Current Topics in the Theory and Application of Latent Variable Models by MacCallum Robert C. Edwards Michael C

Author:MacCallum, Robert C.,Edwards, Michael C.
Language: eng
Format: epub
ISBN: 9780415637787
Publisher: Taylor & Francis (CAM)

8.4 STAT 133 DATA STUDY

Our data set consists of results from the final exam for STAT 133, a business statistics class at The Ohio State University. The exam has 28 multiple-choice questions, each with four response options, and we use binary scores in our analysis; 1 for a correct answer and 0 for an incorrect answer. There were 258 students who took this exam with raw scores ranging from 6 to 28. The item scores, or number of individuals answering the item correctly range from 85 to 255.

The BRugs package in R (Spiegelhalter, Thomas, Best, & Gilks, 1995; R Development Core Team, 2006) was used to fit the 2PLM and 3PLM, with μ0 = 0, σ2θ = 1, μb = − .57, σ2b = 9, va = 3.2, Ωa = 1, αc = 1.5, βc = 4.5. The choice of μβ, is based on our prior knowledge that final exam scores in STAT 133 tend to be higher than 50%, and thus the average difficulty of items should be negative. Initial values for θ were based on the normal quantile of the raw score, and initial values for a, b, and c were generated from the priors. A burn in period of 5,000 updates was used before 20,000 updates were collected with a thinning interval of 10.

A custom R package was developed to fit the nonparametric models. We used many of the same prior distributions as for the parametric models and again used the initial values of θ based on the raw score. Initial values for a, b, and c were generated from the priors while initial values for Z were generated according to a Pólya urn scheme. The other additional prior parameter is M, the prior mass for the Dirichlet process. Here we chose to use M = 5 based on Qin's (1998) exploration. We also used a burn-in period of 5,000 updates and stored 20,000 iterations with a thinning interval of 10.

To obtain the estimated curves with the parametric models, we used the posterior means for aj, bj, and in the case of the 3PL model cj to plot logistic curves. The estimated curves with the NP models were obtained using the posterior expectation of Fj at a sequence of points from —4 to 4. The posterior probability bands were obtained by taking percentiles of the expected curve height. Abilities were estimated under each model with the posterior mean of 0.

For many items, the estimated curves were similar under the different models, but for a few items large differences were evident. Figure 8.1 gives the plots of both the parametric and nonparametric curves for items 12 and 23, two curves where the parametric and nonparametric results gave different results. In Item 12, we see that the 2PLM and NP2M curves differ substantially, but when the guessing parameter is included in the model, the 3PLM and NP3M curves are quite similar. This indicates that the 2PLM does not adequately describe the item functioning for item 12. The more flexible

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