3. IntroductionIntroduction
ā¢ Clinical trials are preāmeditated!ClinicalĀ trialsĀ areĀ pre meditated!
ā¢ WeĀ preāspecifyĀ everything
S i it / i f i itā Superiority/noninferiority
ā PopulationĀ (inclusion/exclusionĀ criteria)
ā PrimaryĀ endpoint
ā SecondaryĀ endpoints
ā AnalysisĀ methods
ā SampleĀ size/power
6. IntroductionIntroduction
ā¢ But changes made before unblinding areButĀ changesĀ madeĀ beforeĀ unblinding areĀ
different
ā¢ Under strong null hypothesis that treatmentā¢ UnderĀ strongĀ nullĀ hypothesis thatĀ treatmentĀ
hasĀ NO effect,Ā blindedĀ dataĀ giveĀ noĀ infoĀ aboutĀ
treatment effecttreatmentĀ effect
ā ImpossibleĀ toĀ cheatĀ evenĀ ifĀ itĀ seemsĀ likeĀ cheating
E if bli d d d t h bi d l di t ib ti itā¢ E.g.,Ā evenĀ ifĀ blindedĀ dataĀ showĀ bimodalĀ distribution,Ā itĀ
isĀ notĀ causedĀ byĀ treatmentĀ ifĀ strongĀ nullĀ isĀ trueĀ
7. Permutation TestsPermutationĀ Tests
ā¢ Permutation tests condition on all data otherPermutationĀ testsĀ conditionĀ onĀ allĀ dataĀ otherĀ
thanĀ treatmentĀ labels
ā¢ Under strong null (D Z ) are independentā¢ UnderĀ strongĀ null,Ā (D,Z )Ā areĀ independent,Ā
whereĀ Z areĀ Ā±1Ā treatmentĀ indicatorsĀ &Ā DĀ areĀ
datadataĀ
ā ObservedĀ dataĀ DĀ wouldĀ haveĀ beenĀ observedĀ
regardless of the treatment givenregardlessĀ ofĀ theĀ treatmentĀ given
ā ItĀ isĀ asĀ ifĀ weĀ observedĀ DĀ FIRST,Ā thenĀ madeĀ theĀ
treatment assignments ZtreatmentĀ assignmentsĀ Z
8. Permutation TestsPermutationĀ Tests
ā¢ Peaking at data changes nothing becausePeakingĀ atĀ dataĀ changesĀ nothingĀ becauseĀ
permutationĀ testsĀ alreadyĀ conditionĀ onĀ D
ā¢ Conditional distribution of test statistic T(Z Y)ā¢ ConditionalĀ distributionĀ ofĀ testĀ statisticĀ T(Z,Y)Ā
givenĀ DĀ isĀ thatĀ ofĀ T(Z,y)Ā whereĀ y isĀ fixed
Di ib i f Z d d d i iā¢ DistributionĀ ofĀ Z dependsĀ onĀ randomizationĀ
methodĀ
ā Simple
ā PermutedĀ block,Ā etc.
10. Permutation TestsPermutationĀ Tests
T C C T C T C T T T C C C T C T
4 8 4 0 1 3 0 4 4 0 2 5 0 2 1 0
T-C T-C T-C T-C
O ll T C
-4.0 3.0 -1.5 0.5
Overall T-C
-0.5
12. Blinded 2āStage ProceduresBlindedĀ 2 StageĀ Procedures
ā¢ Blinded 2āstage adaptive procedures use 1stBlindedĀ 2 stageĀ adaptiveĀ proceduresĀ useĀ 1stĀ Ā
stageĀ toĀ makeĀ designĀ changes
ā SampleĀ sizeĀ (Gould,Ā 1992,Ā Stat.Ā inĀ Med.Ā 11,Ā 55ā66;Ā p ( , , , ;
GouldĀ &Ā Shih,Ā 1992Ā Commun.Ā inĀ Stat.Ā 21,Ā 2833ā
2853)Ā
P i d i ( di li liā PrimaryĀ endpointĀ (e.g.,Ā diastolicĀ versusĀ systolicĀ
bloodĀ pressure)
ā¢ Previous argument shows that if adaptation isā¢ PreviousĀ argumentĀ showsĀ thatĀ ifĀ adaptationĀ isĀ
madeĀ beforeĀ unblinding,Ā aĀ permutationĀ testĀ
on 1st stage data is still validonĀ 1stĀ stageĀ dataĀ isĀ stillĀ valid
13. Blinded 2āStage ProceduresBlindedĀ 2 StageĀ Procedures
ā¢ Careful! Subtle errors are possibleCareful!Ā Ā SubtleĀ errorsĀ areĀ possible
ā¢ E.g.,Ā inĀ adaptiveĀ regression,Ā whichĀ ofĀ theĀ
following is (are) valid?followingĀ isĀ (are)Ā valid?
