1. Unit 3b: From Fixed to Random Intercepts © Andrew Ho, Harvard Graduate School of EducationUnit 3b – Slide 1 http://xkcd.com/588/

2. Contrasting “Total,” “Between,” and “Within” Regression models “Within” Regression as a “Fixed Intercepts/Fixed Effects Model” From Fixed to Random Effects/Intercepts © Andrew Ho, Harvard Graduate School of Education Unit 3b– Slide 2 Multiple Regression Analysis (MRA) Multiple Regression Analysis (MRA) Do your residuals meet the required assumptions? Test for residual normality Use influence statistics to detect atypical datapoints If your residuals are not independent, replace OLS by GLS regression analysis Use Individual growth modeling Specify a Multi-level Model If time is a predictor, you need discrete- time survival analysis… If your outcome is categorical, you need to use… Binomial logistic regression analysis (dichotomous outcome) Multinomial logistic regression analysis (polytomous outcome) If you have more predictors than you can deal with, Create taxonomies of fitted models and compare them. Form composites of the indicators of any common construct. Conduct a Principal Components Analysis Use Cluster Analysis Use non-linear regression analysis. Transform the outcome or predictor If your outcome vs. predictor relationship is non-linear, Use Factor Analysis: EFA or CFA? Course Roadmap: Unit 3b Today’s Topic Area

3. © Andrew Ho, Harvard Graduate School of Education Unit 3b – Slide 3 Contrasting Total, Between, and Within Regressions Regression TypeRegression Model “Total” – Vanilla OLS The total-regression (individual-level) coefficient ignores group membership and endangers statistical inference due to residual correlation. “Between” – Mean Regression The between-regression (mean regression) coefficient ignores within-group variation and differences in group size, risking “ecological fallacies.” “Within” – ANCOVA; Fixed Effects

4. Contrasting “Between,” “Total,” and “Within” Regression Lines © Andrew Ho, Harvard Graduate School of EducationUnit 3b – Slide 4 Illustrating Within- Regression with 2 schools.

5. © Andrew Ho, Harvard Graduate School of EducationUnit 3b – Slide 5 http://bcs.whfreeman.com/ips4e/cat_010/applets/anova.html http://www.rossmanchance.com/applets/Anova/Anova.html http://www.psych.utah.edu/stat/introstats/anovaflash.html

6. © Andrew Ho, Harvard Graduate School of EducationUnit 3b – Slide 6

7. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 7  The first expression is one we’ve already seen. I introduce alternative syntax but recommend the last option.

8. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 8

9. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 9 Within-Regression, Four Ways, Noting Intercepts…  Standard errors for the “de-meaning” approach will be a bit off due to degrees of freedom. Because we haven’t acknowledged that we’ve estimated intercepts. Don’t sweat this too much.  The “ i.SCHNUM ” or “ i.SCHOOLID ” notation creates an “indicator variable” that automatically creates a dummy variable for each school. Estimates a whole ton of intercepts.  areg and xtreg, fe “absorb” the intercepts.  Standard errors for the “de-meaning” approach will be a bit off due to degrees of freedom. Because we haven’t acknowledged that we’ve estimated intercepts. Don’t sweat this too much.  The “ i.SCHNUM ” or “ i.SCHOOLID ” notation creates an “indicator variable” that automatically creates a dummy variable for each school. Estimates a whole ton of intercepts.  areg and xtreg, fe “absorb” the intercepts.

10. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 10 Also, be for between. Next, re for random effects. This estimated correlation between intercepts and covariates (SES) is positive at 0.33.

11. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 11

12. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 12 Recasting the Random Effects Assumption as Generalizability If you were to repeat your experiment, would you keep the same groups or resample them from the population of groups? If groups were states? Schools? Countries? Could they be random effects? People? Teachers?

13. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 13

14. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 14 Four Regression Lines

15. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 15 Fixed vs. Random Effects (Rabe-Hesketh and Skrondal, 2012) Questions:Fixed EffectsRandom Effects Inference for population of clusters? NoYes Minimum number of clusters required? Any number What assumptions are required? Inference for clusters in particular sample? Yes Minimum cluster size required? Is the model parsimonious No, parameters for each intercept, although they can be collapsed over. Yes, one variance parameter for all clusters.

16. © Andrew Ho, Harvard Graduate School of EducationUnit 3b– Slide 16 The Intraclass Correlation ICC of.174 ICC of 0