We know how to compare two samples using a t-test, so we might be tempted to just perform lots of pairwise t-tests. The problem is that doing all of these tests will inflate our Type-I error rate. Instead we want a single test of our single null hypothesis.

The basics of ANOVA. How does ANOVA provide a single test for whether all population means are equal across all of our groups?

An ANOVA uses an F test to calculate a P-value. The F statistics is based on the ratio of among-group variation relative to within-group variation. Here we will go over where all of these numbers come from and how they are calculated.

We can measure the proportion of variation that is explained by our among-group factor using a R-square.

While an ANOVA can tell us whether there are differences among groups, it cannot tell us which means are different from which. To do this, we need to perform a post hoc test.

So far we have talked only about the effects of one factor on a continuous response, but ANOVA can be used to test the effects of multiple factors and their statistical interactions at the same time.

1. In an ANOVA with 4 groups, a rejection of the null hypothesis implies which of the following?

   (A) Some subset of population means differs from some other subset of population means.

   (B) The 4 population means are equal to each other.

   (C) Each population mean differs significantly from all other population means.

2. What do you think the F value will be for the following figure?

3. What is a non-parametric alternative to the ANOVA?