Applied Bayesian Modelling - Free

116 REGRESSION MODELSand a second order one would beg t N(n t , d t t 2 )where n t ˆ g t1 (1 d t =d t1 ) g t2 (d t =d t1 ) (see Fahrmeir and Lang, 2001). If there isequal spacing then the first and second order random walk priors are justg t N(g t1 , t 2 )g t N(2g t1 g t2 , t 2 )With smooths on several regressors x 1t , x 2t , x 3t : : it will be necessary to supply anordering index on each one at observation level O 1t , O 2t , O 3t , etc. A frequent situationis the semi-parametric additive form, with smooths on a subset of p regression variables,with the remainder modelled conventionally. Quite often there would be just a singleregressor with a general additive form and the remainder included in a conventionallinear combination. It may be noted that results may be sensitive to the priors assumedfor the initial smoothing values (e.g. g 1 in a first order random walk) and for theevolution variance t 2 . This is especially so for sparse data, such as for binary outcomesy t . In BUGS convergence may benefit from sampling t 2 directly from its full conditionaldensity.Other smoothness priors have been proposed. For example, Carter and Kohn (1994)consider the signal plus noise modely t ˆ g t e twhere the g t are generated by a differential equationd 2 g tdt 2ˆ t dW tdtwith W t a Weiner process, and t 2 the smoothing parameter. This leads to a bivariatestate vector s t ˆ (g t , dg tdt ) in which s t ˆ F t s t1 u t (3:22)where, with d t defined as above,F t ˆ 1 d t0 1Also, the u t are bivariate Normal with mean zero and covariance t 2 U t , whereU t ˆ d3 t =3 d2 t =2d 2 t =2 d tWood and Kohn (1998) discuss the application of this prior with binary outcomes,which involves applying the latent variable model of Albert and Chib (1993).Example 3.11 Kyphosis and age Consider the kyphosis data from Hastie and Tibshirani(1990) on 81 patients receiving spinal surgery (Appendix Table 2). The binaryoutcome y t relates to the post-surgical presence or otherwise of forward flexion of thespine from the vertical. As mentioned above, this is a sparse data form and so results ofadditive smoothing may be sensitive to priors assumed.