me answers.

In all cases, run 4000 burn-in cycles before monitoring the variables that are of interest, and base your answers on 1000 cycles. Since BUGS has to do adaptation for Metropolis iterations for the 3pl, it won't let us use the first 4000 anyway in problems that concern the 3pl.

1) The first data vector in Dr. Roberts' BUGS data is 1, 1, 1, 1, 0. Replace the 0 with NA.  In BUGS, this means the value of this (potentially observable) variable is not known.  Run BUGS with the rest of data and Dr. Roberts' model as original. Monitor this node, which will be r[1,5]. What is the resulting distribution? How would you interpret it?

2) Replace the entire first vector with NA, NA, N, NA, NA. Run BUGS with the rest of data and Dr. Roberts' model as original. Monitor r[1,5] again. What is the resulting distribution? How would you interpret it? Is it different from the answer to a)? Why or why not?

3) Fit the Rasch model to the original data. Dr. Roberts used the 3 parameter logistic model, in which

p[j,k] <- c[k]+((1-c[k])*(exp(a[k]*(theta[j]-b[k]))/(1+exp(a[k]*(theta[j]-b[k])))))

Here's how to do this:

p[j,k] <- exp(theta[j]-b[k]) / (1+exp(theta[j]-b[k]))

What are the posterior distributions of the five resulting item difficulty parameters, the b's? Compare the resulting posterior standard deviations of your run with those from the original 3PL run, for the five item difficulties and the first five examinees.