Analysis of Variance (ANOVA)

Why do we use ANOVA instead of multiple t-tests when comparing more than two groups?

Because multiple t-tests inflate the Type I error rate (familywise error), ANOVA uses a single test to control this error.

What is familywise error?

The total probability of making at least one Type I error when conducting multiple hypothesis tests on the same research question.

What is the general strategy of ANOVA?

Instead of comparing means directly, ANOVA compares variances using an F-ratio.

What does the F-statistic in ANOVA represent?

The ratio of between-group variance to within-group variance.

What are the two sources of variance in ANOVA?

• Between-subjects variance (MSB): variance of group means

• Within-subjects variance (MSW): variance within each group

What is the null hypothesis in a one-way ANOVA?

H0​:μ1​=μ2​=μ3​=…=μK​

What is the alternative hypothesis in ANOVA?

At least one group mean is different from at least one other.

What happens to the F-statistic if MSB increases?

The F-statistic becomes larger, making it more likely to reject the null hypothesis.

What happens to the F-statistic if MSW increases?

The F-statistic becomes smaller, making it less likely to reject the null.

How is total variance partitioned in ANOVA?

• SSTotal​=SSBetween​+SSWithin​

• dfTotal=dfBetween+dfWithindfTotal​=dfBetween​+dfWithin​

What is the formula for the F-ratio in ANOVA?

F=​MSBetween ​​=SSB​/dfB

——————————​​
MSWithin SSW​/dfW​

How are the degrees of freedom in ANOVA calculated?

• dfBetween=K−1dfBetween​=K−1

• dfWithin=N−KdfWithin​=N−K

What assumptions does ANOVA make?

1. Normality in all groups

2. Homogeneity of variances

What is the 7-step NHST process for ANOVA?

1. Define population distributions

2. State hypotheses

3. State assumptions

4. Define test statistic

5. Specify α and critical values

6. Calculate descriptive stats and check assumptions

7. Calculate F and make decision

What is ω² (omega squared)?

An effect size for ANOVA that estimates the proportion of variance in the dependent variable explained by the independent variable.

What is the formula for ω²?

ω2=​SSB​−MSW​(dfB​)​
SSTotal​+MSW

What are the interpretation guidelines for ω²?

• 0.01 = small

• 0.06 = medium

• 0.14 = large

Does ANOVA tell us which groups differ from each other?

No, it only tells us that at least one group differs. Post-hoc tests are needed for details.