A Note on Modeling the Covariance Structure in Longitudinal Clinical

TrialsDevan V. Mehrotra

Merck Research Laboratories, Blue Bell, PA

FDA/Industry Statistics WorkshopSeptember 18, 2003

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Outline

• Comparative clinical trial

• Typical questions of interest

• Standard analysis

• Simulation results

• Concluding remarks

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Longitudinal Clinical Trial

• Subjects are randomized to receive either treatment A or B. (N = NA + NB)

• Response is measured at baseline (time = 0) and at fixed post-baseline visits (time = 1, 2, … T).

• Yijk = response for time i, trt. j, subject k

ij = E(Yijk)

Note: Due to randomization, 0A = 0B

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Typical Questions of Interest

• Is there a differential treatment effect?What is the magnitude of the difference?

• Typical endpoints for comparing treatments1) 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)

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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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Standard Analysis (REML) Sample SAS Code (time = 0, 1, 2, 3)

PROC MI XED METHOD=REML; CLASS trt time sub; MODEL Y=trt time trt*time; REPEATED time/ SUB=sub(trt) TYPE=CS; ESTI MATE 'L' trt*time -1 0 0 1 1 0 0 -1/ CL; RUN;

For L*, Y0 is removed f rom the response vector and added to the model as a continuous covariate, with an appropriate change to the ESTI MATE statement.

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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)

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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

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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 Zeger’s (1996) “sandwich” estimator via the EMPIRICAL option as insulation against an incorrect covariance structure assumption.

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Standard Analysis (REML) Sample SAS Code (time = 0, 1, 2, 3)

PROC MI XED METHOD=REML EMPIRICAL; CLASS trt time sub; MODEL Y=trt time trt*time; REPEATED time/ SUB=sub(trt) TYPE=CS; ESTI MATE 'L' trt*time -1 0 0 1 1 0 0 -1/ CL; RUN;

For L*, Y0 is removed f rom the response vector and added to the model as a continuous covariate, with an appropriate change to the ESTI MATE statement.

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Simulation Study

NA = NB = 40 randomized subjects per group

True Response Means

Time Point

Hyp. Trt. Grp. 0 1 2 3

A 25 18 14 12

Null B 25 18 14 12

Alt. B 25 23 20 15

Note: higher values are better

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Simulation Study (continued) Covariance structures: CS, AR(1), Unstructured

1

6.1

6.6.1

6.6.6.1

40CS

1

720.1

518.720.1

373.518.720.1

40)1(AR

1

83.1

66.67.1

42.45.57.1

40UN

Note: in each case, mean pairwise correlation = 0.6

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Simulation Study (continued) Missing Data: Subject discontinues if “not responding” - Pr{missing T3, T4} depends on T2 - Pr{missing T4} depends on T3 Note: This is a missing at random (MAR) mechanism.

% of missing data (on average) Time Point

Covariance Trt. Grp. 2 3 AR(1) A 9% 16%

B (Null) 9% 16% B (Alt.) 1% 9%

CS A 13% 24% B (Null) 13% 24% B (Alt.) 3% 14%

UN A 14% 23% B (Null) 14% 23% B (Alt.) 3% 12%

5000 iterations

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Simulation Results

Type I Error Rates ( 05. )

Use data at last time point only

Use all the data

True Naïve t-test TYPE=CS with (without)

EMPI RI CAL option TYPE=UN

L L* L L* L L*

AR(1) .047 .047 .052 (.098) .055 (.078) .045 .047

CS .046 .048 .047 (.045) .051 (.046) .044 .048

UN .045 .047 .048 (.071) .053 (.060) .052 .048

5000 iterations, L=change f rom baseline, L*=ANCOVA

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Simulation Results (continued)

Power (%)

Use data at last time point only

Use all the data

True Naïve t-test TYPE=CS with

EMPI RI CAL option TYPE=UN

L L* L L* L L* AR(1) 55 75 68 81 64 76

CS 70 82 84 90 81 89

UN 49 68 71 83 68 79

5000 iterations, L=change f rom baseline, L*=ANCOVA

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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.

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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.

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References• Box GEP (1954). Annals of Mathematical Statisitcs, 25, 484-498.• Finney, DJ (1990). Statistics in Medicine, 9, 639-644.• Frison L and Pocock SJ (1992). Statistics in Medicine, 11, 1685-1704.• Greenhouse SW and Geisser S (1959). Psychometrika, 24, 95-112.• Huynh H and Feldt LS (1970). JASA, 65, 1582-1589.• Huynh H (1976). Journal of Educational Statistics, 1, 69-82.• Jenrich RI and Schulchter MD (1986). Biometrics, 42, 805-820.• Laird N and Ware JH (1982). Biometrics, 38, 963-974.• Liang NM and Zeger SL (1986). Biometrika, 73, 13-22.