global sensitivity analysis for studies with intermittent...
TRANSCRIPT
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Global Sensitivity Analysis for Studies withIntermittent Missing Data and Death
Chenguang [email protected]
November 30, 2015
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Objectives
Part I: ReviewTo review issues and common analysis methods fordeath-truncated dataPart II: MethodTo learn about a composite endpoint based approachPart III: SoftwareTo introduce a software that implements the proposedmethod
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PART I: REVIEW
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General setting
Consider a two-armed randomized clinical studyThe goal is to evaluate the efficacy of a treatmentOutcomes scheduled to be measured at pre-specifiedpost-randomization time pointsSubjects at (high) risk of deathOutcomes unobserved due to loss to follow-up, withdrawalof consent, out-of-window visits, death, etc.
[Review] 1/53
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Scenarios of unobserved outcomes
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Notation
T = 0,1: treatment assignmentY0: baseline measure at t0Y1, . . . ,YK : functional outcomes at t1, . . . , tKZ = g(Y0, . . . ,YK ): primary functional endpoint
e.g. Z = YK , Z = YK − Y0only defined when ∆K = 1
L: survival time∆k = I(L > tk ): survival status at tkFor survivors (∆K = 1)
τk : missingness indicator of Yk (1: observed, 0: missing)S = (τ1, . . . , τK ): missing patternYobs = {Yk : τk = 1, k ≥ 1}: observed functional outcomeYmis = {Yk : τk = 0, k ≥ 1}: missing functional outcome
X : baseline covariatesY k : (Y0, . . . ,Yk )
[Review] 3/53
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Data truncated by death
Distinction exists between missing data and data truncatedby deathMissing data: exist but not collectedData truncated by death: does not exist and undefinedMissing data imputation methods generally not applicablefor data truncated by death
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Common analysis methods
Evaluate treatment effects conditional on survivalJoint modeling survival and functional outcomesEvaluate causal treatment effects for principal stratumComposite endpoint combining survival and functionaloutcomes
B. F. Kurland, L. L. Johnson, B. L. Egleston, and P. H. Diehr. Longitudinal data with follow-up truncated bydeath: match the analysis method to research aims.Statistical Science, 24(2):211–222, 2009
[Review] 5/53
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Conditional model
Unconditional models not appropriate
E(Yk ) = E(Yk |∆k = 1)P(∆k = 1)+E(Yk |∆k = 0)︸ ︷︷ ︸undefined
P(∆k = 0)
Conditional models evaluate treatment effects at a specifictime tk
fully conditioning on survival time Lor partially conditioning on survival status ∆k
Issue: Selection bias introduced in treatment effectestimation since survival is a post-randomization covariate
M. Shardell and R. R. Miller. Weighted estimating equations for longitudinal studies with death andnon-monotone missing time-dependent covariates and outcomes.Statistics in Medicine, 27:1008–1025, 2008
[Review] 6/53
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Joint model
Comprised of two linked sub-modelssurvival processlongitudinal outcome process
Introduce a set of common latent random effects shared bythe two sub-modelsthe survival process serves as informative censoring forthe longitudinal processthe longitudinal measures can inform the underlyingprocess for survival
A Lawrence Gould, Mark Ernest Boye, Michael J Crowther, Joseph G Ibrahim, George Quartey, Sandrine
Micallef, and Frederic Y Bois. Joint modeling of survival and longitudinal non-survival data: current methods andissues. report of the dia bayesian joint modeling working group.Statistics in medicine, 2014
[Review] 7/53
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Joint model
Allows trajectories of longitudinal outcome after death, notscientifically meaningfulassumptions are made regarding the distributions as wellas the interrelationships between the longitudinal andevent-time datathe models may be most useful as sensitivity analyses,germane exploratory analyses, or informative secondaryanalyses
Dionne L Price and Yan Wang. Commentary on joint modeling of survival and longitudinal non-survival data:current methods and issues.Statistics in medicine, 34(14):2200–2201, 2015
[Review] 8/53
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Principal stratification
Focused on the cohort of subjects who would havesurvived under either treatment arm, i.e. the principlestratification with respect to survival
{∆k (1) = ∆k (0) = 1}
Assess survivor average causal effect (SACE) defined as
SACEk = E(Yk (1)− Yk (0)|∆k (1) = ∆k (0) = 1)
Useful for understanding the mechanistic effect oftreatment on clinical outcomesIssue: whether a subject belongs to the “survivor” stratumunknown
C. E. Frangakis and D. B. Rubin. Principal stratification in causal inference.Biometrics, 58(1):21–29, 2002
[Review] 9/53
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Composite endpoint
A composite endpoint consists of multiple single endpointsFor death truncated data, mix both the survival L and thefunctional outcome ZSimple and useful if
the goal is to evaluate the treatment effect on both thefunctional outcome and survivalthe composite endpoint can be ordered in a meaningful way
Issue: effects of treatment on survival and on thefunctional outcome cannot be easily separated
