Two-way Analysis of Variance

Look at the following graphic:
  1. There is a line connecting each level of the means for A1, A2 and A3. If we did not consider Gender, then this line would be the main effect of A. In Panel 1, the means differ so there would be a main effect of A. This is not the case in Panel 2.
  2. In Panel 3, all the B1s are higher than B2. Thus there is a main effect of B. Note that A has no effect for B1 or B2. So there is no effect of A or an AB interaction. The latter is because A has no differential effect dependent on B.
  3. In Panel 4, A has an effect as Year increases drinking increases. Gender (B) has an effect as the B1 line is higher than the B2 line.
    Big Point Alert!!!!
    The B1 line is parallel to B2 line. Year effects each Gender in the same fashion.
    There is NO INTERACTION!
  4. In Panel 5, A has an effect, drinking increases with age for all groups. B has an effect, as the B1 line is higher than the B2 line.
    Big Point Alert
    The B1 line is STEEPER than the B2 line. Thus the year can have a different effect dependent on your gender. This is an interaction.
  5. In Panel 6 - the slopes are opposite. This is a cute one. If you compute the average for B1 Vs. B2, forgetting about Year, then there is no effect of gender. If you average the Years forgetting about Gender, the years don't differ. You would get NO MAIN EFFECTS. THERE IS ONLY AN INTERACTION. That's why we check it first.

A Big Set of Ideas!!

The analysis is based on the following linear model:

Your score = m+a+b+ab+e

We will Calculate F-ratios for Each of these:

Fa, Fb, FAB

The Idea of these F-ratios
  1. Significant F for Main Effect of A means there are some differences between the A groups (A1, A2 or A3 - Years). Remember that the Genders are combined. An insignificant F means that we did not find any differences between the years with Gender combined. ALERT - ALERT - we need to look at the Interaction - It ain't over yet!

  2. Significant F for Main Effect of B means there are some differences between the B groups (B1 Vs. B2 - Gender). Remember that the Years are combined. An insignificant F means that we did not find any differences between the Genders with Years combined. ALERT - ALERT - we need to look at the Interaction - It ain't over yet!

AB. The Interaction is Significant!!

What this means is that if you now look at the effects of A for a level of B and compare it to the effects of A for another level of B, they may differ.

In English, the effect of year on Drinking would be different for each Gender. Maybe Males increase their drink more over the 3 years than Females. Maybe Males decrease drinking and females increase.

In any case, there can be differences between the years but these differences are different for each gender.