Two-way Analysis of Variance (ANOVA²)

 For this example we will use this data set originally from STAT 414 Introduction to Mathematical Statistics where a physiologist wanted to determine if smoking history affected how a subject responds to different types of stress tests. The time to maximum oxygen uptake (VO_2max) was measured from subjects belonging to three different categories of smoking history (Nonsmoker, Moderate, and Heavy) while performing three different stress tests (Bicycle, Step Test, and Treadmill).

1. Does smoking history affect the time to VO_2max? H₀: α_Nonsmoker = α_Moderate = α_Heavy = 0 H_A: at least one α_i ≠ 0
2. Are there differences between the types of stress tests and the time to reach VO_2max? H₀: β_Bicycle = β_{Step Test} = β_Treadmill = 0 H_A: at least one β_j ≠ 0
3. Is there an interaction between the three types of stress tests and smoking history? H₀: αβ_ij = 0 H_A: αβ_ij ≠ 0

Fit a two-way ANOVA

 After loading our data set into JMP we can fit a two-way ANOVA by selecting Analyze -> Fit Model to bring up a new window where we will specify the parameters of our model. First, we put the Time variable into the Y box, then we add the Smoking History and Test Variables to the Construct Model Effects box. Since we also want to include the interaction term between Smoking History and Test we should also highlight both variables in the Select Columns box and then select Cross in the Construct Model Effects box. We can keep all of the other options at their defaults and select Run to fit our model.

Multiple Comparisons. In the pop-out window we can select the variable we want to run post-hoc tests for in the Choose an Effect box, check the box next to All Pairwise Comparisons - Tukey HSD, then click OK to perform the test.

 The results from the pairwise Tukey HSD test on the Test variable indicate that the mean time to maximum oxygen uptake is significantly different between each of the three stress tests, with Bicycle with the shortest time, then the Treadmill, and then the Step Test with the highest time. For Smoking History, the Tukey HSD results indicate that there is a significantly lower mean time to maximum oxygen uptake between Heavy and the Moderate and Nonsmoker groups while there is not a statistically significant difference between Moderate and Nonsmoker. Therefore, we could conclude that heavy smokers reach their maximum oxygen uptake during high intensity exercise faster than moderate smokers or non-smokers.