1. Intro to Statistics for the Behavioral Sciences PSYC 1900
Lecture 13: ANOVA and Multiple Comparisons

2. Analysis of Variance (ANOVA)
Use is similar to t-tests, but allows comparisons of more than 2 means.
Logic?
With more than 2 means, one difference cannot capture predictions.
Variance of means is similar in concept to difference between 2 means.
If manipulations produce mean differences, than variance of means should increase.
Difference between two means is very similar to a variance measure.
t-tests are simplified case of ANOVA

3. Logic of One-way ANOVA
We calculate 2 estimates of the population variance.
MSwithin or MSerror is calculated as the mean variance of the group variances.
Does not depend on differences in the means.
MSgroup is calculated using the standard error of the group means.
This will equal the population variance only if the groups were drawn from the same population.
If the null is not true, MSgroup will be significantly greater than MSerror
“Significance” is determined using the F test.

4. Calculation Formulae for ANOVA
Notation:
Individual scores for person i in group j:
Means for group j:
Grand mean (mean of all scores):

5. Calculation Formulae for ANOVA
SStotal is the sum of squared deviations of all observations from the grand mean.
SSgroup is the sum of squared deviations of the group means from the grand mean.
SSerror is the sum of squared deviations within each group.

6. Calculation Formulae for ANOVA
Now we convert SS into Mean Squares (MS).
We divide SS by associated df’s.
dftotal=N-1
dfgroup= #groups-1
dferror=(#groups)(n-1)
F is the ratio of these two estimates of population variance.
Critical values of F are found using F(dfgroup, dferror),

7. Example Using Moral Judgment
Is moral judgment objective, or sensitive to actors?
Participants act in an unfair way, or observe another act in an unfair way, and judge fairness of actions.
4 conditions
Self, other, ingroup other, outgroup other
Prediction is that leniency will emerge for self and ingroup other.

8. Example Using Moral Judgment
Null hypothesis is that means of each population are equal (i.e., same population).
Population refers to scores emerging in a given condition.
The null can be invalidated in a number of ways.

10. What does this tell us?
At least one of the means differs from at least one of the others?
But this does not allow us to test our exact hypothesis.

11. Multiple Comparisons
Multiple comparisons are a family of techniques for making comparisons between means subsequent to an ANOVA.
Before choosing which type of comparisons, we must decide what level of Type I errors we are willing to accept.

12. Multiple Comparisons & Familywise Error
Each time we make an individual comparison with an alpha of .05, the chance of making a type I error is .05.
When we make several comparisons, the error rates are cumulative.
For example, the probability for making a type I error in a set of 5 comparisons is approximately .13.

13. Fisher’s Least Significant Difference (LSD) Test
One of the more liberal and common techniques is the LSD.
The LSD (or protected t) is conceptually similar to conducting t-tests on pairs of means.
It is called protected because the ANOVA tell us that at least one difference exists while controlling alpha at .05.
Does not completely keep familywise error at .05, but is acceptable compromise for small numbers of comparisons.

14. Fisher’s Least Significant Difference (LSD) Test
We use the basic t-test formula, but we replace the pooled variance error estimate with MSerror.
Makes sense as the MSerror should be a more accurate estimate of the population variance.
Test using t distribution with dferror degrees of freedom.

15. Bonferroni Procedure
Multiple comparison procedure in which familywise error rate is divided by the number of comparisons.
Keeps error rate at .05; very conservative.
You can then compute t-tests or LSD’s on comparisons, but only interpret differences as significant if the p-value is below the bonferroni critical value.

17. Tukey Test Another common procedure is the Tukey test.
Compares every mean to every other mean while maintaining familywise error rate at alpha.
Only really useful for post hoc or exploratory analyses due to conservative nature.

18. Effect Size
Instead of difference between two means, we look at ratio of explained variance.
Eta squared is simply this ratio:
18% of the variability in judgments is explained by the identity of the actor.

19. Effect Size Eta squared is conceptually simple, but biased.
It tends to overestimate effect size.
Omega squared is much less biased while measuring the same conceptual quantity:
15% of the variability in judgments is explained by the identity of the actor.