Preamble• Midterm
– Review– Room
• Homework• Simple effects
– Simple main effects– Simple interaction effects
The Embarrassing Footnote• With fixed effects analysis, one can’t generalize beyond measured levels of factor– e.g., the influence of expert communicators
• 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 MSwithin does not capture
Random Effects
Example• 4 College x 3 Test
Test is fixed factor, college is random factor
College Test1 Test2 Test3 mean1 28 12 20 202 12 28 20 203 12 28 20 204 12 28 20 20
mean 16 24 20 20
College Test1 Test2 Test3 mean1-250 28 12 20 20
251-500 12 28 20 20mean 20 20 20 20
Ffixed effect = MStest /MSwithin
Frandom effect = MStest /MStest x college
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
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
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
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)”
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
ANOVA Effect Size: Eta
Advantages: conceptual simplicityDisadvantages: 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
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
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)