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Chapter 20: Comparing More than Two Means: Analysis of Variance

INTRODUCTION

As knowledge and clinical theory have developed, clinical researchers have proposed more complex research questions, necessitating the use of elaborate multilevel and multifactor experimental designs. The analysis of variance (ANOVA) is a powerful analytic tool for analyzing such designs, where three or more conditions or groups are compared. The analysis of variance is used to determine if the observed differences among a set of means are greater than would be expected by chance alone. The ANOVA is based on the F statistic, which is similar to t in that it is a ratio of between-groups treatment effects to within-group variability. The test can be applied to independent groups or repeated measures designs.∗

The purpose of this chapter is to describe the application of the analysis of variance for a variety of experimental research designs. An introduction to the basic concepts underlying analysis of variance is most easily addressed in the context of a single-factor experiment (one independent variable) with independent groups. We then follow with discussions of more complex models, including factorial designs and repeated measures designs.

∗As with all parametric tests, the ANOVA is based on the assumption that samples are drawn randomly from normally distributed populations with equal variances. Tests for homogeneity of variance can be performed to validate the latter assumption. With samples of equal size, the analysis of variance is considered "robust" in that reasonable departures from the assumptions of normality and homogeneity will not seriously affect the validity of inferences drawn from the data.1 With unequal sample sizes, gross violations of homogeneity of variance can increase the chance of Type I error. In such cases, a nonparametric analysis of variance can be applied (see Chapter 22), or data can be transformed to a different scale that improves homogeneity of variance within the sample distribution (see Appendix D).

ANALYSIS OF VARIANCE FOR INDEPENDENT SAMPLES: ONE-WAY CLASSIFICATION

In a single-factor experiment, the one-way analysis of variance is applied when three or more independent group means are compared. The descriptor "one-way" indicates that the design involves one independent variable, or factor, with three or more levels.

Although the ANOVA can be applied to two-group comparisons, the t-test is generally considered more efficient for that purpose.†

Statistical Hypotheses

The null hypothesis for a one-way multilevel study states that there is no significant difference among the group means, represented by

image

where k is the number of groups or levels of the independent variable. The alternative hypothesis (H1) states that at least two means will differ.

†The results of a t-test and analysis of variance with two groups will be the same. The t-test is actually a special case of the analysis ...

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