Covariance structures in longitudinal analysis

1. Covariance structures in longitudinal analysis Which one to choose?

2. Repeated Measures

3. Importance of Covariance Structures • variability not explained by the fixed effects are model in the covariance structure • represent the background variability that the fixed effects are tested against • valid inferences for fixed effects parameters

4. Selecting the Appropriate Covariance Structure • Choice of covariance structure is a balance since: • Too simple  Type I error rate increases • Too complex  power and efficiency decreases

5. Example • How does the left atrial dimension change over time in patients newly diagnosed with atrial fibrillation? • Atrial fibrillation is an irregularity of the heart’s rhythm • Due to chaotic electrical activity in the upper chambers (atria), the atria quiver instead of contracting in an organized manner • Atrial enlargement maybe related to how easily a subject can go back to a normal rhythm and the likelihood of a blood clot forming --> stroke

6. Heart Diagram

7. Example - Data • Data source: Canadian Registry of Atrial Fibrillation • Left atrial dimension measured at enrolment, Year 2, Year 4, Year 7 and Year 10 • Fit model with fixed effects only • adjust for age at first diagnosis of atrial fibrillation (AF), gender, hypertension at enrolment and visit year

8. Example

9. Model specification Y = Xb + Zg + e where: Y = response over time X = design matrix for fixed effects b = parameters for fixed effects Z = vector of 1s for the random effects g = parameters for random effects e = within-subject variation Y = Xb + e

10. SAS Code • PROC MIXED < options > ; • CLASS variables ; • MODEL dependent = < fixed-effects > < / options > ; • RANDOM random-effects < / options > ; • REPEATED < repeated-effect > • / TYPE = covariance-structure;

11. Repeat vs Random statement • The RANDOM statement relates to random effects • The REPEATED statement relates to the structure of the within subject errors. • Each statement has a different role…BUT specifying a model with compound symmetry covariance structure can be done with either statement

12. Models with REPEATED Statement only • No random effects specified in model • Assume random effects error is small compared to within subject error • Covariance structure is based only on the within subject error.

13. General covariance structure • Assume homogeneity assumption for practical reasons – reduces the number of parameters estimated • Possible to not assume the homogeneity assumption (can be tested but need sufficient amount of data to specify)

14. Block Diagonal Covariance Matrix r ~ N 0,

15. Covariance structures • Simple (VC – Variance Component) 1 parameter S =

16. Covariance structures • Unstructured (UN) 15 parameters S =

17. Covariance structures • Compound Symmetry (CS) 2 parameters S =

18. Covariance structures • First-order Autoregressive [AR(1)] 2 parameters S =

19. Covariance structures • Toeplitz (TOEP) 5 parameters S =

20. Draftsman’s plots • 2D array of scatterplots for each pair of time lagged observations • For 3 time points: Y1, Y2 and Y3 • Y1 vs. Y2 • Y1 vs. Y3 • Y2 vs. Y3

21. Y2 Y3 Y4 Y1 Y2 Y3 Independence Draftsman’s plot – Simulation examples

22. Autoregressive Compound Symmetry Draftsman’s plot – Simulation examples

23. Example – Draftsman’s plot

24. Example - Correlation matrix

25. Variogram • graphical description of the time/spatial correlation between observations • summarises the relationship between differences in pairs of measurements and the distance of the corresponding points from each other • Equally or unequally spaced observation periods

26. Variogram • Calculate the sample variogram components: vijk = ½ (rij – rik)2 rij=residual uijk = |tij – tik| tij=time • Plot of vijk vs. uijk • Process variance – estimated by the average of ½(rij – rlk)2 for i ≠ l

27. Random Effects ProcessVariance ProcessVariance Measurement Error Variogram - Theoretical Within Subject Correlation Time Lag

28. Variogram – Sitka tree example

29. Example - Variogram

30. Which covariance structure? • Fit model with different covariance structures • Compare goodness-of-fit statistics to choose covariance structure • Goodness-of-fit statistics Bayesian information criterion (BIC) • BIC = -2loglik+ d logn Akaike information criterion (AIC) • AIC = -2loglik+ 2d

31. Estimation method for the covariance parameters • Maximum Likelihood (ML) versus Restricted Maximum Likelihood (REML) • both are based on likelihood principles • properties of consistency, asymptotic normality, and efficiency • differences increase as the number of fixed effects in the model increases

32. ML vs. REML • Goodness-of-fit testing for the two methods differ in what part of the model it assesses • ML: describes the fit of the whole model (fixed and random effects) • REML: describes the fit of the stochastic portion (random effects)

33. Which goodness-of-fit statistic? Bayesian information criterion (BIC) • BIC = -2loglik+ d logn Akaike information criterion (AIC) • AIC = -2loglik+ 2d • The BIC has a higher penalty than AIC for including more parameters  more simple model • a too simple model has inflated Type I error rates • Typically, choose model based on AIC

34. Example • Which covariance structure fits the best?

35. Fixed Effects Parameter Estimates

36. Fixed Effect Parameters – cont’d

37. Likelihood ratio test (LRT) • For nested models, can also test if the additional parameters add a statistically significant improvement in the model • For the example, the LRT for TOEP (5 parameters) vs. CS (2 parameters) ---> choose CS model

38. Summary • Graphical plots to help identify covariance structure • AIC and BIC to choose between covariance structures • LRT to test if additional parameters are warranted

39. References • Dawson, K.S., Gennings, C. and Carter, W.H. 1997. Two graphical techniques useful in detecting correlation structure in repeated measures data. The American Statistician. 51(3). 275-283. • Diggle, P.J., Liang, K.Y. and Zeger, S.L. 1994. Analysis of Longitudinal Data. Oxford. Clarendon Press. • Littell, R.C., Pendergast, J. and Natarajan, R. 2000. Modelling covariance structure in the analysis of repeated measures data. Statistics in Medicine. 19. 1783-1819. • Moser, E.B. 2004. Repeated Measures Modeling with PROC MIXED. Paper 188-29. SUGI 29. • Singer, J.D. 1998. Using SAS PROC MIXED to Fit Multilevel Models, Hierarchichal Models, and Individual Growth Models. Journal of Educational and Behavioral Statistics. 24(40). 323-355. • Singer, J.D. and Willet, J.B. 2003. Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. New York. Oxford Univeristy Press. • Ware, J.H. 1985. Linear models for the analysis of longitudinal studies. The American Statistician. 39(2). 95-101.