Applied Bayesian Modelling - Free

252 ANALYSIS OF PANEL DATAWe obtain 7 the estimates shown in Table 6.8 for time, female gender, age andthe therapy. These compare to the Fitzmaurice and Lipsitz estimates for the effects ofthese covariates of 0.013 (s.e. 0.022), 0.61 (0.42), 0.015 (0.018) and 1.45 (0.45). Thepredictive loss criterion (with w ˆ 1) is 46. The high correlation replicates the work ofFitzmaurice and Lipsitz, who found a strong odds ratio form of dependence betweensuccessive patient self-ratings.An alternative approach is provided by the `one factor model' of Heckman (1981,p. 130) namelyu it ˆ bX it l t a i e itwhere the e it are unstructured errors with variances f t , the a i are latent propensitiesfor good health status with variance s 2 a , and the l t are time-varying loadings. Foridentifiability, it is assumed that l 1 ˆ 1, and f t ˆ f ˆ 1. The variance s 2 a is defined by aU(0, 1) prior onDefining the ratiost ˆ s 2 a =[s2 a f]R t ˆ l 2 t s2 a =[l2 t s2 a f t ]of `permanent' to total variance 8 , the correlation between disturbances at times t and s isthen C ts ˆ A t A s , whereA t ˆ R 0:5tN(1, 1) priors are assumed on the l t (t > 1), and constrained to being positive.Fitting this model 9 enhances the mean therapy effect to 2.2, with 95% interval{0.75, 4.1}, but shows weaker effects than in Table 6.8 for gender or age. The correlationsrange between 0.47 (between times 3 and 4) and 0.74 (between times 1 and 2). Theindividual propensities a i range from 3.3 for patient 4, who records good health statusat all time points, despite being on the placebo treatment, to patient 32 with a score of3.8 and classed as poor status at all points. The predictive loss criterion stands at 45,showing slight gain to using this model. Subject to identifiability more complex onefactor models might improve fit, for example taking f t variable by time.Finally, the method of Albert and Chib (1993) offers a direct approach to samplingthe underlying latent variables in Equation (6.17). It is applied to the model ofTable 6.8 Arthritis status, differential spacing model,parameter summaryMean St. devn. 2.50% 97.50%Intercept 1.86 1.68 1.5 5.17Time 0.022 0.05 0.075 0.12Female 1.14 0.84 2.82 0.49Age 0.017 0.034 0.085 0.052Therapy 1.82 0.79 0.39 3.48g 0.93 0.03 0.86 0.977 A three chain run shows convergence at 1000 iterations and summaries are based on iterations 1000±5000.8 The composite error l t a i multiplies the permanent subject error by a time varying factor.9 A two chain run is taken to 5000 iterations with 1000 burn-in.