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
Outline
•
Comparative clinical trial
•
Typical questions of interest
•
Standard analysis
•
Simulation results
•
Concluding remarks
2
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
3
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)
4
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.
5
Standard Analysis (REML)
Sample SAS Code (time = 0, 1, 2, 3)
PROC MIXED METHOD=REML;
CLASS trt time sub;
MODEL Y=trt time trt*time;
REPEATED time/SUB=sub(trt) TYPE=CS;
ESTIMATE 'L' trt*time -1 0 0 1 1 0 0 -1/CL;
RUN;
For L*, Y0 is removed from the response vector and
added to the model as a continuous covariate, with an
appropriate change to the ESTIMATE statement.
6
Standard Analysis (REML)
•
Assumptions
(1) Multivariate normality of residual vector
(2) Correct specification of the variancecovariance 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 MIXED METHOD=REML EMPIRICAL;
CLASS trt time sub;
MODEL Y=trt time trt*time;
REPEATED time/SUB=sub(trt) TYPE=CS;
ESTIMATE 'L' trt*time -1 0 0 1 1 0 0 -1/CL;
RUN;
For L*, Y0 is removed from the response vector and
added to the model as a continuous covariate, with an
appropriate change to the ESTIMATE 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
 CS
 AR (1)
 1 .6 .6 .6 


1 .6 .6 

 40
1 .6 


1

 1 .720 .518 .373 


518
.
720
.
1


 40
1 .720 


1 

UN
 1 .57 .45 .42 


1 .67 .66 

 40
1 .83 


1 

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)
True

Use data at last
time point only
Naïve t-test
Use all the data
TYPE=CS with (without)
EMPIRICAL option
L*
L
TYPE=UN
L
L*
.055 (.078)
.045
.047
.047 (.045)
.051 (.046)
.044
.048
.048 (.071)
.053 (.060)
.052
.048
L
L*
AR(1)
.047
.047
.052 (.098)
CS
.046
.048
UN
.045
.047
5000 iterations, L=change from baseline, L*=ANCOVA
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Simulation Results (continued)
Power (%)
Use data at last
time point only
Naïve t-test
Use all the data
AR(1)
L
55
L*
75
TYPE=CS with
EMPIRICAL option
L
L*
68
81
CS
70
82
84
90
81
89
UN
49
68
71
83
68
79
True

TYPE=UN
L
64
L*
76
5000 iterations, L=change from 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.
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