1. Analysis of Variance (ANOVA) Statistics for the Social Sciences Psychology 340 Spring 2010

2. PSY 340 Statistics for the Social Sciences Outline Basics of ANOVA Why Computations Post-hoc and planned comparisons Power and effect size for ANOVA Assumptions SPSS –1 factor between groups ANOVA –Post-hoc and planned comparisons

3. PSY 340 Statistics for the Social Sciences Outline Basics of ANOVA Why Computations Post-hoc and planned comparisons Power and effect size for ANOVA Assumptions SPSS –1 factor between groups ANOVA –Post-hoc and planned comparisons

4. PSY 340 Statistics for the Social Sciences Example Effect of knowledge of prior behavior on jury decisions –Dependent variable: rate how innocent/guilty –Independent variable: 3 levels Compare the means of these three groups Clean record Jurors Guilt Rating Criminal record No Information Guilt Rating

5. PSY 340 Statistics for the Social Sciences Analysis of Variance XBXB XAXA XCXC Criminal recordClean recordNo information 1054 716 539 73 843 –Need a measure that describes several difference scores Variance Test statistic Observed variance Variance from chance F-ratio = More than two groups

6. PSY 340 Statistics for the Social Sciences Testing Hypotheses with ANOVA –Step 2: Set your decision criteria –Step 3: Collect your data –Step 4: Compute your test statistics Compute your estimated variances Compute your F-ratio Compute your degrees of freedom (there are several) –Step 5: Make a decision about your null hypothesis Hypothesis testing: a five step program –Step 1: State your hypotheses –Additional tests: Planned comparisons & Post hoc tests Reconciling our multiple alternative hypotheses

7. PSY 340 Statistics for the Social Sciences Null hypothesis: H 0 : all the groups are equal XBXB XAXA XCXC –Step 1: State your hypotheses Hypothesis testing: a five step program Alternative hypotheses (H A ) –Not all of the populations all have same mean The ANOVA tests this one!! The ANOVA tests this one!! Testing Hypotheses with ANOVA Choosing between these requires additional test

8. PSY 340 Statistics for the Social Sciences 1 factor ANOVA XBXB XAXA XCXC Alternative hypotheses (H A ) –Not all of the populations all have same mean Planned contrasts and Post-hoc tests: –Further tests used to rule out the different alternative hypotheses –reject –fail to reject

9. PSY 340 Statistics for the Social Sciences Why do the ANOVA? What’s the big deal? Why not just run a bunch of t- tests instead of doing an ANOVA? –Experiment-wise error (see pg 398, Box 13.1 for discussion) –The type I error rate of the family (the entire set) of comparisons »α EW = 1 - (1 - α) c where c = # of comparisons »e.g., If you conduct two t-tests, each with an alpha level of 0.05, the combined chance of making a type I error is nearly 10 in 100 (rather than 5 in 100) –Planned comparisons and post hoc tests are procedures designed to reduce experiment-wise error

10. PSY 340 Statistics for the Social Sciences Which follow-up test? Planned comparisons –A set of specific comparisons that you “planned” to do in advance of conducting the overall ANOVA Post-hoc tests –A set of comparisons that you decided to examine only after you find a significant (reject H 0 ) ANOVA –Often end up looking at all possible pair-wise comparisons

11. PSY 340 Statistics for the Social Sciences Planned Comparisons General Rule of Thumb –Don’t plan more contrasts than (# of conditions – 1) Different types –Simple comparisons - testing two groups –Complex comparisons - testing combined groups –Bonferroni procedure (Dunn’s test) Use more stringent significance level for each comparison –Divide your desired α-level by the number of planned contrasts

12. PSY 340 Statistics for the Social Sciences Planned Comparisons Basic procedure: 1.Within-groups population variance estimate (denominator) 2.Between-groups population variance estimate of the two groups of interest (numerator) 3.Figure F in usual way

13. PSY 340 Statistics for the Social Sciences Planned Comparisons Example: compare criminal record & no info grps XBXB XAXA XCXC Criminal recordClean recordNo information 1054 716 539 73 843 1) Within-groups population variance estimate (denominator) 2) Between-groups population variance estimate of the two groups of interest (numerator)

14. PSY 340 Statistics for the Social Sciences Planned Comparisons Example: compare criminal record & no info grps Criminal recordClean recordNo information 1054 716 539 73 843 1) Within-groups population variance estimate (denominator) 2) Between-groups population variance estimate of the two groups of interest (numerator) 3) Figure F in usual way F crit (1,12) = 4.75 α = 0.05 Fail to reject H 0 : Criminal record and no info are not statistically different XBXB XAXA XCXC

15. PSY 340 Statistics for the Social Sciences Post-hoc tests Generally, you are testing all of the possible comparisons (rather than just a specific few) –Different types Tukey’s HSD test (only with equal sample sizes) Scheffe test (unequal sample sizes okay, very conservative) Others (Fisher’s LSD, Neuman-Keuls test, Duncan test) –Generally they differ with respect to how conservative they are.

16. PSY 340 Statistics for the Social Sciences Effect sizes in ANOVA The effect size for ANOVA is r 2 –Sometimes called η 2 (“eta squared”) –The percent of the variance in the dependent variable that is accounted for by the independent variable Recall:

17. PSY 340 Statistics for the Social Sciences Effect sizes in ANOVA The effect size for ANOVA is r 2 –Sometimes called η 2 (“eta squared”) –The percent of the variance in the dependent variable that is accounted for by the independent variable

18. PSY 340 Statistics for the Social Sciences ANOVA Assumptions Basically the same as with T-tests –Assumes that the distributions are Normal –Assumes that the distributions have equal variances –In both cases ANOVA analyses are generally robust against violations of these assumptions

19. PSY 340 Statistics for the Social Sciences ANOVA in SPSS Let’s see how to do a between groups 1-factor ANOVA in SPSS (and the other tests too) –Enter the data: similar to independent samples t-test, observations in one column, a second column for group assignment –Analyze: compare means, 1-way ANOVA Observations -> Dependent list Group assignment -> factor – specify any comparisons or post hocs at this time too Planned Comparisons (contrasts): are entered with 1, 0, & -1 Post-hoc tests: make sure that you enter your α-level

20. PSY 340 Statistics for the Social Sciences Analysis of Variance Criminal recordClean recordNo information 1054 716 539 73 843