1. FromĀ ANCOVAsĀ Y=Ī²01+Ī²z+Ī²ixi,Ā i=1,ā¦,k,Ā pickĀ xi
that minimizes MSE; do permutation test onthatĀ minimizesĀ MSE;Ā doĀ permutationĀ testĀ onĀ
winner
2 From ANCOVAs Y=Ī² 1+Ī² x i=1 k pick x that2. FromĀ ANCOVAsĀ Y=Ī²01+Ī²ixi,Ā i=1,ā¦,k,Ā pickĀ xi thatĀ
minimizesĀ MSE;Ā doĀ permutationĀ testĀ onĀ
Y=Ī²01+Ī²z+Ī²*x*,Ā whereĀ x*Ā isĀ winnerĪ²0 Ī² Ī² ,
14. Blinded 2āStage ProceduresBlindedĀ 2 StageĀ Procedures
ā¢ Careful! Subtle errors are possibleCareful!Ā Ā SubtleĀ errorsĀ areĀ possible
ā¢ E.g.,Ā inĀ adaptiveĀ regression,Ā whichĀ ofĀ theĀ
following is (are) valid?followingĀ isĀ (are)Ā valid?
1. FromĀ ANCOVAsĀ Y=Ī²01+Ī²z+Ī²ixi,Ā i=1,ā¦,k,Ā pickĀ xi
that minimizes MSE; do permutation test onthatĀ minimizesĀ MSE;Ā doĀ permutationĀ testĀ onĀ
winner
2 From ANCOVAs Y=Ī² 1+Ī² x i=1 k pick x that2. FromĀ ANCOVAsĀ Y=Ī²01+Ī²ixi,Ā i=1,ā¦,k,Ā pickĀ xi thatĀ
minimizesĀ MSE;Ā doĀ permutationĀ testĀ onĀ
Y=Ī²01+Ī²z+Ī²*x*,Ā whereĀ x*Ā isĀ winnerĪ²0 Ī² Ī² ,
15. Blinded 2āStage ProceduresBlindedĀ 2 StageĀ Procedures
ā¢ Unblinding andĀ apparentĀ Ī±āinflationĀ alsoĀ possibleĀ U b d g a d appa e t Ī± at o a so poss b e
ifĀ strongĀ nullĀ isĀ false
ā¢ E.g.,Ā changeĀ primaryĀ endpointĀ basedĀ onĀ āblindedāĀ g g p y p
dataĀ (X,Y1,Y2),Ā Y1 andĀ Y2 areĀ potentialĀ primariesĀ
andĀ X=levelĀ ofĀ studyĀ drugĀ inĀ blood
ā XĀ completelyĀ unblinds
ā CanĀ thenĀ pickĀ Y1 orĀ Y2 withĀ biggestĀ zāscore
Clearly inflates Ī±ā ClearlyĀ inflatesĀ Ī±
ā Problem:Ā strongĀ nullĀ requiresĀ noĀ effectĀ onĀ ANY
variableĀ examinedĀ (includingĀ X=levelĀ ofĀ studyĀ drug)
16. Blinded 2āStage ProceduresBlindedĀ 2 StageĀ Procedures
ā¢ Claim: the following procedure is validClaim:Ā theĀ followingĀ procedureĀ isĀ valid
ā AfterĀ viewingĀ 1stĀ stageĀ dataĀ D1,Ā chooseĀ testĀ
statistic T1(Y1 Z1) and second stage data to collectstatisticĀ T1(Y1,Z1)Ā andĀ secondĀ stageĀ dataĀ toĀ collect
ā AfterĀ observingĀ D2,Ā chooseĀ T2(Y2,Z2)Ā andĀ methodĀ
of combining T1 and T2, f(T1,T2)ofĀ combiningĀ T1 andĀ T2,Ā f(T1,T2)
ā ConditionalĀ distributionĀ ofĀ f(T1,T2)Ā givenĀ (D1,D2)Ā isĀ
itsĀ stratifiedĀ permutationĀ distributionp
ā StratifiedĀ permutationĀ testĀ controlsĀ conditional,Ā &Ā
