1. Lecture 7: Parametric Models for Covariance Structure (Examples)

2. 1. Model for the mean

3. 2. Model for the covariance matrix

4. 2. Model for the covariance matrix (cont’d)

7. Which model to pick?

8. Which model to pick? (cont’d)

9. Example: CD4+ Level HIV attacks CD4+ cells, which regulate the body’s immuneresponse against infectious agents We have 2376 values of CD4+ cell numbers plotted against time since seroconversion for 369 infected men enrolled in the MAC Study

10. Example: CD4+ Level (cont’d) Goals: 1.Estimate the average time course of CD4+ cell depletion 2.Identify factors which predict CD4+ cell changes 3.Estimate the time course for an individual man taking into account the measurement error in CD4+ cell determinations 4.Characterize the degree of heterogeneity across men in the rate of progression

11. Example: CD4+ Level Goal 1: Estimate average time course of CD4+ cell depletion The model for the covariance matrix is a model of serial correlation.

12. Example: CD4+ Level Goal 2: Identify factors predictive of CD4+ cell changes The model for the covariance matrix is still a model of serial correlation, however we have changed the model for the mean.

13. Example: CD4+ Level Parameter Interpretation

14. Example: CD4+ Level Goal 3: Estimate time course for an individual man, accounting for measurement error in CD4+ cell counts This is a model with a random intercept and slope + serial correlation + measurement error.

15. Example: CD4+ Level Parameter Interpretation

16. Random Effects Model: Interpretation of coefficients Heterogeneity between subjects at baseline Heterogeneity between subjects in rate of change

17. Example: CD4+ Level Goal 4: Characterize degree of heterogeneity across men in progression rate (From the previous slide)

18. Example: Protein contents of milk samples

19. Barley (25 cows) Mixed (27 cows) Lupins (27 cows)

20. Example: Protein contents of milk samples (cont’d)

21. 0.02

22. Example: Protein contents of milk samples Model for the Mean

23. Example: Protein contents of milk samples Model for the Covariance Matrix

24. Table 5.1 (b)

26. Example: Protein contents of milk samples Does diet affect the mean response profile?

27. Example: Protein contents of milk samples Is there a rise in the mean response towards the end of the experiment?

28. Example: Protein contents of milk samples Is there a rise in the mean response towards the end of the experiment? (cont’d)

30. 0.02

31. Example: Body weight of 26 cows Dataset consists of body weights of 26 cows, measured at 23 unequally-spaced times over a period of about 22 months. Treatments were allocated in a 2x2 factorial design: Control (4) Iron-dosing (4) Infection (9) Iron + Infection (10)

33. Control N=4 Iron N=4 Infection N=9 Iron + Infection N=10 Log Y: Variance-stabilizing transformation

34. Example: Body weight of 26 cows (cont’d) Look at your data Estimate empirical variogram What do you see? Measurement variance small Substantial between-cow variability Gaussian correlation model appropriate

35. Small measurement error Experimental correlation Random effects Empirical variogram of the OLS residuals from a saturated model for the mean response

36. Example: Body weight of 26 cows Model for the Mean

38. Example: Body weight of 26 cows Model for the Covariance Matrix

39. Example: Body weight of 26 cows Questions Q1: Can we conclude linear (vs. quadratic) growth? …The quadratic curve is appropriate. Q2: Is there a main effect for iron?…NO

40. Example: Body weight of 26 cows Questions (cont’d) Q3: Is there a main effect for infection?…YES Q4: Is there an interaction between iron and infection?…NO

41. Example: Body weight of 26 cows Questions (cont’d) We re-fit the model with only the infection term: Conclusions: Highly significant effect of infection No significant effect of iron No significant effect of interaction