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

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1994). A major difficulty with posterior predictive methods concerns double use of thedata (Evans 2007). Specifically, the observed data x is used both to fit the model givingrise to the posterior density π(θ | x) and then is used in the comparison of y i versus x. Forthis reason, some authors prefer a cross-validatory approach (Gelfand, Dey and Chang1992) where the data x = (x 1 , x 2 ) are split such that x 1 is used for fitting and x 2 is usedfor validation.We take the view that in assessing a Bayesian model, the entire model ought to beunder consideration, and the entire model consists of both the sampling model of thedata and the prior. We also want a methodology that does not suffer from double use ofthe data. For the models proposed here, we recommend an approach that is similar tothe posterior predictive methods but instead samples “model variates” y from the priorpredictive density∫f(y) = f(y | θ) π(θ) dθ (8)where π(θ) is a proper prior density. This approach was advocated by Box (1980) beforesimulation methods were common. It is not difficult to write R code to simulate y 1 , . . . , y Nfrom the prior predictive density in (8). It is then a matter of deciding how to compare they i ’s against the observed data matrix X. In our application, the data are high dimensional,and we advocate a comparison of “features” that are of direct interest. This is an intuitiveand simple approach which is not part of current statistical practice. For example, onemight compare observed subject means ¯X i = ∑ mj=1 X ij /m with subject means generatedfrom the prior predictive simulation. A simple comparison of these vectors can be easilycarried out through the calculation of Euclidean distances. Naturally, as the priors becomemore diffuse, it becomes less likely to find evidence of model inadequacy. We do not viewthis as a failing of the methodology. Rather, if you really want to detect departures froma model, it is necessary that you have strong prior opinion concerning your model.20
To provide a more stringent test, we consider a modification of our model where subjectivepriors µ j ∼ Uniform(2, 5) and Σ G = 0.01I are introduced. We assess goodness-offit on the SFU data discussed in section 4. With N simulated vectors from the prior predictivedistribution, there are ( N+12 ) Euclidean distances of interest; N of these distancesare between the observed mean vector and the simulated vectors, and the remaining ( N 2 )distances correspond to distances between simulated vectors. These distances are displayedin a histogram with the N = 20 distances highlighted in Figure 3. Since thesedistances appear typical, there is no evidence of lack of fit. In fact, we observe that mostof the Euclidean distances involving observed data lie on the left side of the histogram.This suggests that the most extreme variates arose from the prior-predictive distribution.Clearly, graphical displays for alternative features can also be produced.Another approach to Bayesian goodness-of-fit which appears promising in the contextof the proposed model is due to Johnson (2007). Let θ consist of all model parameters, letX i be the vector of discrete responses for the ith respondent and let β i = b i (µ+a i 1−31)+31denote the mean of the corresponding latent variable Y i , i = 1, . . . , n. Under the “true”θ, we then note that S(X i , θ) = (Y i − β i ) ′ Σ −1 (Y i − β i )/b 2 i is distributed as a Chi-squarevariable with m = 15 degrees of freedom. Following Johnson (2007), S(X i , θ) is pivotalin the sense that its conditional distribution does not depend on θ and there are n = 75values of S(X i , θ) that can be calculated for a given θ. For a single sampled value θ fromthe MCMC simulation, Figure 4 provides a plot of the ordered values of S(X i , θ) versusthe theoretical Chi-square quantiles. The plotted points appear to be roughly scatteredabout the line y = x and hence provide no strong indication of lack of fit.21
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 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: Table 2: Posterior means and poster
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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