thereforeĀ unconditionalĀ typeĀ IĀ errorĀ rateĀ
17. Focus of Rest of TalkFocusĀ ofĀ RestĀ ofĀ Talk
ā¢ Permutation tests are asymptoticallyPermutationĀ testsĀ areĀ asymptoticallyĀ
equivalentĀ toĀ tātests
ā¢ Suggests that adaptive t tests might be valid ifā¢ SuggestsĀ thatĀ adaptiveĀ tātestsĀ mightĀ beĀ validĀ ifĀ
adaptiveĀ permutationĀ testsĀ are
W id i bā¢ WeĀ considerĀ connectionsĀ betweenĀ
permutationĀ andĀ tātests,Ā andĀ validityĀ ofĀ
d i f d i iadaptiveĀ tātestsĀ fromĀ adaptiveĀ permutationĀ
testsĀ
18. OneāSample CaseOne SampleĀ Case
ā¢ CommunityĀ randomizedĀ trialsĀ sometimesĀ pairĀ Co u ty a do ed t a s so et es pa
matchĀ &Ā randomizeĀ withinĀ pairs
ā¢ E.g.,Ā COMMITĀ trialĀ usedĀ communityĀ interventionĀ g y
toĀ helpĀ peopleĀ quitĀ smokingā11Ā matchedĀ pairs
ā¢ D=differenceĀ inĀ quitĀ ratesĀ betweenĀ treatmentĀ (T)Ā
&Ā controlĀ (C)
T CĀ Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā D=TāC
PairĀ iĀ Ā Ā Ā Ā Ā Ā Ā Ā 0.30Ā Ā Ā Ā Ā 0.25Ā Ā Ā Ā Ā Ā Ā Ā +0.05
19. OneāSample CaseOne SampleĀ Case
ā¢ CommunityĀ randomizedĀ trialsĀ sometimesĀ pairĀ Co u ty a do ed t a s so et es pa
matchĀ &Ā randomizeĀ withinĀ pairs
ā¢ E.g.,Ā COMMITĀ trialĀ usedĀ communityĀ interventionĀ g y
toĀ helpĀ peopleĀ quitĀ smokingā11Ā matchedĀ pairs
ā¢ D=differenceĀ inĀ quitĀ ratesĀ betweenĀ treatmentĀ (T)Ā
&Ā controlĀ (C)
C TĀ D=TāC
PairĀ iĀ Ā Ā Ā Ā Ā Ā Ā Ā 0.30Ā Ā Ā Ā Ā 0.25Ā Ā Ā Ā Ā Ā Ā Ā ā0.05
20. OneāSample CaseOne SampleĀ Case
ā¢ Permuting labels changes only sign of DPermutingĀ labelsĀ changesĀ onlyĀ signĀ ofĀ D
ā¢ PermutationĀ testĀ conditionsĀ onĀ |Di|=Ā di
+;Ā
d + d d + ll lik lādi
+Ā andĀ di
+ areĀ equallyĀ likely
ā¢ The permutation distribution of ļDi is dist. ofTheĀ permutationĀ distributionĀ ofĀ ļDi isĀ dist.Ā of
21w p1where /ZdZ ļļ½ļ„ ļ«
21w.p.1
21w.p.1where,
/
/ZdZ iii
ļ«
ļ½ļ„
21. OneāSample CaseOne SampleĀ Case
ā¢ InĀ 1st stage,Ā adaptĀ basedĀ onĀ |D1|,ā¦,|Dn|Ā (blinded)g , p | 1|, ,| n| ( )
ā E.g.,Ā increaseĀ stageĀ 2Ā Ā sampleĀ sizeĀ becauseĀ |Di|Ā isĀ veryĀ
large
ā¢ What is conditional distribution of 1st stage sumā¢ WhatĀ isĀ conditionalĀ distributionĀ ofĀ Ā 1st stageĀ sumĀ
Ī£Di givenĀ |D1|=d1
+,ā¦,|Dn|= dn
+ andĀ theĀ
adaptation?adaptation?
