Analysis of Variance (ANOVA)

Usually a two-sample t test is applied to test for a significant difference between two population means based on the two samples. For example, consider the data in Table 1. Twenty patients with high blood pressure are randomly assigned to two groups of 10 patients. Patients in Group 1 are assigned to receive placebo, while patients in Group 2 are assigned to receive Drug A. Patients’ systolic blood pressures (SBPs) are measured before and after treatment, and the differences in SBPs are recorded in Table 1. A two-sample t test would be an efficient method for testing the hypothesis that drug A is more effective than placebo when the differences in before and after measurements are normally distributed. However, there are usually more than two groups involved for comparison in many fields of scientific investigation. For example, extend the data in Table 1 to the data in Table 2. Here the study used 30 patients who are randomly assigned to placebo, Drug A, and Drug B. The goal here is to compare the effects of placebo and experimental drugs in reducing SBP. But a two-sample t test is not applicable here as we have more than two groups. Analysis of variance (ANOVA) generalizes the idea of the two-sample t test so that normally distributed responses can be compared across categories of one or more factors.

Since its development, ANOVA has played an indispensable role in the application of statistics in many fields, such as biology, social sciences, finance, pharmaceutics, and scientific and industrial research. Although ANOVA can be applied to various statistical models, and the simpler ones are usually named after the number of categorical variables, the concept of ANOVA is based solely on identifying the contribution of individual factors in the total variability of the data. In the above example, if the variability in SBP changes due to the drug is large compared with the chance variability, then one would think that the effect of the drug on SBP is substantial. The factors could be different individual characteristics, such as age, sex, race, occupation, social class, and treatment group, and the significant differences between the levels of these factors can be assessed by forming the ratio of the variability due to the factor itself and that due to chance only.