The term analysis of variance refers to a very large body of statistical methods for regressing a dependent variable against indicator covariates associated with one or more categorical variables. Much of the literature on this topic is concerned with sophisticated study designs that could be evaluated using the mechanical or electric calculators of the last century. These designs are now of reduced utility in medical statistics. This is, in part, because the enormous computational power of modern computers makes the computational simplicity of these methods irrelevant, but also because we are often unable to exert the level of experimental control over human subjects that is needed by these methods. As a consequence, regression methods using categorical variables have replaced classical analyses of variance in many medical experiments today.

In this chapter we introduce traditional analysis of variance from a regression perspective. In these methods, each patient is observed only once. As a result, it is reasonable to assume that the model errors for different patients are mutually independent. These techniques are called fixed-effects methods because each observation is assumed to equal the sum of a fixed expected value and an independent error term. Each of these expected values is a function of fixed population parameters and the patient's covariates. In Chapter 11, we will discuss more complex designs in which multiple observations are made on each patient, and it is no longer reasonable to assume that different error terms for the same patient are independent.

One-way analysis of variance

A one-way analysis of variance is a generalization of the independent t test. Suppose that patients are divided into k groups on the basis of some categorical variable.