P. Diehr, D. L. Patrick, S. Hedrick, M. Rothman, D. Grembowski, T. E. Raghunathan, and S. Beresford.
Including deaths when measuring health status over time.Medical Care, 33:AS164 – AS172, 1995
[Review] 10/53
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PART II: METHOD
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Goal
To propose a composite endpoint approach that handles bothdeaths and intermittent missing data among subjects alive atthe assessment times
[Proposal] Goal 11/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Ranking
Assume no missing data at this timeAssume that higher values of Z denote better outcomes
[Proposal] Ranking 12/53
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Ranking
Consider two subjects i and jSubject i ranked better than j if
both alive at tK (∆i,K = ∆j,K = 1) and Zi > Zjboth dead at tK (∆i,K = ∆j,K = 0) and Li > Ljsubject i alive at tK and subject j dead at tK
Subject i ranked the same as j ifboth alive at tK (∆i,K = ∆j,K = 1) and Zi = Zjboth dead at tK (∆i,K = ∆j,K = 0) and Li = Lj
[Proposal] Ranking 13/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Treatment effect
Treatment effect is measured by the probability that the rank fora random individual with T = 0 is less than the rank of arandom individual with T = 1 minus the probability that the rankfor a random individual with T = 0 is greater than the rank of arandom individual with T = 1
[Proposal] Treatment effect 14/53
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Treatment effect
R: rank of a subject among all the study participantsR(0): rank for a random subject on T = 0R(1): rank for a random subject on T = 1The treatment effect is measured by
θ = P(R(1) > R(0))− P(R(1) < R(0))
[Proposal] Treatment effect 15/53
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Treatment effect
Why not θ∗ = P(R(1) > R(0))?Let δ be the probability that two subjects are ranked thesameThen, θ = 2θ∗ − 1 + δ
Under the null hypothesis of no treatment effectθ = 0 regardless of ηθ∗ = (1− η)/2
The definition of θ handles ties
[Proposal] Treatment effect 16/53
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Estimation of θ
In the absence of missing data, estimate θ by
θ̂ =1
n0n1
∑i:Ti=0
∑j:Tj=1
{I(Ri < Rj)− I(Ri > Rj)}
where n0 =∑
i(1− Ti) and n1 =∑
i Ti
[Proposal] Treatment effect 17/53
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Variance of θ̂
Var(θ̂)
=
(1
n0n1
)2 n0∑
i=1
n1∑j=1
{I(Ri < Rj) + I(Rj < Ri)}
+n0 − 1
n0
n0∑i=1
n0∑i ′=1,i 6=i ′
n1∑j=1
{I(Ri < Rj ,Ri ′ < Rj) + I(Ri > Rj ,Ri ′ > Rj)
− I(Ri < Rj ,Ri ′ > Rj)− I(Ri > Rj ,Ri ′ < Rj)}
+n1 − 1
n1
n0∑i=1
n1∑j=1
n1∑j ′=1,j 6=j ′
{I(Ri < Rj ,Ri < Rj ′) + I(Ri > Rj ,Ri > Rj ′)
