Applied Bayesian Modelling - Free

382 SURVIVAL AND EVENT HISTORY MODELSwhere B i is a function of covariates x i . Assuming h 0 (0) ˆ 0, the cdf for subject i isapproximated as( )X MF(s i ) ˆ 1 exp e B id j (s i a j1 )(9:16)where (u) ˆ u if u > 0, and is zero otherwise.For a subject exiting or finally censored in the jth interval, the event is taken to occurjust after a j1 , so that (s i a j1 ) ˆ (s i a j1 ). The likelihood for a completed durations i in the jth interval, i.e. a j1 < s i < a j , is thenP ij ˆ F(x i , a j ) F(x i , a j1 )where the evaluation of F refers to individual specific covariates as in Equation (9.16),as well as the overall hazard profile. A censored subject with final known follow up timein interval j has likelihoodS ij ˆ 1 F(x i , a j )Example 9.10 Leukaemia remission To illustrate the application of this form of prior,consider the leukaemia remission data of Gehan (1965), with N ˆ 42 subjects andobserved t i ranging from 1±35 weeks, and define M ˆ 18 intervals which define theregrouped times s i . The first interval (a 0 , a 1 ] includes the times t ˆ 1, 2; the secondincluding the times 3,4 . . up to the 18th (a 17 , a 18 ] including the times 35,36. The midintervals are taken as 1.5 (the average of 1 and 2), and then 3.5, 5.5, . . and so on up to35.5. These points define the differences a 1 a 0 , a 2 a 1 , : : as all equal to 2. The Gehanstudy concerned a treatment (6-mercaptopurine) designed to extend remission times; thecovariate is coded 1 (for placebo) and 0 for treatment, so that end of remission shouldbe positively related to being in the placebo group.It may be of interest to assess whether a specific time dependence (e.g. exponential orWeibull across the range of all times, as opposed to piecewise versions) is appropriate ifthis involves fewer parameters; a non-parametric analysis is then a preliminary tochoosing a parametric hazard. One way to gauge this is by a plot of log {S(u)} againstu, involving plots of posterior means against u, but also possibly upper and lower limitsof log (S) to reflect varying uncertainty about the function at various times. A linearplot would then support a single parameter exponential.In WINBUGS it is necessary to invoke the `ones trick' (with Bernoulli density for thelikelihoods) or the `zeroes trick' (with Poisson density for minus the log-likelihoods).With B i ˆ b 0 b 1 Placebo, a diffuse prior is adopted on the intercept, and an N(0, 1)prior on the log of hazard ratio (untreated vs. treated). Following Chen et al. (2000,Chapter 10), a G(a j a j1 , 0:1) prior is adopted for d j , j ˆ 1, : : , M 1 and aG(a j a j1 , 10) prior for d M .With a three chain run, convergence (in terms of scaled reduction factorsbetween 0.95 and 1.05 on b 0 , b 1 and the d j ) is obtained at around 1300 iterations.Covariate effects in Table 9.11 are based on iterations 1500±5000, and we find a positiveeffect of placebo on end of remission as expected, with the remission rate about3.3 times (exp(1.2)) higher. The plot of log S as in Figure 9.2 is basicallysupportive of single rate exponentials for both placebo and treatment groups, thoughthere is a slight deceleration in the hazard at medium durations for the placebogroup.jˆ1