ā TheĀ adaptationĀ isĀ aĀ functionĀ ofĀ |D1|,ā¦,|Dn|Ā
ā TheĀ nullĀ distributionĀ ofĀ Ī£Di givenĀ |D1|=d1
+,ā¦,|Dn|=Ā dn
+
i g | 1| 1 , ,| n| n
ISĀ itsĀ permutationĀ distribution
ā Conclusion:Ā permutationĀ testĀ onĀ stageĀ 1Ā dataĀ stillĀ valid
23. OneāSample CaseOne SampleĀ Case
ā¢ Asymptotically,Ā permutationĀ distributionĀ isĀ sy ptot ca y, pe utat o d st but o s
normalĀ withĀ thisĀ meanĀ andĀ varianceĀ (Lindebergā
FellerĀ CLT)
ā¢ I.e.,Ā conditionalĀ distributionĀ ofĀ ļDi givenĀ , i g
|D1|=d1
+,ā¦,|Dn|=Ā dn
+ isĀ asymptoticallyĀ N(0,ļdi
2)
ā¢ DependsĀ onĀ |D1|=d1
+,ā¦,|Dn|=Ā dn
+ onlyĀ throughĀ
L2=ļdi
2L ļdi
26. OneāSample CaseOne SampleĀ Case
ā¢ Begs question, is this true for all sample sizesBegsĀ question,Ā isĀ thisĀ trueĀ forĀ allĀ sampleĀ sizesĀ
underĀ normalityĀ assumption?
ā¢ if Di are iid N(0,ļ³2), then canifĀ Di areĀ iid N(0,ļ³ ),Ā thenĀ can
?fti d db' 2
ļ„ļ„ i
D
D
T ?oftindependenbe' 2
2 ļ„
ļ„
ļ„ļ½ i
i
i
D
D
T
ā¢ SeemsĀ crazy,Ā butĀ itāsĀ true!
27. OneāSample CaseOne SampleĀ Case
ā¢ One way to see that Tā is independent of ļDi
2OneĀ wayĀ toĀ seeĀ thatĀ T isĀ independentĀ ofĀ ļDi
usesĀ Basuās theorem:Ā
ā¢ RecallĀ SĀ isĀ sufficient forĀ Īø ifĀ F(y|s)Ā doesĀ notĀ
d d Īø i i l if { ( )} f ll ĪødependĀ onĀ Īø;Ā itĀ isĀ complete ifĀ E{g(S)}=0Ā forĀ allĀ Īø
impliesĀ g(S)ā”0Ā withĀ probabilityĀ 1
ā¢ A is ancillary if its distribution does not dependā¢ AĀ isĀ ancillary ifĀ itsĀ distributionĀ doesĀ notĀ dependĀ
onĀ Īø
ā¢ Basu,Ā 1955,Ā Sankhya 15,Ā 377ā380:
IfĀ SĀ isĀ aĀ complete,Ā sufficientĀ statisticĀ andĀ AĀ
isĀ ancillary,Ā thenĀ SĀ andĀ AĀ areĀ independent
29. OneāSample CaseOne SampleĀ Case
ā¢ Same argument shows that the usual tāSameĀ argumentĀ showsĀ thatĀ theĀ usualĀ t
statisticĀ isĀ independentĀ ofĀ ļDi
2
2 2ā¢ UnderĀ Di iid N(0,ļ³2)Ā withĀ ļ³2Ā unknown
āļDi
2 isĀ completeĀ andĀ sufficient
ā UsualĀ tāstatisticĀ T=Ā ļDi/(ns2)1/2 isĀ ancillary
ā By Basuās theorem T and ļD 2 are independentā ByĀ Basu s theorem,Ā TĀ andĀ ļDi areĀ independentĀ
(Ā Shao (2003):Ā MathematicalĀ Statistics,Ā Springer)Ā
30. OneāSample CaseOne SampleĀ Case
ā¢ This result is important for adaptive sample sizeThisĀ resultĀ isĀ important forĀ adaptiveĀ sampleĀ sizeĀ
calculations
ā Stage 1 with n1= half of original sample size: changeStageĀ 1Ā withĀ n1 Ā halfĀ ofĀ originalĀ sampleĀ size:Ā changeĀ
secondĀ stageĀ sampleĀ sizeĀ toĀ n2=n2(Ī£Di
2)
ā Conditioned on Ī£D 2:ā ConditionedĀ onĀ Ī£Di :Ā
ā¢ TestĀ statisticĀ T1 hasĀ exactĀ tādistributionĀ withĀ n1ā1Ā d.f.
