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

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Dunson and Gelfand 2009, Gill and Casella 2009). The nonparametric prior specificationin our model and the associated clustering of subjects provides another essential differencebetween our approach and that of Rossi, Gilula and Allenby (2001).3 COMPUTATIONThe model described in section 2 is generally referred to as a Dirichlet process mixturemodel, and various Markov chain methodologies have been developed to facilitatesampling-based analyses (Neal 2000).sophistication on the part of the programmer.However, these algorithms require considerableA goal in this paper is to simplify the programming aspect of the analysis by carryingout computations in WinBUGS. The basic idea behind WinBUGS is that the programmerneed only specify the statistical model, the prior and the data. The Markov chaincalculations are done in the background whereby the user is then supplied with Markovchain output. Markov chain output is then conveniently averaged to give approximationsof posterior means.To implement the analysis of our model in WinBUGS, we make use of the constructivedefinition of the Dirichlet process given by Sethuraman (1994). The definition is knownas the stick breaking representation, and in the context of our problem, it is given asfollows: Generate a set of iid atoms (a ∗ i , b ∗ i ) from tr-Normal 2 (µ G , Σ G ) and generate a setof weights w i = y i∏ i−1j=1(1 − y j ) where the y i are iid with y i ∼ Beta(1, α) for i = 1, . . . , ∞.Thenwhere δ (a ∗i ,b ∗ i ) is the point mass at (a ∗ i , b ∗ i ).∞∑G = w i δ (a ∗i ,b ∗ i ) (6)i=110
For programming in WinBUGS, the Sethurman (1994) construction is most useful asit allows us to approximately specify the prior. We see that the stick breaking mechanismcreates smaller and smaller weights w i . This suggests that at a certain point we cantruncate the sum (6) and obtain a reasonable approximation to G (Muliere and Tardella1998). Ishwaran and Zarepour (2002) suggest that the number of truncation points be nwhen the sample size is small and √ n when the sample size is large. The stick breakingconstruction clearly shows that a generated G is a discrete probability distribution whichimplies that there is non-negligible probability that (a i , b i )’s generated from the same Ghave the same value. This facilitates the clustering of personalities in ordinal survey data.We note that the original definition of the Dirichlet process (Ferguson 1973) does notprovide a WinBUGS-tractable expression for the prior.4 EXAMPLES4.1 Course Evaluation Survey DataThe proposed model is fit to data obtained from teaching and course evaluations in theDepartment of Statistics and Actuarial Science at Simon Fraser University (SFU). Thestandard questionnaire at SFU contains m = 15 questions with responses on a five-pointscale ranging from 1 (a very negative response) to 5 (a very positive response) wherethe specific interpretation of responses are question dependent. The survey questions aregiven as follows:1. The course text or supplementary material was2. I would rate this course as3. The assignments and lectures were4. The assignments and exams were on the whole11
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: our rationale for the Uniform(0, 6)
Page 13 and 14: Markov chain simulations. The resul
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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