Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Independent Increments in Group SequentialTests: A Review

KyungMann Kimkmkim@biostat.wisc.edu

University of Wisconsin-Madison, Madison, WI, USA

July 13, 2013

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Outline

Early Sequential Analysis

Independent Increments

Group Sequential Tests with Increment Increments

Conclusion

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Why do I look the way I do?

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Sequential Probability Ratio Test by Wald

I Test simple hypotheses H0 against H1

I Decisions D0 and D1 favoring H0 and H1, respectivelyI After each observation, with Ln0 and Ln1, the likelihoods of

H0 and H1, respectively, stop sampling when

Ln1/Ln0 ≥ (1− β)/α

andLn1/Ln0 ≤ β/(1− α),

with decisions D1 and D0, respectively

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

SPRT by Wald

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Sequential Methods for Clinical Trials

I Armitage (1960)I Ethical imperatives in considering early termination in

clinical trialsI Restricted plan (Armitage, 1957) as a modification of the

two-sided version of SPRTs by Sobel and Wald (1949)I Repeated significant test (RST) by Armitage, McPherson &

Rowe (1969) as a means to adjust the critical value toaccount for multiple testing leading to a constant criticalvalue for repeated significance tests

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Sequential Tests by Armitage

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Group Sequential Methods for Clinical Trials

I Wald (1947, pp 101–3) refers to taking groups ofobservations and applying SPRTs for binary outcome

I Elfring & Schultz (1973) was the first to specifically use theterm “group sequential" for clinical trials with binaryoutcome

I Pocock (1977), following the repeated significance test byArmitage, McPherson & Rowe (1969), popularized thegroup sequential tests for clinical trials with Gaussianoutcome

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Why Independent Increments?I In group sequential tests, one has to solve the following

multivariate integral∫ b1

−b1

· · ·∫ bK

−bK

f (s1, . . . , sK )ds1 · · · dsK = 1− α

where f is the joint density of the sequentially computedtest staistics

I However, if the following holds

Cov(Sk ,Sk ′) = Var(Sk ), k < k ′

i.e., if the test statistics have independent increments, themultivariate integration above becomes univariateintegration involving recursion

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Armitage, McPherson & Rowe (1969)I Repeated significant testsI Assume that Y1,Y2, . . . are independent Gaussian

observations with mean µ and variance 1, and letSk =

∑kj=1 Yj

I To test H0 : µ = 0 against H1 : µ 6= 0, sampling isterminated the first time when

|Sk | > bk

I Denote by fk the probability density function of Sk in thesequential procedure:

fk (s;µ) =

∫ bk−1

−bk−1

fk−1(u;µ)φ(s − u − µ)du

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

I The probability of stopping at or before k is

Pk = Pr(k? ≤ k ;µ) = 1−∫ bk

−bk

fk (u;µ)du

I The probability of stopping, i.e. exit probability, at k? = k is

Pk−Pk−1 =

∫ bk−1

−bk−1

fk−1(u;µ){1−Φ(bk−u;µ)+Φ(−bk−u;µ)}du

I This can be done by application of a Newton-Cotesformula of the second order, i.e. Simpson’s rule

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Classical Group Sequential Tests

I Pocock (1977)I O’Brien & Fleming (1979)I Slud & Wei (1982)I Lan & DeMets (1983)

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Intuition about Independent Increments

I With Gaussian outcome, it is intuitive that group sequentialtest statistics would have independent increments, thusallowing application of the classical group sequentialmethods

I With time to event outcome, it is unclear since each subjectcontributes follow-up data possibly multiple times overgroup sequential tests

I Conjecture by Armitage (1975, pp 140–2) for time to eventdata subject to random censoring

I With longitudinal outcome, again it is unclear since eachsubject contributes follow-up data multiple timeslongitudinally

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Tsiatis (1981)

I The asymptotic joint distribution of the efficient scores testfor the proportional hazards model calculated over time

I The score statistic of the proportional hazards modelcalculated at different points in time is shown to convergeasymptotically to a multivariate Gaussian process withindependent increments when individuals enter randomlythroughout the course of the study

I This allows group sequential methods to be based on thelogrank test in clinical trials with time to event outcomesubject to random censoring, thus proving Armitage’sconjecture

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Tsiatis (1982)

I Generalization of the results in Tsiatis (1981) to a generalclass of nonparametric test statistics characterized by

I Tarone and Ware (1977)I Prentice and Marek (1979)

I The asymptotic joint distribution of the sequentiallycomputed test statistics, within this general class ofnonparametrict ests, is derived, showing independentincrements

I Again this allows group sequential methods to be based onthis general class of nonparametric tests in clinical trialswith time to event outcome subject to random censoring

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Kim & Tsiatis (1990)

I The first in a series of joint publications on groupsequential methods

I Based on Tsiatis (1981)I Propose a unified design procedure for group sequential

clinical trials with time to event outcom subject to randomcensoring and staggered patient entry

I Provide a foundation for the launch of the commercialproduct EaSt by Cytel Corp.

I Kim, Boucher & Tsiatis (1995)I Maximum information vs duration designs

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

At Cytel Corp. for EaSt Version 2 in April 1996

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Tsiatis, Boucher & Kim (1995)

I Joint distribution of sequentially computed score and Waldtests for a general parametric model for time to eventoutcome, including a cure rate model

I In particular, score and Wald tests for a parameter ofinterest in the presence of nuisance paramters haveindependent increments

I Seed for group sequential methods based onsemiparametric efficient test statistics by Scharfstein,Tsiatis & Robins (1997)

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Lee, Kim & Tsiatis (1996)

I An independent increments structure of sequentiallycomputed test statistics based on the generalisedestimating equations of Liang & Zeger (1986) forlongitudinal data

I The limiting distribution has uncorrelated increments whenthe “working variances” are equal to or consistentlyestimate the true variances

I This simplifies the computational procedure for groupsequential boundaries to one involving recursive univariateintegrations and allows the use of standard groupsequential methods

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Scharfstein, Tsiatis & Robins (1997)

I Joint distributions of many group sequential statistics usedto analyze data arising from clinical trials with time to eventand longitudinal outcomes are multivariate normal with anindependent increments covariance structure

I This limiting distribution arises naturally when one uses anefficient test statistic to test a single parameter in asemiparametric or parametric model

I Most general results based on semiparametric efficiencttests

I Jennison & Turnbull (1997) for general regression models

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis

Early Sequential Analysis Independent Increments Group Sequential Tests with Increment Increments Conclusion

Conclusion

I Congratulations to Butch!I A very prolific run from 1981 to 1997 and beyondI Established many important results for group sequential

methods for clinical trialsI Set the standards for pratice of group sequential methods

for clinical trialsI Many thanks to Butch and former students

I Sandro PampallonaI Hélène BoucherI Sandra Lee

Independent Increments in GSTs A Symposium Honoring Professor Tsiatis