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 data when three or more groups or conditions are compared with independent groups or repeated measures. The ANOVA uses the F statistic, named for Sir Ronald Fisher, who developed the procedure.
The purpose of this chapter is to describe the application of the ANOVA for a variety of research designs. Although statistical programs can run the ANOVA easily, understanding the basic premise is important for using and interpreting results appropriately. 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. Discussion will then follow with more complex models, including factorial designs, repeated measures and mixed designs.
Recall from our discussion of the t-test (Chapter 24) that differences between means are examined in relation to the distance between group means as well as the error variance within each group—the signal to noise ratio. The analysis of variance is based on the same principle, except that it is a little more complicated because the ratio must now account for the relationships among more than two means.
Also like the t-test, the ANOVA is based on statistical assumptions for parametric tests, including homogeneity of variance, normal distributions, and interval/ratio data. When assumptions for parametric statistics are not met, there are nonparametric analogs that can be applied (see Chapter 27).
The ANOVA can be applied to two-group comparisons, but the t-test is generally considered more efficient for that purpose. The results of a t-test and ANOVA with two groups will be the same. The t-test is actually a special case of the analysis of variance, with the relationship F = t2
One-Way Analysis of Variance
The one-way analysis of variance is used to compare three or more independent group means. The descriptor “one-way” indicates that the design involves one independent variable, or factor, with three or more levels.
➤ CASE IN POINT #1
Consider a hypothetical study of the differential effect of four intervention strategies to increase pain-free range of motion (ROM) in patients with elbow tendinitis. Through random assignment, we create 4 independent groups treated with ice, a nonsteroidal anti-inflammatory drug (NSAID), a splint support, or rest. We measure ROM at baseline and after 10 days of intervention, using the change in pain-free ROM as our outcome. We recruit a sample of 44 patients with comparable histories of tendinitis and randomly assign 11 subjects to each of the four groups.
In this study, the independent variable modality has four levels, ...