

Threeway ANOVA, also called threefactor ANOVA, determines how a response is affected by three factors, for example:
•Treated vs. control
•Male vs. female
•Pretreatment with low vs. high dose
This example has two levels of each of the three factors, so there are 2x2x2=8 different treatment groups. This diagram might help this make sense.
Threeway ANOVA tests seven null hypotheses so reports seven P values. Yes seven! Threeway ANOVA is complicated.
Three of the P values test main effects:
•Null hypothesis 1: On average, the measured value is the same in males and females. So this P value compares the red vs. the blue cubes above.
•Null hypothesis 2: On average, the measured value is the same for treated and control. This P value compares the striped vs solid cubes above.
•Null hypothesis 3: On average, the measured value is the same when the pretreatment is low or high dose. This P value compares the dark colored cubes with the light colored cubes above.
Three of the P values test twoway interactions, and one tests a three way interaction. Here are the null hypotheses:
•Null hypothesis 4: Pooling male and female, the effect of treatment vs. control is the same for pretreatment with low and high dose.
•Null hypothesis 5: Pooling treated and control, is the effect of pretreatment with low and high dose the same for males and females.
•Null hypothesis 6: Pooling pretreatment with low and high dose, the effect of treatment vs. control is the same for males and females.
•Null hypothesis 7: There is no three way interaction among all three factors. This one is hard to understand.