ā¢ TestĀ statisticĀ T2 hasĀ exactĀ tādistributionĀ withĀ n2ā1Ā d.f. andĀ isĀ 2 2
independentĀ ofĀ T1
ā¢ PāvaluesĀ P1 andĀ P2 areĀ independentĀ U(0,1)
ā¢ Y={n 1/2Ī¦ā1(P )+n 1/2Ī¦ā1(P )}/(n +n )1/2 is N(0 1) under Hā¢ Y={n1
1/2Ī¦ 1(P1)+n2
1/2Ī¦ 1(P2)}/(n1+n2)1/2 isĀ N(0,1)Ā underĀ H0
31. OneāSample CaseOne SampleĀ Case
ā¢ Reject if Y>zRejectĀ ifĀ Y>zĪ±
ā¢ ConditionedĀ onĀ Ī£Di
2,Ā typeĀ IĀ errorĀ rateĀ isĀ Ī±
ā¢ UnconditionalĀ typeĀ IĀ errorĀ rateĀ isĀ Ī± asĀ well
ā¢ Most other twoāstage procedures are onlyMostĀ otherĀ two stageĀ proceduresĀ areĀ onlyĀ
approximate
32. OneāSample CaseOne SampleĀ Case
ā¢ CouldĀ evenĀ makeĀ otherĀ adaptationsĀ likeĀ changingĀ p g g
primaryĀ endpoint
ā¢ LookĀ atĀ Ī£Di
2 forĀ eachĀ endpointĀ andĀ determineĀ
whichĀ oneĀ isĀ primaryĀ Ā
ļ 2ā E.g.,Ā pickĀ endpointĀ withĀ smallestĀ ļDi
2
ā¢ Slight generalization of our result shows thatā¢ SlightĀ generalizationĀ ofĀ ourĀ resultĀ showsĀ thatĀ
conditionalĀ distributionĀ ofĀ TĀ givenĀ adaptation isĀ
stillĀ exactĀ tĀ
33. OneāSample CaseOne SampleĀ Case
ā¢ Shows that conditional type I error rate givenShowsĀ thatĀ conditionalĀ typeĀ IĀ errorĀ rateĀ givenĀ
adaptationĀ isĀ controlledĀ atĀ levelĀ Ī±
ā¢ Unconditional type I error rate must also beā¢ UnconditionalĀ typeĀ IĀ errorĀ rateĀ mustĀ alsoĀ beĀ
controlledĀ atĀ levelĀ Ī±
D i i l i i liā¢ DerivationĀ assumesĀ multivariateĀ normalityĀ
withĀ variance/covarianceĀ notĀ dependingĀ onĀ
mean
34. TwoāSample CaseTwo SampleĀ Case
ā¢ CanĀ useĀ sameĀ reasoningĀ inĀ 2āsampleĀ settingĀ Ca use sa e easo g sa p e sett g
ā¢ WithĀ equalĀ sampleĀ sizes,Ā theĀ numeratorĀ is
ļ„ļ„ļ„ YZYY
ā¢ Permutation distribution is distribution of
ļ„ļ„ļ„ ļ½ļ ii
C
i
T
i YZYY
PermutationĀ distributionĀ isĀ distributionĀ ofĀ
ļ„ļ„ ļ½ļ±ļ½ 0,1each, iiii ZZyZ
ā¢ LetĀ sL
2 beĀ ālumpedāĀ varianceĀ ofĀ allĀ dataĀ
(treatment and control)(treatmentĀ andĀ control)Ā
37. SummarySummary
ā¢ Permutation tests are often valid even inPermutationĀ testsĀ areĀ oftenĀ validĀ evenĀ inĀ
adaptiveĀ settingsĀ ifĀ blindĀ isĀ maintained
ā¢ There is a close connection betweenā¢ ThereĀ isĀ aĀ closeĀ connectionĀ betweenĀ
permutationĀ testsĀ andĀ tātests
C d d lidi f d i fā¢ CanĀ deduceĀ validityĀ ofĀ adaptiveĀ tātestsĀ fromĀ
validityĀ ofĀ adaptiveĀ permutationĀ tests