STAT 778 – Homework 4
You need only answer 5 of the following 6 questions. If you answer all 6 I will count the best 5.
1) In class we discussed each of the three types of polytomous IRT models in terms of their defining
probability statements:
Graded Response Model:
Partial Credit Model:
Continuation Ratio Model:
P[Ui≥x | ]
P[Ui=x | Ui=x or Ui=x-1 ]
P[Ui ≥ x | Ui ≥ x-1 ]
The parametric polytomous models are made by setting the probability statement equal to a function similar to a
Rasch or 2PL model. You could also make nonparametric versions by simply setting them equal to increasing
functions, and it was mentioned in class that the nonparametric graded response model was the most general of
the polytomous models we discussed.
Choose either Master’s partial credit model or Tutz’s continuation ratio model (the Rasch like ones) and show
that P[Ui≥x | ] for that model is an increasing function of  for any x greater than 0.
2) Consider the example on pages 160-163 of “A Generalized Partial Credit Model” by Muraki. How many
categories does item 1 have? At what  value do the item category response functions for scores of 3 and 4
cross?
3) Consider the questionnaire described on Table 1 (page 20) of the Roberts, Donoghue, and Laughlin article
“A General Item Response Theory Model for Unfolding Unidimensional Polytomous Responses”. Imagine
that one researcher is using just items 16-20 and the estimated parameters on page 20 (so they have a form like
figure 2 on page 7). Imagine another researcher fits Muraki’s partial credit model to items 16-20 (so they look
like figure 2 on page 162 of that article).
Briefly say what each of these models would likely predict for an examinee with ability =12. Also say why
this seemingly odd result would rarely be of practical concern.
4) Consider the estimated 2PL and 3PL item parameters for the reading test described on the first take home
exam: http://www.stat.sc.edu/~habing/courses/data/rexam2pl.txt and
http://www.stat.sc.edu/~habing/courses/data/rexam3pl.txt. Explain what effect the choice of 2PL
or 3PL model seems to have on the estimated item information curves by plotting a few chosen examples and
briefly describing the differences.
5) Which of items 1-5 does the best job of estimating ability for examinees of ability 1? What is the estimated
standard deviation of the ability estimate for that single item? Briefly describe how that item compares to the
other 4 items at other ability levels.
6) The r-code at http://www.stat.sc.edu/~habing/courses/740/methast.txt is an example of a very
basic MCMC algorithm. It assumes that you have three observations (-8, 8, and 17) from a Cauchy distribution.
The goal of the code is to estimate the posterior distribution of the distributions parameter  using a normal
prior distribution with mean 0 and standard deviation 10.
If you cut and paste the code in, you will get the following output. The vector theta will contain the elements
of the chain (the random walk through the posterior). A histogram of all 10,001 observations (it is supposed to
approximate the real posterior distribution) with a curve that is the actual posterior distribution (this is easy
enough we can find it directly) drawn over it.
Briefly, how well does the histogram of estimates match with the actual posterior? What is the EAP estimate of
 using all of the elements of the chain? Use the command plot(theta) to see all of the chain elements in
order; why is it obvious that you want to make sure you use a long chain instead of a very short one?