In this chapter, we will study sample size issues in dose-response trials. As indicated in 21 CFR 312.21, Phase I clinical investigation provides an initial introduction of an investigational new drug in humans. The primary objectives are to (1) determine the metabolism and pharmacological activities of the drug, the side effects associated with increasing dose and early evidence in effectiveness and (2) obtain sufficient information regarding the drug's pharmacokinetics and pharmacological effects to permit the design of well-controlled and scientifically valid Phase II clinical studies. Thus, Phase I clinical investigation includes studies of drug metabolism, bioavailibility, dose ranging, and multiple dose. For dose-escalation studies, clinical researchers usually start with low dose which is unlikely to present any harmful effects to subjects. Then, several cohorts of subjects are treated at progressively higher doses until a predetermined level of drug-related toxicity is reached. The level of drug-related toxicity is usually referred to as dose-limiting toxicity (DLT). In practice, since the test drug for progressive disease such as oncology is usually toxic and only a small number of patients are allowed for the study due to ethical consideration, the efficiency of the design is not only measured by its power but also the number of DLTs and accuracy for determination of maximum tolerated dose (MTD). In Phase II, dose-response studies are focused on the efficacy. Four questions are often of interest (Ruberg, 1995a, b): (1) Is there any evidence of the drug effect? (2) What doses exhibit a response different from the control response? (3) What is the nature of the dose-response? (4) What is the optimal dose? The first question is often essential and most hypothesis test methods try to answer this question; The second question can usually be addressed by Williams' test for minimum effective dose. The third question can be addressed by model-based approaches, either of frequentist or Bayesian type. The last question is multiple dimensional involving at least efficacy and safety components. We will limit our discussion on the sample size calculations for answering the first three questions. Specifically, in next section, some general concepts regarding sample size calculation are briefly reviewed. Section 14.3 provides a formula for sample size calculation for various study endpoints under multiple-arm response trials, including Williams' test for minimum effective dose for normal response, Cocharan-Armitage trend test for binary response, and a newly derived contrasts test for survival endpoint. Sample size estimation and related operating characteristics of Phase I dose escalation designs are discussed in Section 4. A brief concluding remark is given in the last section.

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