Estimating Individual-Level and Population-LevelCausal Effects of Organ Transplantation
Treatment Regimes
David M. Vock, Jeffrey A. Verdoliva Boatman
Division of BiostatisticsUniversity of Minnesota
September 14, 2017
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Deceased Donor Organ Offers in the United States
I More patients in need organ transplant than organs available→ In 2015, 122,071 waiting at year end, 30,975 transplantsperformed, and 15,068 donors recovered
I United Network for Organ Sharing (UNOS) responsible fororderly offering deceased donor organs
I Order in which deceased donor organs offered to recipients adeterministic function (urgency, benefit, geography, organquality, fairness)
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Donor Lung Offers in the United States
Lung Allocation in the United States. Left figure from Colvin-Adams(2012)
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Organ Acceptance
I Each time an organ is offered to a recipient it may be turneddown
I Many competing factors go into organ acceptance
1. Patient2. Surgeon3. Transplant Center4. Organ Procurement Organization5. Other Patients on Waiting List
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Treatment Regimes for Organ Transplantation
Patients awaiting organ transplantation face a difficult decision ifoffered a low-quality organ:
I Accept the organ
I Decline the organ, hope to be offered a higher quality one inthe future
Testing the effect of personalized decision rules (dynamictransplant regimes) would be helpful here.
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Transplant Regimes
Treatment regimes of interest: Rules which dictate when anoffered organ should be declinedTreatment regimes for transplantation tend to be “poorly-defined”:
I There is not a 1-to-1 correspondence between patientcovariate history and treatment
I Transplants are not available at any time
I We cannot easily specify treatment regimes such as “Get atransplant as soon as LAS > 50” because these regimes arenot clinically meaningful
I Avoid regimes which dictate if an organ should be accepted
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Time-varying Treatment and Time-varying Confounding
I No randomized trials of different transplant regimes
I Necessitates the use observational datasets and appropriatemethods
I Inverse probability of compliance estimators →
1. Patient’s follow-up time is considered only while she is“compliant” with a regime of interest
2. Once non-compliant, her follow-up time is artificially censored
3. Observations are weighted according to the inverse ofprobability of compliance to correct for the potential selectionbias introduced by the artificial censoring
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Limitations of Existing Methods
Problem:
I Anticipated survival for a given DTR depends on the qualityand availability of organs, and these depend on the strategiesthat other patients follow to accept or decline an organ
I Patients’ probability of transplant depends on regimes thatother patients follow
I Conceptually similar, although not identical, to the “spillover”effect described in other contexts
Existing methods have a major limitation here:
I They estimate the causal effect of a treatment regime for asingle random participant who adopts the treatment regime
I But the causal effect if all patients were to adopt thetreatment regime may have more public health relevance
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Goal and Outline
Develop weighted causal estimators for both cases:
I “One follows”
I “All follow”
Develop estimators:
I Assuming we have data from an observational registry
I Avoid modeling the many random processes: patient arrival,organ arrival, etc...
