Let $M(x)$ denote the expected value at level $x$ of the response to a certain experiment. $M(x)$ is assumed to be a monotone function of $x$ but is unknown to the experimenter, and it is desired to find the solution $x = \theta$ of the equation $M(x) = \alpha$, where $\alpha$ is a given constant. We give a method for making successive experiments at levels $x_1,x_2,\cdots$ in such a way that $x_n$ will tend to $\theta$ in probability.
In recent work in the area of employment discrimination statistics we have noticed that a basic formula quantifying the degree of underadjustment bias in regression coefficients seems not to have been clearly stated and proved. We believe the underadjustment phenomenon to be of great importance in assessing the validity of multiple regression studies of wage disparities in equal employment litigation, and of observational studies involving two populations in general. Since the argument is based entirely on some simple mathematical features of the linear model, the result should be of interest in other fields of application as well.
Publisher Summary This chapter discusses sequential design of comparative clinical trials. If two treatments A and B of unknown efficacy are available and a large number N of patients is to be treated, then the trial consists of pairwise allocation of treatments to n pairs of patients. A conclusion can be made as to which treatment is superior so that the remaining N − 2n patients not on trial can be given the apparently superior treatment based on the results of the first n pairs. In 1963, Anscombe proposed a decision-theoretic approach to the problem of choosing n when N is given. The chapter describes the continuous-time problem and a class of stopping rules.
For testing which of two normally distributed treatments with a common variance is “superior” (has a larger mean response), we give a class of sequential probability ratio tests whose error probability functions are essentially independent of the sampling rule used. It is shown that the expected number of observations on the “inferior” treatment must be at least one-half that required by pairwise sampling, and a class of sampling rules is given which for some parameter values and significance levels almost attains this theoretical bound.