− I(Ri < Rj ,Ri > Rj ′)− I(Ri > Rj ,Ri < Rj ′)}
+(n0 − 1)(n1 − 1)
n0n1
n0∑i=1
n1∑j=1
n0∑i=1,i 6=i ′
n1∑j=1,j 6=j ′
{I(Ri < Rj)− I(Rj < Ri)}2− θ̂2
[Proposal] Treatment effect 18/53
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Quantiles
Quantiles (e.g. median) of the composite endpoint,(L,∆K Z ), may further quantify the treatment effectMore straight forward for interpretationNecessary supplement to the primary rank analysis
[Proposal] Treatment effect 19/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Missingness
To determine ranks, need to impute Ymis for non-completers
[Proposal] Benchmark missing data assumptions 20/53
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Benchmark assumptions
f (Ymis|∆K = 1,Yobs,Y0,X ,T ,S = s)
= f (Ymis|∆K = 1,Yobs,Y0,X ,T ,S = 1) ∀s 6= 1
1: a K dimensional vector of 1’sS = 1: “completers”Complete case missing value (CCMV) restrictions appliedto the missing data patterns for patients alive at tK
Roderick JA Little. Pattern-mixture models for multivariate incomplete data.Journal of the American Statistical Association, 88(421):125–134, 1993
[Proposal] Benchmark missing data assumptions 21/53
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CCMV vs MAR
j = 1 j = 2 j = 3 j = 4
S = 1 p1(y1) p1(y2|y1) p1(y3|y1, y2) p1(y4|y1, y2, y3)S = 2 p2(y1) p≥2(y2|y1) p2(y3|y1, y2) p2(y4|y1, y2, y3)S = 3 p3(y1) p≥2(y2|y1) p≥3(y3|y1, y2) p3(y4|y1, y2, y3)S = 4 p4(y1) p≥2(y2|y1) p≥3(y3|y1, y2) p4(y4|y1, y2, y3)
[Proposal] Benchmark missing data assumptions 22/53
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Data transformation
For imputation, it is important to utilize an approach thatrespects bounds on the functional outcomesFailure to do so can result in non-sensical imputationsMay consider a data tranformation
Y †k = φ(yk ) = log{
yk − BL
BU − yk
},
where (BL,BU) denote the lower and upper bound of YThere is a one-to-one mapping
f (Y K |∆K = 1,Y0,X ,T ,S = 1) =
h(Y†K |∆K = 1,Y0,X ,T ,S = 1)
∣∣∣∣∣K∏
k=1
dφ(Yk )
dYk
∣∣∣∣∣[Proposal] Benchmark missing data assumptions 23/53
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Modeling strategy
Sequential factorization
f (Y K |∆K = 1,Y0,X ,T ,S = 1)
=K∏
k=1
f (Yk |∆K = 1,Y k−1,X ,T ,S = 1)
Specify
Yk |Y k−1,Y0,X ,T = t ,S = 1
= µk ,t (Y k−1,X ;αk ,t ) + εk ,t
µk,t : mean function, e.g.
µk,t (Y k−1,X ;αk,t ) = αk,t,0 + αk,t,1Y k−1 + αk,t,2Y0 + αk,t,3X
εk,t : residuals
[Proposal] Benchmark missing data assumptions 24/53
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Estimation
αk ,t : estimated using least square estimator
α̂k ,t = argmin
{n∑
i=1
I(Ti = t)∆K ,i
(K∏
k=1
τk ,i
)ε2k ,t ,i
}εk ,t ∼ Fk ,t
Fk,t = N(0, σ2k,t ) under normality assumption
Fk,t estimated by kernel density estimator, e.g.
