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Factorial ANOVA: Within-within example

This is an excerpt from Statistics in Kinesiology 6th Edition by Joseph P. Weir,Anthony B. Ciccone,Jacob A. Siedlik,William J. Vincent.

A researcher wanted to know whether physiological differences existed in response to exercise on a traditional treadmill compared with exercise on a stair-step treadmill. Thirteen healthy male college students were randomly selected and asked to report to the laboratory on two different days with at least one rest day in between. On the first day, subjects were asked to perform a graded submaximal exercise test on a treadmill. The test consisted of four increasing stages of work. Heart rate responses were recorded at each stage as dependent variables. The amount of physical work performed on the treadmill at each stage was calculated.

On the second day, the same subjects performed an identical amount of physical work over four increasing stages on a stair-step treadmill device. Hence, the same subjects performed on two different modes of exercise (factor A—within) and over four equivalent stages of work (factor B—within). This design may be categorized as a 2 × 4 within–within factorial ANOVA. The data layout for the example is shown in table 14.9. (Note: Set up the computer database for this problem as follows in table 14.9.)

The analysis of this design is similar to the analysis of between–within, but because both factors are repeated measures on the same subjects, the assumptions of sphericity must be met for both factors. The analysis produces three F values: main effect of mode of exercise (factor A), main effect for stages of exercise (factor B), and interaction. If interaction is not significant, then subjects respond in a similar manner on both modes of exercise to the identical graded exercise stimulus. If the interaction is significant, then physiological responses are different between the two modes of exercise. The mean values for heart rate are presented in table 14.10, and the factorial ANOVA summary is presented in table 14.11.

Step-Down Analysis

The F for the main effect for mode (factor A) is not significant; therefore, no further analysis is justified. We can conclude that the heart rate response, averaged across the four stages, is similar between treadmill and stair exercise. The F value for the main effect for stages (factor B) is significant (p [less than] .001). This indicates that the heart rate values, averaged across the two modes of exercise, vary across the different stages. A post hoc test may be used to determine which stages differ. Tukey’s HSD at p [less than] .01 is ~10. Because every stage for both modes is more than 10 beats per minute higher than the previous stage, we can conclude that heart rate increases significantly at every stage. The F value for interaction is not significant, indicating no differences in the slopes of the lines. Figure 14.9 presents the data in graphic format. As always, when interpreting and reporting the results, for pairwise comparisons, include calculation of the 95% CI for the mean differences and an estimate of effect size, such as Cohen’s d.

Conclusions

1. The F value for mode is not significant; therefore, we concluded that the heart rate did not differ on the two exercise modes.
2. The F value for stages is highly significant. Although a small violation of circularity existed for stages, it was not sufficient to alter the p value. The increase in heart rate was expected because physical work increased stage by stage. Indeed, if we did not see a significant increase in heart rate over stages, we would suspect an error. Tukey’s HSD indicates that a statistically significant increase in heart rate occurred between each of the four stages.
3. The F value for interaction is not significant. This confirms the insignificant F value for mode and provides further evidence that the subjects responded physiologically in the same manner on both modes of exercise over all four stages.
4. Note: If more than one physiological variable is included (e.g., heart rate, V̇O₂, blood pressure, respiratory exchange ratio, and so on), the data should be analyzed using MANOVA (see chapter 18).

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