1. PSY 1950 Fixed and Random Effects October 20, 2008

2. Preamble Midterm –Review –Room Homework Simple effects –Simple main effects –Simple interaction effects

3. The Embarrassing Footnote With fixed effects analysis, one can’t generalize beyond measured levels of factor –e.g., the influence of expert communicators

4. The Embarrassing Footnote

5. Fixed effects and random effects analyses treat different variables as randomly sampled –For FE, randomly sampled variable is subject –For RE, randomly sampled variable is factor Fixed effects and random effects analyses use different error term (i.e., the denominator in an F- ratio) –For FE, error term is within-group variance –For RE, error term is interaction term Why? –If interactions are present, random sampling of levels introduces additional variability that MS within does not capture Random Effects

6. Example 4 College x 3 Test Test is fixed factor, college is random factor F fixed effect = MS test /MS within F random effect = MS test /MS test x college

7. Fixed vs. Random Effects Generalization –FE: no generalization beyond measured levels –RE: generalization beyond measured levels Selection of levels –FE: nonrandom –RE: random Interest in levels –FE: focused (e.g., planned/post-hoc tests) –RE: general Replication –FE: same levels –RE: different levels

8. The Weak Test of Generality RE analyses sacrifice power for generality –Reduction in F-ratio –Reduction in df One on hand… fixed effects –Power, but no generality On the other hand… random effects –Generality, but no power

9. Minimizing the Dilemma RE model Huge main effect Small interaction effect Many levels of random factor FE model Representative levels of ordered factor e.g., age, angle of rotation Exhaustive levels e.g., gender

10. Item analyses Prof. Snedeker: “Psycholinguists do one thing which is different from most areas of psychology. We do all of our analyses twice: once with subject as the random variable (averaging across items), and once with item as the random variable (averaging across subjects). The goal is to understand whether the results generalize both to the population of possible participants and to the population of possible items (words, sentences etc). I don't expect the stats course to cover this (though it might help the students grasp the notion of a random variable)”

11. t-test is Special Case of ANOVA (k=2)

12. Contrast Weighting w/ Zero With odd number of groups, contrast weights for some trends require weight of zero –e.g., linear trend w/ 3 groups: -1, 0, 1

13. a1a1 a2a2 a3a3 01 M1M1 M2M2 M3M3 234 a1M1a1M1 a2M2a2M2 a3M3a3M3 -204

14. ANOVA Effect Size: Eta Advantages: conceptual simplicity Disadvantages: biased, depends on other factors/effects, depends on design/blocking Advantages: does not depend on other factors/effects Disadvantages: biased, conceptually complexity, depends on design/blocking

15. ANOVA Effect Size: Beyond Eta Omega-squared (  2 ) and partial omega- squared (partial  2 ) –Not biased estimators of population effect size –Better than eta for inferential purposes Generalized eta and omega –cf. Bethany’s presentation –Correct/control for research design Independent measures ANOVA and dependent measures ANOVA designs that investigate the same effect produce comparable effect sizes

16. ANOVA Assumption #1 Normality of sampling distribution of means –Not normality of raw sample data –Not normality of population –CLT says that sampling distribution of means is normal if: Population is normal Sample size is large (>30)

20. ANOVA Assumption #2 Independence of errors –Group –Time/sequence –Space

21. ANOVA Assumption #3 Homogeneity of variances –Population variances, not sample variances