Applied Bayesian Modelling - Free

422 MODELLING AND ESTABLISHING CAUSAL RELATIONS2Log Odds relative to SBP of 90-951.510.5090 140 190SBP240 290Figure 10.4SBP parameterst 0:5g is reduced to 0.15, but the shape of the log-odds curve is very similar to Figure 10.4.From Figure 10.4, a final knot point at around 250 might be selected and the spline inthe upper category restricted to be linear (see Greenland, 1998a) to avoid the implausiblefall in risk at very high SBP.Example 10.7 Cumulative mortality in relation to dose-time The previous example hasillustrated how standard model assumptions (e.g. linearity in dose effects) may need tobe critically examined. As mentioned above, dose-response modelling one may alsoneed to consider the joint action of dose with other (confounding) factors, as well asdepartures from standard sampling assumptions (e.g. clustering as a source of binomialoverdispersion).To illustrate an alternative modelling structure, drawing on survival analysis conceptsto represent the joint effects of dose and a (confounding) time index, consider an animalexperiment reported by Pack and Morgan (1990). This involves deaths over a 13 dayperiod among flour beetles sprayed with the insecticide pyrethrins B, where the focus is notonly on endpoint or total mortality by the end of the period, but on cumulative mortalityat days 1, 2, 3, . . , up to 13. Four dosage levels (mg=cm 2 ) were applied, 0.20, 0.32, 0.50 and0.80. The relation of mortality to dose-time is expected to be differentiated by the sex k ofthe beetle. It may be noted that the structure of this example could be applied withsuitable modifications in Example 10.6 (with SBP as the dose, and age parallel to time).The probability of death at dose X r in the jth time interval {t j1 , t j }, wheret 0 ˆ 0, t 1 ˆ 1, : : t 13 ˆ 13 is modelled aswherep rj ˆ C(t j , X r ) C(t j1 , X r )(10:9a)1=C(t, X) ˆ [1 exp { b 1 b 2 log (X)}] [1 t b 3b 4 ] (10:9b)Thus, separate dose-response and time-response models are present in Equation (10.9b).As an illustration of the potential benefits of parameter transformation, an alternative