Profit based phase II sample size determination when adaptation by design is adopted

Background. Adaptation by design consists in conservatively estimating the phase III sample size on the basis of phase II data, and can be applied in almost all therapeutic areas; it is based on the assumption that the effect size of the drug is the same in phase II and phase III trials, that is a very common scenario assumed in product development. Adaptation by design reduces the probability on underpowered experiments and can improve the overall success probability of phase II and III trials, but increases phase III sample size enlarging time and cost of drug development, also reducing the potential time on market of the drug under study. In this work, the aim is to build a profit model under the assumption that adaptation by design is applied, in order to compute the dependence of profit from phase II sample size and conservativeness and to appropriately size phase II on the basis of profit behavior. Methods. Recent theoretical results on adaptation by design (viz. conservative sample size estimation) provide the probabilistic distribution of phase III sample size and the probability of launching phase III. The moments of phase III sample size, viewed as a random variable, can be computed, in function of the phase II sample size and of the amount of conservativeness. Results. The modeled revenue depends on: income per patient, annual incidence, time on market, market share, phase III success probability. The modeled cost depends on: fixed cost of the two phases, cost per patient under treatment. Profit is revenue minus cost, and it depends on the random phase III sample size. So profit moments depend on phase II sample size and conservativeness. To consider expected profit and profit volatility is mandatory, in agreement with modern evaluation of investment performances. The utility, a linear function of profit expectation and volatility, is therefore evaluated. Phase II sample size can be determined on the basis of utility, for example optimizing utility, or achieving a given utility value. An application shows how profit expectation, volatility and utility depend on phase II sample size and conservativeness, and how phase II sample size can be determined. Conclusion. It has been shown how profit and profit utility depend on phase II sample size, amount of conservativeness, launching rule. A suitable setting of these adaptation by design operational parameters can improve profit and increase profit utility. Adaptation by design can be adopted in many different statistical problems. Consequently, the profit evaluations here proposed can be widely applied, together with profit based phase II sample size determination.