Generalized pairwise comparisons of prioritized outcomes

Marc Buyse, ScD

marc.buyse@iddi.com

• The Wilcoxon test, and generalizations• Generalized pairwise comparisons• Universal measure of treatment effect• An example• Conclusions

Outline

General Setup

Let Xi be the continuous outcome of the i th subject in T (i = 1, … , n )

REligible subjects

Control (C )Treatment (T )Let Yj be the continuous outcome ofthe j th subject in C (j = 1, … , m )

The Wilcoxon test statistic can be derived from all possible pairs of subjects, one from T and one from C.Let

Wilcoxon-Mann-Whitney test statisticW

The Mann-Whitney form of the Wilcoxon test

The Wilcoxon test can be generalized to the case of censored outcomes. Letting   and denote censored observations, the pairwise comparison indicator is now

Gehan generalized the Wilcoxon test

Now let Xi and Yj be observed outcomes for ANY outcome measure (continuous, time to event, binary, categorical, …)

First, generalize the test further for a single outcome measure

Yj Xi

favors T (favorable) favors C(unfavorable) neutral uninformative

pairwisecomparison

Binary outcome measure

Pairwise comparison Pair isXi = 1, Yj = 0 favorableXi = 1, Yj = 1 or Xi = 0, Yj = 0 neutralXi = 0, Yj = 1 unfavorableXi orYj missing uninformative

Pairwise comparison Pair isXi Yj > * favorable Xi Yj ≤ * neutralXi Yj < * unfavorableXi orYj missing uninformative

* chosen to reflect clinical relevance; = 0 is Wilcoxon test

Continuous outcome measure

Time to event outcome measure

Pairwise comparison Pair isXi Yj > * or Yj > * favorable Xi Yj ≤ * neutralXi Yj < * or Xi < * unfavorableotherwise uninformative* chosen to reflect clinical relevance; = 0 is Gehan test

Let Xi and Yj be VECTORS of observed outcomes for any number of occasions of a single outcome measure, or any number of outcome measures.We assume that the occasions and/or the outcome measures can be prioritized.

Generalized pairwise comparisons

Next, generalize the test to prioritized repeated observations of a single outcome measure…

Occasion with higher priority Occasion with lower priority Pair isfavorable ignored favorableunfavorable ignored unfavorableneutral ignored neutraluninformative favorable favorableuninformative unfavorable unfavorableuninformative neutral neutraluninformative uninformative uninformative

Last, generalize the test to severalprioritized outcome measures…

Outcome with higher priority Outcome with lower priority Pair isfavorable ignored favorableunfavorable ignored unfavorableneutral ignored neutraluninformative favorable favorableuninformative unfavorable unfavorableuninformative neutral neutraluninformative uninformative uninformative

Extend the previous definition of Uij

U is the difference between the proportion of favorable pairs and the proportion of unfavorable pairs. We call this general measure of treatment effect the « proportion in favor of treatment » ().

A general measure of treatment effect

is a linear transformation of the probabilistic index, P (X > Y ) :The proportion in favor of treatment ()

Situation P (X > Y )

T uniformly worse than C 0 1T no different from C 0.5 0T uniformly better than C 1 +1

For a binary variable, is equal to the difference in proportionsFor a continuous variable , is related to the effect size dFor a time-to-event variable, is related to the hazard ratio and the proportion of informative pairs f

The proportion in favor of treatment ()

A re-randomization test for

The test statistic U (or ) no longer has known expectation and variance. An empirical distribution of can be obtained through re-randomization.Tests of significance and confidence intervals follow suit.

The proportion in favor of treatment for the l th prioritized outcome (l = 1, . . . , L ) is given by

and the cumulative proportion is

Cumulative proportions for prioritized outcomes

Early breast cancer

two combination chemotherapies plus herceptin

R3,222 patients after curative resection of HER2+ breast cancer

AdriamycinCyclophosphamideTaxotere (ACT)TaxotereCarboplatinHerceptin (TCH)

standard chemotherapy

1,0731,075

main efficacy endpoints disease recurrence or death main safety endpoint congestive heart failure

AdriamycinCyclophosphamideTaxotereHerceptin (ACTH)

1,074

87%

81%78%

75%

92%

87%84%

81%

93%

88%86%

84%

Disease-free survival

Prioritized outcomes

Priority Outcomes1 Time to death from any cause2 Time to second malignancy3 Time to distant metastases4 Time to locoregional relapse5 Time to congestive heart failure

Difference in ACTH better ACTbetter Cumulative

P-value *Time to death 4.97% 2.87% 2.09% 0.006Time to second tumor 1.20% 1.21% 2.08% 0.022Time to distant mets 7.03% 3.46% 5.66% < 0.001Time to relapse 1.82% 1.01% 6.47% < 0.001

Time to CHF 0.62% 1.83% 5.25% < 0.001

Prioritized outcomes

GENERALIZED PAIRWISE COMPARISONSACTH vs. ACT

* Unadjusted for multiplicity

Difference in TCH better ACTbetter Cumulative

P-value *Time to death 5.05% 3.49% 1.56% 0.059Time to second tumor 1.22% 0.72% 2.05% 0.029Time to distant mets 7.18% 3.96% 5.26% < 0.001Time to relapse 1.75% 1.47% 5.55% < 0.001

Time to CHF 0.63% 0.71% 5.47% < 0.001

Prioritized outcomes

GENERALIZED PAIRWISE COMPARISONSTCH vs. ACT

* Unadjusted for multiplicity

Difference in TCH better ACTHbetter Cumulative

P-value *Time to death 3.04% 3.57% -0.53% 0.46Time to second tumor 1.29% 0.74% 0.02% 0.98Time to distant mets 3.84% 4.36% -0.50% 0.68Time to relapse 1.04% 1.63% -1.09% 0.40

Time to CHF 1.97% 0.74% 0.14% 0.93

Prioritized outcomes

GENERALIZED PAIRWISE COMPARISONSTCH vs. ACTH

* Unadjusted for multiplicity

1. are equivalent to well-known non-parametric tests in simple cases2. allow testing for differences thought to be clinically relevant3. allow any number of prioritized outcomes of any type to be analyzed simultaneously4. naturally lead to a universal measure of treatment effect, , which is directly related to classical measures of treatment effect (difference in proportions, effect size or hazard ratio)

Generalized Pairwise Comparisons

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.