Bayesian analysis of ordinal survey data using the Dirichlet process ...

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5. The marking scheme was on the whole6. How informative were the lectures7. The Instructor’s organization and preparation were8. The Instructor’s ability to communicate material was9. The Instructor’s interest in the course content appeared to be10. The Instructor’s feedback on my work was11. Questions during class were encouraged12. Was the Instructor accessible for extra help13. Was the Instructor responsive to complaints/suggestions14. Overall, the Instructor’s attitude towards students was15. I would rate the Instructor’s teaching ability asData were collected from n = 75 students pertaining to an introductory Statisticscourse. Posterior means and standard deviations corresponding to the parameter µ aregiven in Table 1. These are based on a MCMC simulation using WinBUGS with aburn-in period of 1000 iterations followed by 4000 iterations, taking roughly 2 hours ofcomputation on a personal computer. The WinBUGS code is provided in the Appendix.The highest posterior mean was recorded for the 9th question which asked about “theInstructor’s interest in the course material”. The smallest mean was recorded for the10th question which asked about “the Instructor’s feedback on work”. These resultsare consistent with past surveys taken in the same course with the same Instructor. Inparticular, the Instructor does not grade assignments and this yields some criticisms fromthe students. Note that the posterior standard deviations are sufficiently small such thatwe can sensibly discuss the posterior means.To investigate the clustering effect, we recorded the number of clusters in each of the12
Markov chain simulations. The resulting histogram is given in Figure 1. We observe thatthere are quite a few clusters, and there is considerable uncertainty about the number ofclusters. More specifically, the number of clusters appears roughly uniform between 3 and18, with approximately 10 clusters on average. This implies that there is a substantialnumber of personality types amongst the n = 75 students. The clustering effect is corroboratedin Figure 2 where we provide a plot of the posterior means of the (a i , b i ) pairs.By looking closely along both vertical and horizontal strips, there are approximately 10classes of personalities, some of which do not differ greatly. In Figure 2, the clustering ismore difficult to distinguish with respect to the disposition parameter a, suggesting morevariability in a than in b. We observe that roughly 50% of the a i ’s are greater than 0.0,and roughly 50% of the b i ’s are greater than 1.0.For small values of b, the corresponding students tend to have responses which areoften the same, and this may be due to a desire to finish the questionnaire as quicklyas possible. With b = 1.0, the interpretation is that these students do not distort theirresponses in an inflationary/deflationary sense, and there are roughly 20 students of thistype. For large values of b, the corresponding students inflate their responses; they makeharsher decisions near both ends of the scale (1’s and 5’s).Note that one of the respondents provided a score of 5.0 for all m = 15 questions. Itturns out that the corresponding posterior mean of (a i , b i ) for this student was (0.36, 1.09).Based on an average posterior response ¯µ = 4.1, this student’s mean latent Y-score is1.09(¯µ + 0.36 − 3) + 3 = 4.59 which rounds to a respondent X-score of 5.0 accordingto the cut-point model (1). This provides some evidence that the (a i , b i ) parametersare estimated sensibly. For this student, we note that the variance 1.09Σ of the latentresponse Y i which includes personality traits exceeds the variance Σ of the standardizedresponse Z i . This is because the student in question is “extreme”, and has the capacity13
Page 1 and 2: Bayesian Analysis of Ordinal Survey
Page 3 and 4: survey where there is a respondent
Page 5 and 6: validity of the model. The proposed
Page 7 and 8: and 4’s. We instead consider a st
Page 9 and 10: our rationale for the Uniform(0, 6)
Page 11: For programming in WinBUGS, the Set
Page 15 and 16: Table 1: Posterior means and poster
Page 17 and 18: our model gave D = 891. In both the
Page 19 and 20: Table 2: Posterior means and poster
Page 21 and 22: To provide a more stringent test, w
Page 23 and 24: Brooks, S. P. and Gelman, A. (1997)
Page 25 and 26: Parducci, A. (1965). “Category ju
Page 27 and 28: proportion0.00 0.02 0.04 0.06 0.08
Page 29 and 30: elative frequency0.0 0.1 0.2 0.3●

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