1 Inference in Covariate Inference in Covariate - - Adaptive Adaptive allocation allocation Elsa Valdés Márquez & Nick Fieller MPS Research Unit, University of Reading & Department of Probability and Statistic, University of Sheffield EFSPI Adaptive Randomisation Meeting Brussels, 7 December 2006
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Inference in Covariate-AdaptiveAdaptive allocation · 2007-10-28 · 1 Inference in Covariate-AdaptiveAdaptive allocation Elsa Valdés Márquez & Nick Fieller MPS Research Unit, University
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Inference in CovariateInference in Covariate--AdaptiveAdaptiveallocationallocation
Elsa Valdés Márquez & Nick Fieller
MPS Research Unit, University of Reading &
Department of Probability and Statistic, University of Sheffield
(prognostic factors of previous patients & new patient)Treatment allocation probability
adjusted to balance covariates
Treatment A
Treatment B
Measure
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MinimizationMinimization aims to reduce marginal imbalance over each factoreither deterministically (Taves,1974)or with bias in random element (Pocock & Simon, 1975)
DS-Optimum designminimizes variance of treatment effect by deterministic allocation(Begg & Iglewicz, 1980)
Different factors and samplesDifferent factors and samples
1,000 group of patients
Covariate adaptive methods always moreefficient than complete randomisation
method with random element (PS)only efficient for larger sample sizes
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InferenceInference
“Randomization models the outcome variable of interest asfixed & the treatment assignment (designs points) as random;
in a population model we traditionally treat the variableof interest as random at fixed values of the design points”
Rosenberger and Lanchin (2002)
Randomization model
The distribution of the test statistic dependson the randomization procedure
Population model
The use of complete randomization or covariate-adaptive allocationmethods does not influence the inferential procedure
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Randomization ModelRandomization Model
Permutation tests
The reallocation of the patients to treatments shouldbe in a manner consistent with original assignment
Two situations:
• (Exact A) Patients arrive by chance in any order andthey could equally well arrive in any other
• (Exact B) The arrival order of the patients is importantFixed order
Past work on permutation tests:Metha, et.al. (1988), Smythe & Wie (1983),Hollander & Peña (1988), France (1998)
+ Ebutt et. al. (1997) considers both fixed & random orders of subjects
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Randomization ModelRandomization Model
• (Exact A) Patients arrive by chance in any order andthey could equally well arrive in any other order
Randomisation distribution obtained by permuting the orderof subjects & reapplying minimisation algorithm
• (Exact B) The arrival order of the patients is importantCondition on order and then randomisation distribution obtained onlyfrom random elements in allocation(including resolution of ties in deterministic methods such as Taves)
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Toy ExampleToy Example
Objective: Test effect of Captopril on kidney functionin insulin-dependent diabetic patients with nephropathy
16 subjects selected from larger data set unrealistically small number of subjects but illustrates main features
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Toy example : Randomisation DistributionToy example : Randomisation Distribution(Deterministic allocation methods)(Deterministic allocation methods)
5,000 times
only discrete choice of achievablep-values with assumption B
more p-values available withOptimal Design based methods
p-values from t-distribution severely biased
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Toy example : Randomisation DistributionToy example : Randomisation Distribution((allocation methods with random elements)allocation methods with random elements)
5,000 times
t-distribution now a good approxto randomisation distribution
t-approximation more biased
fixed randomelement
random elementdepends on imbalance
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Toy example (16 subjects): pToy example (16 subjects): p--valuesvalues
-1.5806-2.3239-1.5362-2.4248Value of teststatistic
0.07400.12400.03920.0642Exact B
0.06320.01460.07580.0140Exact A
0.06810.01760.07340.0147Classical
DADSPSTVMethod:-
p-values with fixed order of subjects (assumption B)very different from those given by classical t-distribution
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Example 2 Cirrhosis : pExample 2 Cirrhosis : p--valuesvalues(with adjustment of test statistic for covariates)(with adjustment of test statistic for covariates)
1.15181.81332.33382.3669Value of teststatistic
0.04540.05460.01240.0592Exact B
0.03680.01380.01160.0018Exact A
0.07680.02440.02250.0024Classical
DADSPSTVDistribution
5 prognostic factors and 50 patients
Similar features with larger data set &test statistic adjusted for covariate differences
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Confidence intervalsConfidence intervals
Guaranteed method for constructing randomisationdistribution based confidence intervals as inverse ofsignificance tests is very time consumingcan be obtained only by ‘trial & error’
CI is set of points not rejected by a test
usual standard error measures variability undercomplete randomisation (e.g. note that s2 is unbiased for population variance only under CR)
Need an approximate method based on‘pseudo standard errors’ (or effective s.e.)
pseudo (or effective) s.e. measures variability underrestricted randomisation used in the permutation tests
then use this with t-distribution for approx ‘pseudo CIs’