1. Chapter 10 Introduction to the Analysis of Variance
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

2. Analysis of Variance Abbreviated as “ANOVA”
Used to compare the means of more than two samples
Null hypothesis is that all populations being studied have the same mean
Reject null if at least one population has a mean that differs from the others
Actually works by analyzing variances
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

3. Two Different Ways of Estimating Population Variance
Estimate population variance from variation within each sample
Is not affected by whether or not null hypothesis is true
Estimate population variance from variation between each sample
Is affected by whether or not null hypothesis is true
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

4. Two Important Questions
How can you estimate population variation from variance between samples?
How is that estimate affected by whether or not the null is true?
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

5. Estimate population variance from variation between means of samples
First, variation among means of samples is related directly to the amount of variation within each population from which samples are taken
The more variation within each population, the more variation in means of samples taken from those populations
Note that populations on the right produce means that are more scattered
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

6. Estimate population variance from variation between means of samples
And second, when null is false there is an additional source of variation
When null hypothesis is true (left), variation among means of samples caused by
Variation within the populations
When null hypothesis is false (right), variation among means of samples caused by
And also by variation among the population means
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

7. Basic Logic of ANOVA
ANOVA entails a comparison between two estimates of population variance
Ratio of between-groups estimate to within-groups estimate called an F ratio
Compare obtained F value to an F distribution
Analogy of signal-to-noise ratio…
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

8. Carrying Out an ANOVA
Step 1: Estimate population variance from variation of scores within each group
Pool variance estimates from each sample
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

9. Carrying Out an ANOVA
Step 2: Estimate population variance from variation of scores within each group
Estimate the variance of distribution of means
Figure the estimated variance of the population
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

10. Carrying Out an ANOVA Step 3: Compute F ratio
Divide between-groups estimate of the population variance by the within-groups estimate of the population variance
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

11. Carrying Out an ANOVA Step 4: Compare obtained F to an F distribution
Compute degrees of freedom for both between-groups estimate and within-groups estimate
Consult an F table
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

12. Assumptions of an ANOVA
Populations follow a normal curve
Populations have equal variances
As for t tests, ANOVAs often work fairly well even when those assumptions are violated
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

13. Rejecting the Null Hypothesis
A significant F tells you that at least one of the means differs from the others
Does not indicate how many differ
Does not indicate which one(s) differ
For more specific conclusions, a researcher must conduct follow-up t tests
Problem: Lots of t tests increases the chances of finding a significant result just by chance (i.e., increases chances beyond p = .05)
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

14. Effect Size
To determine effect size of an ANOVA, divide square root of F by square root of n, the number of scores in each group
Small = .10
Medium = .25
Large = .40
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

15. Factorial ANOVA
Procedure that allows one to examine two or more variables in the same study
Efficient
Allows for examination of interaction effects
An ANOVA with only one variable is a one-way ANOVA, an ANOVA with two variables is a two-way ANOVA, and so on
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

16. Main Effects vs. Interactions
A main effect refers to the effect of one variable, averaging across the other(s)
An interaction effect refers to a case in which the effect of one variable depends on the level of another variable
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall

17. Main Effects vs. Interactions
In a two-way ANOVA, there can be two main effects and one interaction
Any combination of significant and non-significant results is possible
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall