Biostatistics Case Studies 2008

1. Biostatistics Case Studies 2008 Session 5: Choices for Longitudinal Data Analysis Peter D. Christenson Biostatistician http://gcrc.labiomed.org/biostat

2. Case Study

3. Study Goal - General

4. Specific Primary Aim The “ANCOVA” would be a t-test, if we ignored the baseline values and the different centers. The outcome is change in HAM-A. The groups are drug and placebo. The signal:noise ratio is ……

5. Comparison of Change Means with t-test Strength of Treatment Effect: Signal:Noise Ratio t= Observed Δ SDΔ√(1/N1 + 1/N2) Δ = Drug - Placebo Mean (Final-Base) Diff in HAM-A changes SD = Std Dev of within group HAM-A changes N1 = N2 = Group size | t | > ~1.96 ↔ p<0.05

6. Comparison of Change Means with t-test (Actually adjusted for baseline and center) Strength of Treatment Effect: Signal:Noise Ratio t= Observed Δ SDΔ√(1/N1 + 1/N2) -11.8 - (-10.2) (?)√(1/134 + 1/132) = = -1.10 → p=0.27

7. More than Two Visits How can we get one signal:noise ratio incorporating all visits? Perhaps we want to detect treatment effect at any visit.

8. Suppose Only Three Visits - Weeks 0, 4, 8 Two Treatment Differences in Changes: Δ1 = D1 - P1 Δ2 = D2 - P2 P1 P2 D1 D2 Total Effect: Δ12 + Δ22

9. Comparison of Change Means with ANOVA Strength of Treatment Effect: Signal:Noise Ratio F= Observed (Δ12 + Δ22 ) √V V involves SDΔ1 and SDΔ2 and the 1/Ns. Large F ↔ Δ12 + Δ22 too large to be random ↔ p<0.05

10. The previous slide is “classical” repeated measures ANOVA. • Could have many groups and many time points. • If the overall “total” effect is significant, then we would examine which Δs are the cause. • Same conclusions if changes from baseline, not sequential changes were used. • Since the signal or effect Δ12+Δ22 equally weights the two Δ, we must know all changes for a subject. If we do not (missing data), then that subject is completely removed from the analysis. Repeated Measures ANOVA

11. Mixed Models for Repeated Measures (MMRM) • “Classical” repeated measures ANOVA uses only subjects with no missing visits. • MMRM overcomes that limitation by making a signal:noise ratio as the weighted average of signals or effects from sets of subjects with the same missing visit pattern. • MMRM still provides the overall ratio, as in the classical ANOVA that cannot handle missing visits.

12. Mixed Models for Repeated Measures (MMRM) The next four slides use a simpler example to give the idea of how the weighting is done in MMRM. These four slides can be skipped to get to the bigger picture of longitudinal analyses.

13. Consider a crossover (paired) study with 6 subjects. Subject 5 missed treatment A and subject 6 missed B. MMRM Example* LOCF Difference 8 2 -1 8 0 0 2.83 Completer analysis would use IDs 1-4; trt diff=4.25. Strict LOCF analysis would impute 22,17; trt diff=2.83. *Brown, Applied Mixed Models in Medicine, Wiley 1999.

14. MMRM Example Cont’d ΔW=4.25 Paired ΔB=5 Unpaired Mixed model gets the better* estimate of the A-B difference from the 4 completers paired mean Δw=4.25. It gets a poorer unpaired estimate from the other 2 subjects ΔB = 22-17 = 5. How are these two “sub-studies” combined? *Why better?

15. MMRM Example Cont’d ΔW=4.25 Paired ΔB=5 Unpaired The overall estimated Δ is a weighted average of the separate Δs, inversely weighting by their variances: Δ = [ΔW/SE2(ΔW) + ΔB/SE2(ΔB)]/K = [4.25/4.45 + 5.0/43.1]/(1/4.45 + 1/43.1) = 4.32 The 4.45 and 43.1 incorporate the Ns and whether data is paired or unpaired.

16. MMRM - More General I The example was “balanced” in missing data, with information from both treatments A and B in the unpaired data. What if all missing data are for A, and none for B? The unpaired A mean is compared with the combined A and B mean, giving an estimate of half of the A - B difference. It is appropriately weighted with the paired A - Bestimate.

17. Competing Conclusions The next three slides show differences obtained by using different repeated measures approaches. These three slides can be skipped to get to other approaches for longitudinal analyses.

18. Competing Conclusions

19. Imputation with LOCF • Ignores potential progression; conservative; usually attenuates likely changes and ↑ standard deviations. • No correction for using unobserved data as if real. 30 denotes imputed: N=63/260 HAM-A Score Completer Individual Subjects 0 0 1 2 3 4 6 8 Week Use all 260 values as if observed here.

20. Completer vs. LOCF vs. MMRM Analysis LOCF Analysis Δ b/w groups = 1.8 N=260: 197 actual, 63 imputed Completer Analysis Δ b/w groups = 2.5 N=197: 197 actual (Week 8 or earlier) MMRM uses all available visits for all 260. No imputation

21. MMRM vs. Classical: Why Distinguish? Doesn’t distinguishing MMRM and classical seem to be about a minor technical point about weighting? Why make such a big deal? The MMRM is not in many basic software packages. It is not obvious how to perform it in software that does have it. So, it is not user-friendly yet. If you have missing data, ask a statistician to set it up in software correctly.

22. Other Approaches to Longitudinal Data So far, we have considered all sequential changes or changes from baseline. What other outcomes could be of interest?

23. Some Other Goals with Longitudinal Data • Use one visit at a time: • Compare treatments at each time separately - doesn’t look at changes in individuals. • Compare treatments at end of study. • Create summary over time: • Compare average over time - trends unimportant. • Specific pattern features, as in pharmacokinetic studies of AUC, peak, half-life, etc. • Compare treatments on rate of change over time.

24. Average over Time - Trends Unimportant Area Under the Curve (AUC), divided by total length of time, is an average outcome, weighted for time. Larger weights are given to the larger time intervals, since AUC is just a sum of trapezoids. . . . . . . AUC

25. “Growth” Curves Parabola or line or equation based on theory describes time trend. The idea is to compare treatment groups on a parameter describing the pattern, e.g., slope.

26. “Growth” Curves The logic is to compare treatment groups by finding means over subjects in each group for a parameter describing the pattern, e.g., slope. Next slide for correct method.

27. “Growth” Curves The idea is to compare treatment groups on a parameter describing the pattern, e.g., slope. Conceptually, we could just fit a separate regression line for each subject, get the slopes, and compare mean slopes between groups with a t-test. But subjects may have different numbers of visits, and the slope might be correlated with the intercept (e.g., start off higher → smaller slope). So, another form of “mixed models” is more accurate: “random coefficient” models. They give slopes also. Like MMRM, they are not very user-friendly in software, so ask a statistician to set up.

28. Summary on Mixed Models Repeated Measures • Currently one of the preferred methods for missing data. • Does not resolve bias if missingness is related to treatment. • Requires more model specifications than is typical. • Mild deviations from assumed covariance pattern do not usually have a large influence. • May be difficult to apply objectively in clinical trials where the primary analysis needs to be detailed a priori. • Can be intimidating; need experience with modeling; software has many options to be general and flexible.