Applied Bayesian Modelling - Free

376 SURVIVAL AND EVENT HISTORY MODELS10.90.80.7Probability0.60.50.40.30.20.100 20 40 60 80 100WeeksMean '2.5% '97.5%120 140 160 180 200Figure 9.1Weibull mixture analysiswith survivor functionS(t, x) ˆ exp ( L 0 (t)e bz )where the integrated hazard is denoted by… tL 0 (t) ˆ l 0 (u)du (9:13)0Then the conditional probability of surviving through the jth interval given that asubject has survived the previous j 1 intervals is" … #ajq j ˆ exp e bz l 0 (u)dua j1ˆ exp e bz {L 0 (a j ) L 0 (a j1 )}whileh j ˆ 1 q jis the corresponding hazard rate in the jth interval [a j1 , a j ).The total survivor function until the start of the jth interval isDefiningS j ˆ exp [ e bz L 0 (a j1 )]g j ˆ ln [L 0 (a j ) L 0 (a j1 )]the likelihood of an event in interval [a j1 , a j ) given survival until then, can be written(Fahrmeir and Tutz, 2001; Kalbfleisch and Prentice, 1980) as