Question

How would you know if a two way ANOVA resulted in the rejection of equal means? Does this mean that the means of the two factors were not equal?

Answer 1

In two way ANOVA there are two factors with different levels of each factors.

First we need to understand the hypothesis in two way ANOVA.

There are three sets of hypothesis with the two-way ANOVA.

The null hypotheses for each of the sets are given below.

The population means of the first factor are equal.

This is like the one-way ANOVA for the row factor.

The population means of the second factor are equal.

This is like the one-way ANOVA for the column factor.

There is no interaction between the two factors.

This is similar to performing a test for independence with contingency tables.

Here we are interesting the meaning of rejection of equal means.

If we reject null hypothesis corresponding to the row factor then the means of row factor are different. Note that the number of levels included in row factor are same as the number of means of testing of row factor.

If we reject null hypothesis corresponding to the column factor then the means of column factor are different. Note that the number of levels included in column factor are same as the number of means of testing of row factor.

From the above discussion it is clear that the rejection of means does not mean that the means of the two factors were not equal.