Remarks on the dose selection procedure

The method of Bedrick et al. (1996) of eliciting prior distributions for the dose-response curves parameters assumes that the beta prior distributions elicited at dose levelsd−1 and

d0are independent. This assumption simplifies the mathematics but as noted in Whitehead et al. (2006), it has the undesired consequences that it is possible for βE < 0 or βT <

0 when it is believed that βE ≥ 0 and βT ≥ 0. This is because assuming the elicited beta distributions at dose levelsd−1 andd0 are independent, for example implies that the probability that the probability of efficacy at dose leveld−1is higher than the probability of efficacy at dose leveld0ford−1 < d0is not zero even when it is believed efficacy improves

5.6. REMARKS ON THE DOSE SELECTION PROCEDURE 97

with dose level. This in turn means it is possible to have βE < 0 when it is believed

βE ≥0. To partly address this problem the beta prior distributions are elicited at locations that are far from each other. Also as in Whitehead et al. (2006), since we are interested in the posterior means of the conditional powers associated with continuing with different set of doses, negative parameter values for the slope parameters will not have undesired effects on the predictive power. Further, since we obtain the posterior distributions by updating the prior distributions using all the phase II clinical trial data, for the posterior distributions, the slope parameters are unlikely to be negative when the slope parameters are actually positive.

The use of conditional efficacy and toxicity models (5.1) and (5.2) may raise concern about the association between efficacy and safety. At each dose level, we are assuming independence between the probabilities of efficacy and toxicity to obtain the predictive power. However, because we are using more than one experimental dose, this does not imply marginal independence between efficacy and toxicity. To demonstrate this, using odds ratio as a measure of the association, first we give the expression for the odds ratio and then give the implied odds ratio for some scenarios. As above, letpEj andpTj respectively denote the probability of efficacy and of toxicity at dosej (j = 1, ..., k1). Further letpRj denote the probability of a patient being randomized to dosej (j = 1, ..., k1). Using law of total probability, the marginal probabilities of efficacy (pE) and toxicity (pT) and the probability of efficacy and toxicity (pET) assuming independence of safety and efficacy at each dose level are expressed as:

pT = k1 X j=1 pTjpRj, pE = k1 X j=1 pEjpRj and pET = k1 X j=1 pEjpTj·pRj so that the marginal odds ratio is given by

pET(1−pE−pT +pET)

(pT −pET)(pE−pET)

. (5.13)

effect probability of efficacy in the next chapter. We refer to the three scenarios as the refer- ence scenario, Scenario 2 and Scenario 3. The dose-response curves for the three scenarios are given in Figure 5.2. In the three scenarios, the dose-response curve for the probability of toxicity is the same with(αT, βT) = (−2.5782,0.1621)and is given by the continuous line (—). The three scenarios differ in the parameter vector(αE, βE). For the reference sce- nario, the parameter vector(αE, βE) = (−1.4867,0.2720)and the dose-response curve is given by the dashed line (- - -). In Scenario 2,(αE, βE) = (−2.6226,0.3187)and the dose- response curve is given by the dotted line (_{· · ·}). For Scenario 3,(αE, βE) = (−0.8473,0) and the dose-response curve is given by the dashed and dotted line (_·-_·-_·).

Assuming a new drug is tested at the marked dose levels on the x-axis of Figure 5.2, that is dose levels 10.5mg, 35mg, 87.5mg, 262.5mg, 700.0mg and 1050.0mg, the marginal odds ratios for reference scenario, Scenario 2 and Scenario 3 respectively are 1.13, 1.14 and 1.0. In Scenario 3 the probability of toxicity increases with dose level and probability of efficacy does not change with the dose level so that an odds ratio of 1 would not be a bad assumption. Scenario 2 has a higher odds ratio than the reference scenario which is what we would desire. This is made possible since we assume some dose-response curves. The marginal odds ratio expression (5.13) holds even when probabilities of efficacy and toxicity are not modelled using some dose-response curves. Modelling the probabilities at each dose level independently may result in instances where the marginal odds ratio for Scenario 2 is less than the odds ratio for the reference scenario. By using different dose levels, as would be expected, we observed that the modelled odds ratios for the reference scenario and Scenario 2 are higher when: (1) patients are allocated to more dose levels, and (2) the experimental dose level are further apart.

To conclude, by modelling the probabilities of efficacy and the probabilities of tox- icity as described above, we assume that the probability of efficacy is independent of the probability of toxicity given dose subject to a given marginal odds ratio. The marginal odds ratio is induced by assuming some dose-response curves for the probabilities of ef-

5.6. REMARKS ON THE DOSE SELECTION PROCEDURE 99 Dose (mg) p(d) 10.5 35.0 87.5 262.5 700.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Toxicity Reference Scenario Scenario 2 Scenario 3

Figure 5.2: Different scenarios of dose response curves used to give examples of implied marginal associations.

ficacy and of toxicity. Thus, although we assume independence at each dose level, there is a restriction of the values the probabilities of efficacy and of toxicity can take. If there is correlation between efficacy and toxicity, we reduce the set of values probabilities of efficacy and of toxicity can take at each dose level so that the independence assumption is less strong compared to modelling outcomes (efficacy and toxicity) at each dose level independent and also outcomes at a dose level independent of the outcomes in other dose levels.