A Note on Modeling the Covariance Structure in Longitudinal Clinical Trials

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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

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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 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)

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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)

• 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
• TypeCS 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 TypeCS 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 TypeCS but use
  Liang and Zegers (1996) sandwich estimator via
  the EMPIRICAL option as insulation against an
  incorrect covariance structure assumption.

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Simulation Study
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Simulation Study (continued)
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Simulation Study (continued)
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Simulation Results
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Simulation Results (continued)
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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 TYPECS with the EMPIRICAL option is an
  attractive default approach. It usually provides
  more power than using TYPEUN, 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.