f̂k,t (x) ∝n∑
i=1
I(Ti = t)∆K ,i
(K∏
k=1
τk,i
)φ
(x − εk,t,i
h
)where h is the bandwidth
[Proposal] Benchmark missing data assumptions 25/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Sensitivity analysis
Benchmark assumptions (CCMV) untestableSensitivity analysis essential to evaluate the robustness ofinferences to deviations from benchmark assumptions
Panel on Handling Missing Data in Clinical Trials; National Research Council. The Prevention and Treatmentof Missing Data in Clinical Trials.The National Academies Press, 2010.ISBN 9780309158145.URL http://www.nap.edu/openbook.php?record_id=12955
[Proposal] Sensitivity analysis 26/53
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Exponential tilting
Exponential tilting model
f ′(y) ∝ eβy f (y)
Constructs a neighborhood of distributions f ′(y)
centered around benchmark distribution f (y)indexed by (sensitivity) parameter β
Closed form can be derived for (multivariate) normaldistribution
Y ∼ N(µ,Σ)Y ′ ∼ N(µ + Σβ,Σ)
[Proposal] Sensitivity analysis 27/53
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Example: Exponential tilting
Y ∼ N(
0,[
1 0.50.5 1
])
−4 −2 0 2 4
−4
−2
02
4
Y1
Y2
β = − 1β = 0β = 1
[Proposal] Sensitivity analysis 28/53
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Dimension of sensitivity parameters
Recall: benchmark assumptions
f (Ymis|∆K = 1,Yobs,Y0,X ,T = t ,S = s)
= f (Ymis|∆K = 1,Yobs,Y0,X ,T = t ,S = 1)
Sensitivity parameters typically introduced as follows:
f (Ymis|∆K = 1,Yobs,Y0,X ,T = t ,S = s)
∝ exp{βt ,sYmis}f (Ymis|∆K = 1,Yobs,Y0,X ,T = t ,S = 1)
Sensitivity parameters βt ,sdepends on treatment and missing patterndimension too highdifficult to set sensitivity analysis scenariosdifficult to interpret and summarize results
[Proposal] Sensitivity analysis 29/53
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Sensitivity analysis assumption
f (Ymis|∆K = 1,Yobs,Y0,X ,T = t ,S = s)
∝ exp{βtZ}f (Ymis|∆K = 1,Yobs,Y0,X ,T = t ,S = 1)
Z : primary endpoint, clinical interestβt : treatment specific, dimension 2 regardless of Kβt = 0: benchmark assumptions|βt |: distance (in the units of Z ) from benchmarkassumptions
[Proposal] Sensitivity analysis 30/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Benchmark assumptions
S τ1 τ2 Y1 Y2s1 0 0 x xs2 0 1 x y2s3 1 0 y1 xs4 1 1 y1 y2
Assumptions:
f (Y2,Y1|∆2 = 1,Y0,X ,T ,S = (0,0))
= f (Y2,Y1|∆2 = 1,Y0,X ,T ,S = 1)
f (Y1|∆2 = 1,Y2,Y0,X ,T ,S = (0,1))
= f (Y1|∆2 = 1,Y2,Y0,X ,T ,S = 1)
f (Y2|∆2 = 1,Y1,Y0,X ,T ,S = (1,0))
= f (Y2|∆2 = 1,Y1,Y0,X ,T ,S = 1)
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Modeling
f (Y1,Y2|∆2 = 1,Y0,X ,T = t ,S = 1)
= f (Y1|∆2 = 1,Y0,X ,T = t ,S = 1)︸ ︷︷ ︸model 1
× f (Y2|∆2 = 1,Y1,Y0,X ,T = t ,S = 1)︸ ︷︷ ︸model 2
[Proposal] Bivariate case (example) 32/53
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Sensitivity analysis assumptions
Let Z = Y1 + Y2
Assumptions
f (Y2,Y1|∆2 = 1,Y0,X ,T = t ,S = (0,0))
∝ exp{βt (Y1 + Y2)}f (Y2,Y1|∆2 = 1,Y0,X ,T ,S = 1)
f (Y1|∆2 = 1,Y2,Y0,X ,T ,S = (0,1))
∝ exp{βtY1}f (Y1|∆2 = 1,Y2,Y0,X ,T ,S = 1)
f (Y2|∆2 = 1,Y1,Y0,X ,T ,S = (1,0))
∝ exp{βtY2}f (Y2|∆2 = 1,Y1,Y0,X ,T ,S = 1)
[Proposal] Bivariate case (example) 33/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Numerical sampling
Goal: to draw samples of Ymis for each individual with∆K = 1 and S 6= 1 from
f (Ymis|∆K = 1,Yobs,Y0,X ,T ,S = s)
∝ exp(βT Z )f (Ymis|∆K = 1,Yobs,Y0,X ,T ,S = 1)
Close form only available whenµk,t : linearεk,t : normally distributedZ = g(Y0, . . . ,YK ): linear
Numerical sampling necessary in generalPropose to apply a random-walk Metroplis-Hastingsalgorithm
[Proposal] Imputation 34/53
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Sampling steps
