Generalized pairwise comparisons of prioritized outcomes Marc Buyse, ScD marc.buyse@iddi.com
Outline • The Wilcoxon test, and generalizations • Generalizedpairwisecomparisons • Universalmeasure of treatmenteffect • An example • Conclusions
General Setup Eligible subjects R Treatment (T ) Control (C ) Let Xibe the continuous outcome of the i thsubject in T (i = 1, … , n ) Let Yjbe the continuous outcome of the j thsubject in C (j = 1, … , m )
The Mann-Whitney form of the Wilcoxon test The Wilcoxon test statisticcanbederivedfromall possible pairs of subjects, one fromT and one fromC. Let Wilcoxon-Mann-Whitney test statisticW
Gehan generalized the Wilcoxon test The Wilcoxon test canbegeneralized to the case of censoredoutcomes. Letting and denotecensored observations, the pairwisecomparisonindicatorisnow
First, generalize the test further for a single outcome measure Now let Xi and Yjbeobservedoutcomes for ANYoutcomemeasure (continuous, time to event, binary, categorical, …) pairwise comparison Xi Yj favorsC (unfavorable) favorsT (favorable) uninformative neutral
Binary outcome measure
Continuous outcome measure
Time to event outcome measure
Generalized pairwise comparisons Let Xi and YjbeVECTORS of observedoutcomes for anynumber of occasions of a single outcomemeasure, or anynumber of outcomemeasures. We assume that the occasions and/or the outcomemeasurescanbeprioritized.
Next, generalize the test to prioritized repeated observations of a single outcome measure…
Last, generalize the test to severalprioritized outcome measures…
A general measure of treatment effect Extend the previousdefinition of Uij Uis the differencebetween the proportion of favorable pairs and the proportion of unfavorable pairs. We call thisgeneralmeasure of treatmenteffect the « proportion in favor of treatment » ().
The proportion in favor of treatment () is a linear transformation of the probabilistic index, P (X > Y ):
The proportion in favor of treatment () For a binary variable, isequal to the difference in proportions For a continuous variable , isrelated to the effect size d For a time-to-event variable, isrelated to the hazard ratio and the proportion of informative pairs f
A re-randomization test for The test statisticU (or ) no longer has known expectation and variance. An empirical distribution of canbeobtainedthroughre-randomization. Tests of significance and confidence intervalsfollow suit.
Cumulative proportions for prioritized outcomes The proportion in favor of treatment for the l th prioritized outcome (l = 1, . . . , L ) is given by and the cumulative proportion is
Early breast cancer 3,222 patients after curative resection of HER2+ breast cancer R 1,075 1,073 1,074 Adriamycin Cyclophosphamide Taxotere Herceptin (ACTH) Taxotere Carboplatin Herceptin (TCH) Adriamycin Cyclophosphamide Taxotere (ACT) two combination chemotherapies plus herceptin standard chemotherapy main efficacy endpoints disease recurrence or death main safety endpoint congestive heart failure
Disease-free survival 93% 92% 88% 87% 86% 84% 84% 81% 87% 81% 78% 75%
Prioritized outcomes
Prioritized outcomes GENERALIZED PAIRWISE COMPARISONS ACTH vs. ACT *Unadjusted for multiplicity
Prioritized outcomes GENERALIZED PAIRWISE COMPARISONS TCH vs. ACT *Unadjusted for multiplicity
Prioritized outcomes GENERALIZED PAIRWISE COMPARISONS TCH vs. ACTH *Unadjusted for multiplicity
Generalized Pairwise Comparisons • are equivalent to well-known non-parametric tests in simple cases • allowtesting for differencesthought to beclinically relevant • allowanynumber of prioritizedoutcomes of any type to beanalyzedsimultaneously • naturally lead to a universalmeasure of treatmenteffect, , whichisdirectlyrelated to classicalmeasures of treatmenteffect (difference in proportions, effect size or hazard ratio)
References Buyse M. Generalized pairwise comparisons for prioritized outcomes in the two-sample problem. Statistics in Medicine 29:3245-57, 2010. Buyse M. Reformulating the hazard ratio to enhance communication with clinical investigators. Clinical Trials 5:641-2, 2008.
