What is the logic underlying the ratio used in simple ANOVA? Explain, conceptually, how this test is used to evaluate the differences between means (i.e., what does the ratio look like when the test is statistically significant? Not significant?).

Question

What will be an ideal response?

Answer 1

Ans: Varies.

Simple ANOVA works by comparing systematic variance (or variance that results from differences between means) against error variance (or variance that is within-groups). Systematic variance reflects differences between groups who experienced different levels of an independent variable--it is the type of variance that researchers wish to accentuate through strong manipulations and it is in the numerator of the ratio. The denominator is the error variance. The error variance estimates how different the individuals in each group are from each other. Because each type of variance estimates population variance, we would expect the two to be similar where the null hypothesis is true. Thus, under the null hypothesis, between-group variance is equal to error variance, and the F (which is calculated using the ratio) is 1. Variation from this is expected by chance alone. If the test is not statistically significant, then researchers would expect their observed F given the null case of 1. If the extent to
which an observed F differs from 1 is
unlikely where the null is true, researchers
have a statistically significant result.