1. Set j = 0. Choose arbitrary initial values for Ymis, denotedby Y (0)
mis. Let Z (0) be the primary functional endpoint withdata (Yobs,Y
(0)mis)
2. Set j = j + 13. Generate Y ′mis from a (multivariate) Gaussian distribution
with mean Y (j−1)mis and variance Σ
[Proposal] Imputation 35/53
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Sampling steps
4. Calculate the acceptance ratio as
a =exp{βT Z ′}f (Y ′mis|∆K = 1,Yobs,Y0,X ,T ,S = 1)
exp{βT Z (j−1)}f (Y (j−1)mis |∆K = 1,Yobs,Y0,X ,T ,S = 1)
=exp{βT Z ′}f (Y ′mis,Yobs|∆K = 1,Y0,X ,T ,S = 1)
exp{βT Z (j−1)}f (Y (j−1)mis ,Yobs|∆K = 1,Y0,X ,T ,S = 1)
where Z ′ and Z (j−1) are the primary functional endpointswith data (Yobs,Y ′mis) and (Yobs,Y
(j−1)mis ), respectively
[Proposal] Imputation 36/53
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Sampling steps
5. Accept Y (j)mis = Y ′mis with probability min(1,a) and
Y (j)mis = Y (j−1)
mis with probability 1−min(1,a)
6. Repeat Steps 2-5 until the Markov chain converges
7. Draw random samples from the set {Y (j0)mis ,Y
(j0+1)mis , . . .} as
the imputed missing values, where j0 corresponds to thenumber of burn-in
[Proposal] Imputation 37/53
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Multiple imputation
Draw M copies Ymis for each individual with ∆K = 1 andS 6= 1Create M complete datasetsFor each complete dataset m, estimate θ by θ̂m
Overall estimator of θ
θ̃ =1M
M∑m=1
θ̂m
Confidence intervals constructed by non-parametricbootstrap
[Proposal] Imputation 38/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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HT-ANAM 302 study
Randomized, double-blind, placebo-controlled Phase IIIstudyIntent-to-treat population: advanced non-small cell lungcancer subjectsTo evaluate the efficacy of drug anamorelinFunctional outcome lean body mass (LBM) scheduled tobe measured at baseline (Y0), 6 weeks (Y1) and 12 weeks(Y2)
Primary functional endpoint: Z = (Y2+Y1)2 − Y0
[Proposal] Case study 39/53
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Death and missingness
Placebo Anamorelinn = 157 n = 322
Died Prior to Wk 12 24 (15.3%) 54 (16.8%)Survivors with complete data 93 (59.2%) 185 (57.5%)Survivors missing only Wk 6 3 (1.9%) 17 (5.3%)
Survivors missing only Wk 12 17 (10.8%) 31 (9.6%)Survivors missing both Wks 6, 12 20 (12.7%) 35 (10.9%)
[Proposal] Case study 40/53
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Survival
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
Time
Sur
viva
l pro
babi
lity
Placebo (n=157)Anamorelin (n=322) p−val=0.6621
[Proposal] Case study 41/53
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Baseline covariates
Covariates LevelsECOG 0:{0,1}, 1:{2}
AGE 0:≤ 65, 1:> 65GENDER 0:M, 1:F
BMI 0:≤ 18.5, 1:> 18.5WEIGHT LOSS 1 0:≤ 10%, 1:> 10%
Y0 Continuous
1in prior 6 months[Proposal] Case study 42/53
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Modeling
Specify µk ,t (Y k−1,X ;αk ,t ) as follows:
µ1,t ,i = α1,t ,1 + α1,t ,2Y0,i + α1,t ,3ECOGi + α1,t ,4AGEi
+ α1,t ,5Gi + α1,t ,6BMIi + α1,t ,7WLi
µ2,t ,i = α2,t ,1 + α2,t ,2Y0,i + α2,t ,3ECOGi + α2,t ,4AGEi
+ α2,t ,5Gi + α2,t ,6BMIi + α2,t ,7WLi
+ α2,t ,8Y1,i
[Proposal] Case study 43/53
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Model fitting diagnosis
30 40 50 60 70
−10
−5
05
Placebo,k=1
Fitted values
Res
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02
Placebo,k=1
Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q−Q
140
301
114
30 40 50 60 70
−4
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02
4
Placebo,k=2
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23
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Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q−Q
