1. Conditions of application Assumption checking

2. Assumptions for mixed models and RM ANOVA Linearity  The outcome has a linear relationship with all of the predictors Homoscedasticity  The residuals are equally variable at any level of the predictors Normality of the residuals

3. Fitting the model Fit the same model from last week:  SIGNAL = (b 0 + u 0 ) + b 1 ACCELERATION + b 2 COIL + b 3 COILxACCELERATION + ε  Be sure to use the long dataset, and ALL values of RESOLUTION  ACCELERATION is a covariate, not a factor  SAVE the residuals and the predicted values

4. Residual plots Create a histogram of the residuals (Analyze → Descriptive Statistics → Frequencies → Chart), and a scatterplot of the residuals v.s. the predicted values (Graphs → Chart builder). What are we testing for?  Linearity (no pattern)  Homoscedasticity (constant variance)  Normality of residuals (bell-shaped histogram)

5. Analyze the residual plots Do our plots look okay?  Scatterplot Looks decent  Histogram Looks plausibly normal, given the sample size Weird bi-modality

6. Normality tests Go to Analyze → Descriptive Statistics → Explore

7. Tests of normality Check the Shapiro-Wilk and the Kolmogorov- Smirnov Neither value is statistically significant What does that mean?  We have no evidence of non-normality  We pass!  Be careful, though: these tests are poor at finding bimodality

8. Analyze the Q-Q plot What are we testing?  Normality of the residuals How does it look?  Decent, except for the extreme tails  Probably okay

9. RM ANOVA Back to the wide file! Fit an RM ANOVA. Ignore RESOLUTION SAVE Cook's Distance

10. Checking Cook's D Transform the dataset to long form Only keep ID and the Cook's D variables Plot Cook's D v.s. ID

11. How does it look? Not too bad None of the points are wildly farther than the others It looks none of the points were wildly influential  Subject 23 had a big impact, though

12. Now... Check the conditions of application for the same models, only now only for subjects with RESOLUTION = 2

13. Tests of normality Uh-oh We fail our tests! And they don't have a lot of power with small samples, so this might be really bad

14. Q-Q plots Not looking good. There's something bad happening in the tails

15. Residual histogram That spike in the middle is problematic

16. Residuals v.s. predicted At least this looks okay

17. Residuals v.s. acceleration This explains it: it's the weird interaction between acceleration and coil that we noticed before

18. Questions? About the homework?