A Note on Modeling the Covariance Structure in Longitudinal Clinical Trials Devan V. Mehrotra Merck Research Laboratories, Blue Bell, PA FDA/Industry Statistics Workshop September 18, 2003

2 Outline Comparative clinical trial Typical questions of interest Standard analysis Simulation results Concluding remarks

3 Longitudinal Clinical Trial Subjects are randomized to receive either treatment A or B. (N = N A + N B ) Response is measured at baseline (time = 0) and at fixed post-baseline visits (time = 1, 2, … T). Y ijk = response for time i, trt. j, subject k ij = E(Y ijk ) Note: Due to randomization, 0A = 0B

4 Typical Questions of Interest Is there a differential treatment effect? What is the magnitude of the difference? Typical endpoints for comparing treatments 1) Response at last time point (L) 2) Average of all responses over time (A) 3) Slope, or linear component of the treatment x time interaction (S) Our focus in this talk is on endpoint (1)

5 Typical Questions of Interest (continued) Null Hypothesis: TA = TB Equivalent to ( TA - 0A ) = ( TB - 0B ) because 0A = 0B under randomization Two common analyses - Change from baseline (L) - ANCOVA: baseline is a covariate (L*) Note: L and L* test the same hypothesis and estimate the same parameter.

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7 Standard Analysis (REML) Assumptions (1)Multivariate normality of residual vector (2)Correct specification of the variance- covariance matrix of the residual vector For this talk, we assume (1) is ~ true and focus on potential departures from assumption (2)

8 Comments on the Covariance Structure PROC MIXED BC Type=CS is implicit in classic linear model analyses of longitudinal data (split-plot, variance component ANOVA models with compound symmetry structure) Box (1954), Huynh & Feldt (1970) etc., noted that classic analyses can provide incorrect inference if Type=CS assumption is violated Greenhouse & Geisser (1959), Huynh & Feldt (1976) provided approximate alternative tests based on adjusted d.f. Note: Finney (1990) refers to the classic mixed model ANOVA as a dangerously wrong method

9 Comments on the Covariance Structure (continued) PROC MIXED AD Laird & Ware (1982), Jenrich & Schlucter (1986), etc. suggested using prior experience or the current data to select an appropriate covariance structure. PROC MIXED provides several choices, including CS, AR(1), Toeplitz, and UN. Frison & Pocock (1992) looked at data from several trials, covering a variety of diseases and quantitative outcome measures. They reported no major departure from the compound symmetry assumption Our alternative strategy: specify Type=CS but use Liang and Zegers (1996) sandwich estimator via the EMPIRICAL option as insulation against an incorrect covariance structure assumption.

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11 Simulation Study

12 Simulation Study (continued)

13 Simulation Study (continued)

14 Simulation Results

15 Simulation Results (continued)

16 Concluding Remarks Incorrect specification of the covariance structure can result in Type I error rates that are far from the nominal level. Using the Liang and Zeger sandwich estimator via the EMPIRICAL option insulates us from an incorrect covariance structure assumption. Using TYPE=CS with the EMPIRICAL option is an attractive default approach. It usually provides more power than using TYPE=UN, particularly for small trials.

17 Concluding Remarks (continued) Analysis with baseline as a covariate usually provides notably more power than the corresponding change from baseline analysis. The (not uncommon) naïve t-test approach (same as complete case approach) should be abandoned for longitudinal trials. It can result in a substantial loss of power, especially when there are missing values.

18 References Box GEP (1954). Annals of Mathematical Statisitcs, 25, Finney, DJ (1990). Statistics in Medicine, 9, Frison L and Pocock SJ (1992). Statistics in Medicine, 11, Greenhouse SW and Geisser S (1959). Psychometrika, 24, Huynh H and Feldt LS (1970). JASA, 65, Huynh H (1976). Journal of Educational Statistics, 1, Jenrich RI and Schulchter MD (1986). Biometrics, 42, Laird N and Ware JH (1982). Biometrics, 38, Liang NM and Zeger SL (1986). Biometrika, 73,