Question 709:

Asked on April 30, 2009

Tags: ANOVA , 2-Way ANOVA , Factorial ANOVA

1

Answer:

No answer provided yet.For one independent variable with 3 or more levels, the One-Way ANOVA is the way to go. With more than one independent variable you need to use the Factorial Analysis of Variance. For 2 independent variables, there is a special Factorial ANOVA called the 2-Way ANOVA. 

Factorial ANOVAs have 3 general advantages.

1. They allow for a broader interpretation of the results than say multiple 1-Way ANOVAs since they allow you to see the full range of results for all independent variables.
2. They provide interaction effects, that is show when certain levels of one variable result in higher means with certain combinations of other levels of variables.
3. Economy or getting more with fewer participants. Since we are examining multiple variables and their interactions simultaneously,  we don't need to conduct multiple 1-Way ANOVAs, so we in short can get more results wit the same sample size as a 1-Way ANOVA.

For all the ANOVA's lauded power, the major draw back is that should we get a significant p-value, we don't know which means are significantly different, we only know that at least 1 means is significantly different than another. We then need to conduct paired comparison tests such as the t-test.

Of course, should we need to go to the t-test anyway, this begs the question of why not just conduct several t-tests instead of bothering with the ANOVA. The two major drawbacks are we won't be able to see interactions, and the more means we need to compare, the more we capitalize on chance and inflate our Type I error probability (saying there is a significant difference when one really doesn't exist).

When the tests are planned ahead of time, the inflation of chance is not really there. This is really a problem when we begin testing all combinations of means to check for significant differences.  These so called post-hoc tests (post hoc from Latin meaning after the fact) come in a variety of forms. The 2 most common are the Tukey test or the Sheffe' Test, along with many others. They are mostly variants on the t-test which attempt to control for the inflation of the aforementioned Type I error rate.