Analysis of Variance (ANOVA)

Definition

A set of procedures that estimate and attribute variance in a data set to different sources and determine the probability, under the null hypothesis, of obtaining the differences between the variance estimates by chance.

Distinctive Features

In ANOVA, variance attributable to differences between groups of scores is compared with an average variance attributable to differences between the scores within each group. Between-group variance is defined with respect to differences between the group means, and within-group variance is defined with respect to differences between the individual scores and their group mean. If the nature of the groups influences the scores more than chance fluctuation, then the between-group variance estimate will exceed the within-group variance estimate. If this difference between the variance estimates is sufficiently large, then the null hypothesis that all group means are equal is rejected.

The t-test compares the means of two experimental conditions. However, when there are more than two groups or conditions, more than one t-test is needed to compare all of the conditions. Unfortunately, the likelihood of obtaining a significant result by chance increases with the number of statistical tests carried out (that is, hypotheses tested). A Type 1 error is committed when the null hypothesis is rejected erroneously. Therefore, the Type 1 error rate increases with the number of statistical tests applied to any data set. ANOVA was developed to assist in the analysis of data obtained from agricultural experiments with any number of experimental conditions without any increase in Type 1 error. ANOVA procedures appropriate for an extensive variety of experimental designs are now available (e.g. Kirk, 1995).

Nevertheless, despite ANOVA being developed for use in experimental research, it may be applied to any data organized by categories. However, as with any statistical procedure, interpretation of the ANOVA results will depend upon the data collection methodology and the conformity of the data to the statistical assumptions underlying the analysis (e.g. Rutherford, 2001).

Evaluation

When regression, ANOVA and ANCOVA (analysis of covariance) are expressed in matrix algebra terms, a commonality is evident. Indeed, the same matrix algebra equation is able to summarize all three of these analyses. As regression, ANOVA and ANCOVA can be described in an identical manner, clearly they follow a common pattern. This common pattern is the GLM (general linear modelling) conception. It is said that regression, ANOVA and ANCOVA are particular instances of the GLM or that the GLM subsumes regression, ANOVA and ANCOVA. Unfortunately, the ability of the same matrix algebra equation to describe regression, ANOVA and ANCOVA has resulted in the inaccurate identification of the matrix algebra equation as the GLM. However, just as a particular language provides a means of expressing an idea, so matrix algebra provides only one notation for expressing the GLM.