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

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five-point scale. When 0 ≤ b i < 1, the extremism parameter has the effect of pulling thelatent response Y ij closer to the middle. A respondent whose b i ≈ 0 might be described asmoderate and we impose the constraint b i > 0 to avoid nonidentifiability. Note that theparameter σ i in (2) addresses variability which is somewhat different from our concept ofextremism.To provide a little more clarity, when Z ij + a i − 3 > 0, the ith respondent is positivelyinclined towards survey question j. When Z ij + a i − 3 < 0, the i-th respondent isnegatively inclined towards survey question j. The quantity Z ij + a i − 3 is then scaled byb i to account for extremism on the part of the i-th respondent. The personality differentialb i (Z ij + a i − 3) is then added to 3 to yield the latent variable Y ij . Note that whereas azero score for b i (Z ij + a i − 3) represents ambivalence (neither agree nor disagree in theLikert setting), Y ij = 3 represents ambivalence in the latent variable. Having adjusted forrespondent personalities, we are interested in the average response µ for the m questionsand the corresponding correlation structure Σ. We recognize that not all individuals sharethe same temperment. The i-th respondent is characterized by the parameters a i and b iwhere a i is the disposition parameter and b i is the extremism parameter.As the proposed approach is Bayesian, prior distributions are required for the modelparameters in (3). Specifically, we assign moderately diffuse priorsΣ −1 ∼ Wishart m (I, m)µ j ∼ Uniform(0, 6)where the components of µ = (µ 1 , . . . , µ m ) ′ are apriori independent. The Wishart distributionis the standard and conjugate prior distribution for the inverse covariance matrixin normal models (Bernardo and Smith 1994) where the identity matrix and degrees offreedom parameter m are convenient choices in the absence of subjective prior information.Regarding the parameters µ j , although it is tempting to assign flat improper priors,8
our rationale for the Uniform(0, 6) prior distribution is based on the observed responseX ij constrained to the five-point scale. It is thought that X ij represents the rounded scoreof the continuous latent variable Y ij whose mean is µ j when b i = 1 and a i = 0. For thepersonality parameters a i and b i , the prior assignment is based on the supposition thatthere are classes of personalities. We therefore consider the Dirichlet process(a i , b i ) ′ iid ∼ GG ∼ DP(α, tr-Normal 2 (µ G , Σ G ))(5)for i = 1, . . . , n. The specification in (5) states that (a i , b i ) arises from a distribution Gbut G itself arises from a distribution of distributions known as the Dirichlet process. TheDirichlet process in (5) consists of the concentration parameter α and baseline distributiontr-Normal 2 (µ G , Σ G )) where tr-Normal refers to the truncated bivariate Normal whosesecond component b i is constrained to be positive. The baseline distribution serves as aninitial guess of the distribution of (a i , b i ) and the concentration parameter determines ourconfidence in the baseline distribution with large values of α > 0 corresponding to greaterdegrees of belief. Prior distributions can be assigned to the hyperparameters in (5). Ouranalyses involving course evaluation surveys on a five-point scale give sensible resultswith α ∼ Uniform(0.4, 10) (Ohlssen, Sharples and Spiegelhalter, 2007), µ G = (0, 1) ′ andΣ G = (σ ij ) where σ 11 = 1.0, σ 22 = 0.5 and σ ij = 0 for i ≠ j. Note that the choice of1.0 and 0.5 are sufficiently diffuse in the range of parameters a i and b i . The key aspectof the Dirichlet process in our application is that the personality parameters (a i , b i ) havesupport on a discrete space and this enables the clustering of personality types. Anadvantage of the Dirichlet process approach is that clustering is implicitly carried out inthe framework of the model and the number of component clusters need not be specified inadvance. Once a theoretical curiousity, the Dirichlet process and its extensions are findingdiverse application areas in nonparametric modelling (e.g. Qi, Paisley and Carin 2007,9
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 4’s. We instead consider a st
Page 11 and 12: For programming in WinBUGS, the Set
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)
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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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