One-Way ANOVA

Analysis of variance, which is abbreviated as ANOVA, was developed by Sir Ronald Fisher (1925) to determine whether the means of one or more factors and their interactions differed significantly from that expected by chance. “One-way” refers to an analysis that involves only one factor. “Two-way” (or more ways) refers to an analysis with two or more factors and the interactions between those factors. A factor comprises two or more groups or levels. Analysis of variance is based on the assumption that each group is drawn from a population of scores that is normally distributed. In addition, the variance of scores in the different groups should be equal, or homogeneous.

The scores in the different groups may be related or unrelated. Related scores may come from the same cases or cases that have been matched to be similar in certain respects. An example of related scores is where the same variable (such as job satisfaction) is measured at two or more points in time. Unrelated or independent scores come from different cases. An example of unrelated scores is where the same variable is measured in different groups of people (such as people in different occupations). An analysis of variance consisting of only two groups is equivalent to a t-test where the variances are assumed to be equal.

A one-way analysis of variance for unrelated scores compares the population estimate of the variance between the groups with the population estimate of the variance within the groups. This ratio is known as the F-test, F-statistic, or F-value in honor of Fisher and is calculated as

The bigger the between-groups variance is in relation to the within-groups variance estimate, the larger F will be and the more likely it is to be statistically significant. The statistical significance of F is determined by two sets of degrees of freedom: those for the between-groups variance estimate (the numerator or upper part of the F ratio) and those for the within-groups variance estimate (the denominator or lower part of the F ratio). The between-groups degrees of freedom are the number of groups minus 1. The within-groups degrees of freedom are the number of cases minus the number of groups.

In a one-way analysis of variance for related scores, the denominator of the F ratio is the population estimate of the within-groups variance minus the population estimate of the variance due to the cases. The degrees of freedom for this denominator are the number [Page 765]of cases minus 1 multiplied by the number of groups minus 1.

Where the factor consists of more than two groups and F is statistically significant, which of the means differ significantly from each other needs to be determined by comparing them two at a time. Where there are good grounds for predicting the direction of these differences, these comparisons should be carried out with the appropriate t-test. Otherwise, these comparisons should be made with a post hoc test. There are a number of such tests, such as the scheffé test.

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