426
430
346
30 35 40 45 50 55 60
−10
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05
10
Anamorelin,k=1
Fitted values
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Theoretical Quantiles
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dize
d re
sidu
als
Normal Q−Q
206
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02
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Theoretical Quantiles
Sta
ndar
dize
d re
sidu
als
Normal Q−Q
165
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[Proposal] Case study 44/53
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Analysis under benchmark assumptions
10 imputed datasets generated200 bootstrap samples
Table: Hypothesis testing
θ̂ (95% CI) p-valueHT-ANAM 302 Study 0.30(0.19,0.40) < 0.0001
Table: Median
p̂50 (95% CI)HT-ANAM 302 Study Anamorelin 0.67( 0.45, 0.89)
Placebo -0.92(-1.43,-0.28)
[Proposal] Case study 45/53
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Cumulative plot
Composite Endpoint
Pro
babi
lty
Survival Functional3 84−6.9 11
0.00
0.25
0.50
0.75
1.00 Placebo
Anamorelin
[Proposal] Case study 46/53
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Choice of sensitivity parameters
−10 −5 0 5 10 15
0.00
0.05
0.10
0.15
Placebo
End Point
Den
sity
β = −0.5β = 0β = 0.5
−10 −5 0 5 10 15
0.00
0.05
0.10
0.15
Anamorelin
End Point
Den
sity
Change in E(Z ) about 1.5 lb at βT = 0.5 and βT = −0.5Set βT = {−0.5,−0.4, . . . ,0, . . . ,0.5}
[Proposal] Case study 47/53
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Sensitivity analysis: Rank
−0.4 −0.2 0.0 0.2 0.4
−0.
10.
00.
10.
20.
30.
4
Rank
Placebo
Test
Sta
tistic
Anamorelin: −0.5
Anamorelin: 0
Anamorelin: 0.5
[Proposal] Case study 48/53
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Sensitivity analysis: Median
−0.4 −0.2 0.0 0.2 0.4
−1.
0−
0.5
0.0
0.5
Median
β
Med
ian
PlaceboAnamorelin
[Proposal] Case study 49/53
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Sensitivity analysis: Contour of p-values
0.000
0.005
0.010
0.015
0.020
−0.5 −0.3 −0.1 0.0 0.1 0.2 0.3 0.4 0.5
−0.5
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
1e−04
0.00
1
0.0
1 Placebo
Ana
mor
elin
[Proposal] Case study 50/53
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Conclusion
There is a significant difference between the Placebo and theAnamorelin arms in their composite endpoints of survival andaverage LBM change. The difference favors the Anamorelinarm
[Proposal] Case study 51/53
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Outline
Goal
Ranking
Treatment effect
Benchmark missing data assumptions
Sensitivity analysis
Bivariate case (example)
Imputation
Case study
Summary
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Summary
Propose a composite endpoint approach for evaluatingtreatment effects in randomized clinical trials with deathand missingnessApply complete case missing-variable restrictions (CCMV)for handling missing data in survivorsApply exponential tilting model for sensitivity analysisIntroduce a parsimonious way of introducing sensitivityparameters
[Proposal] Summary 52/53
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PART III: SOFTWARE
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Web application
Currently available athttp://sow.familyds.com/shiny/composite/
Major componentsupload study datagraphical data presentationgenerate imputed datasetbootstrap analysis
[Software] 53/53
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Recapitulation
Issues and common analysis methods for death-truncateddataProposal: a composite endpoint based approachWeb-application
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THE END