The experiments we analyzed in Chapter 8 were ones in which the subjects were divided according to a single factor, such as mental status or diet. Although many experiments can be analyzed using that design, there are also times when one wishes to divide the experimental subjects into groups according to two factors. For example, in Chapter 8, we tested the hypothesis that circulating cortisol levels differed among normal, nonmelancholic depressed, and melancholic depressed people. Now, suppose that we wanted to investigate whether or not the results depended on the gender of the person studied. In this case, we have a two-factor, or two-way, analysis-of-variance problem in which each individual experimental subject is classified according to two factors (Table 9-1). Although fundamentally the same as the single-factor analysis of variance discussed in Chapter 8, the two-factor analysis of variance provides a different perspective on the data.
Table 9-1Schematic Diagram of Design of Hypothetical Two-Factor Analysis of Variance of Depression Effect on Cortisol Level, Also Taking into Account the Gender of the Individuals |Favorite Table|Download (.pdf) Table 9-1 Schematic Diagram of Design of Hypothetical Two-Factor Analysis of Variance of Depression Effect on Cortisol Level, Also Taking into Account the Gender of the Individuals
| ||Diagnosis |
|Gender ||Healthy ||Nonmelancholic Depressed ||Melancholic Depressed |
|Male || || || |
|Female || || || |
First, two-way analysis of variance allows us to use the same data to test three hypotheses about the data we collected. We can test whether or not the dependent variable changes significantly with each of the two factors (while taking into account the effects of the other factor) as well as test whether or not the effects of each of the two factors are the same regardless of the level of the other one. This third hypothesis is whether or not there is a significant interaction effect between the two factors. In terms of the hypothetical study of the dependence of cortisol concentration on mental state and gender, these three hypotheses would be
Cortisol concentration does not depend on mental state (the same hypothesis as before).
Cortisol concentration does not depend on gender.
The effect of mental state on cortisol does not depend on gender.
Second, all things being equal, a two-factor analysis of variance is more sensitive than multiple single factor analyses of variance looking for similar effects. When one sets up an experiment or observational study according to the analysis of variance paradigm, the experimental subjects are assigned at random to the different treatment groups, so the only difference between the different groups is the treatment (or, in an observational study, the presence of a certain characteristic of interest). The randomization process is designed to eliminate systematic effects of other potentially important characteristics of the experimental subjects by randomly distributing these characteristics across all the sample groups. It is this assumption ...