A General Framework

This chapter distinguishes between hypothesis testing (Section 2.1) and parameter estimation (Section 2.2). We begin with simple settings in which the test statistic and treatment effect estimator are a sum and mean, respectively, of independent and identically distributed (i.i.d.) random variables.We show that in less simple settings, the test statistic and treatment effect estimator behave as if they were a sum and mean, respectively, of i.i.d. random variables. This leads naturally to the concept of a sum process (S-process) behaving like a sum and an estimation process (E-process) behaving like a sample mean. Following the approach of Lan and Zucker (1993) [LZ93] and Lan and Wittes (1988) [LW88], we show the connection between S-processes, E-processes, and Brownian motion. We use Brownian motion to approximate the joint distribution of repeatedly computed test statistics over time for many different trial settings, including comparisons of means, proportions, and survival times, with or without adjustment for covariates. Because of our extensive use of Brownian motion, we were tempted to subtitle this chapter “Brown v. the Board of Data Monitoring.”

This chapter, which presents the general framework for the rest of the book, is necessarily long. The reader may prefer to read the first three sections containing the essential ideas applied to tests of means, proportions, and survival, and then proceed to the next chapter showing how to apply Brownian motion to compute conditional power. The reader may then return to this chapter to see how to use the same ideas in more complicated settings such as maximum likelihood or minimum variance estimation, or even mixed models. While digesting the next sections, the reader should keep in mind the essential idea throughout this chapter—test statistics and estimators behave like sums and sample means, respectively, of i.i.d. random variables.