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Potential Outcomes and Target of Interest
I T ∗(∞): survival time from listing if the patient were to neverreceive a transplanted organ
I T ∗(b,q): survival time if the patient were to receive an organb days after listing with organ characteristics q
I X ∗(b) to be the covariates collected b days after listing
I Inferring the distribution of T ∗(b, q) is not of primary interest
I T ∗(g , g ′): survival time if she followed regime g for decliningoffered organs and all other patients follow regime g ′
I Inferring the distribution of T ∗(g , g ′) IS of primary interest
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Potential Outcomes and Target of Interest
I Density of T ∗(g , g ′) (fT∗(g ,g ′)(t)) is a mixture distribution ofwell-defined counterfactual survival times
∑x(t)
fT∗(∞)|X∗(t) {t|x(t)}
t∏s=1
1−∑q
ρ
(g,g′
){s, q|x(s)}
fX∗(t) {x(t)} (1)
+t∑
b=1
∑q
∑x(b)
fT∗(b,q)|X∗(b) {t|x(b)}
b−1∏s=1
1−∑q
ρ
(g,g′
){s, q|x(s)}
ρ(g,g′) {b, q|x(b)} fX∗(b) {x(b)} ,
I ρ(g ,g′) {b,q|x(b)}: probability of receiving a transplant b days
after listing with organ characteristics q given she isuntransplanted b − 1 days after listing with covariate historyx(b) and the patient follows regime g while all others followregime g ′
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Transplant Regime Spillover
I “Spillover” typically refers to situations in which thedistribution of well-defined potential outcomes depends on thetreatment assignment of others
I Here the probability of initiating treatment depends on thetreatment regime other patients follow
I Refer to this as transplant regime spillover
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Notation
Notation:
I Li : listing time for ith patient
I Nij ,Yij : failure and at-risk indicators for ith patient on jthstudy day
I Xij : patient characteristics including confounders
I Aijk : indicator for accepting kth organ on the jth study day
I Aij : treatment history through the jth day
I Oijk : indicator for offered kth organ on jth study
I E ·jk to be the collection of information on all subjects prior toassigning the kth organ on the jth study day day
Causal estimand: λt(g , g′): hazard of death t days after entering
the waiting list, and the associated survival St(g , g′)
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Components of the weights
π(∅,∅)ij
(Aij ,E ·jSj
)=∏j
j ′=1
∏Sj′k=1 π
(∅,∅)ijk
(Aij ′k ,E·j ′k
): The probability
of the observed treatment history assuming all patients followregime ∅ (no changes in propensity to accept or decline organs)
π(g ,g ′)ij
(Aij ,E ·jSj
)=∏j
j ′=1
∏Sj′k=1 π
(g ,g ′)ijk
(Aij ′k ,E·j ′k
):The
probability of the observed treatment history in the counterfactualworld where ith patient follows regime g and all others follow g ′
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Estimating λt(g , g′)
We can estimate λt(g , g′) by solving the estimating equation
m∑j=1
n∑i=1
π(g ,g ′)ij
(Aij ,E ·jSj
)π(∅,∅)ij
(Aij ,E ·jSj
) {Nij − Yijλt(g , g′)}I (j − Li = t) = 0
Intuition for the weightsπ(g,g′)ij
(Aij ,E ·jSj
)π(∅,∅)ij
(Aij ,E ·jSj
) :
I Similar to IPCW
I Standardize to the target population where probability of Aij
is different
I Mean zero estimating function → estimator for λt(g , g′) is
CAN
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Estimating π(∅,∅)ij
(Aij ,E ·jSj
)with the Observed Data
π(∅,∅)ij
(Aij ,E ·jSj
)is a function of the probability of being offered,
and the probability of accepting if offered:
I Pr(Aij = 1) = Pr(Oij = 1) · Pr(Aij = 1|Oij = 1)
I Pr(Aij = 0) = 1− Pr(Oij = 1) · Pr(Aij = 1|Oij = 1)
Estimating Pr(O = 1) is straightforward provided I have a modelfor Pr(A = 1|O = 1)
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Estimating π(∅,∅)ij
(Aij ,E ·jSj
)with the Observed Data
ID LAS A Pr(A = 1|O = 1) Pr(O = 1) Pr(A)
763 52.54 0 0.563 1.000 0.437197 43.33 1 0.339 0.437 0.148553 41.60 0 0.302 0.289 0.91366 33.38 0 0.160 0.202 0.968
592 33.31 0 0.158 0.169 0.973
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Estimating π(g ,g ′)ij
(Aij ,E ·jSj
)with the Observed Data
This is more challenging.
The numerator is the expectation of the probability of beingoffered and accepting organs in the hypothetical world where allpatients follow regime g ′ given the observed data.
π(g ,g ′)ijk (1,E·jk) = E (π
(g ,g ′)ijk (1,E
(g ,g ′)·jk )|E·jk) If all patients follow
regime g ′:
I We don’t know which patients would be on the waiting listI We don’t know what their characteristics would beI The probability of being offered an organ can’t be directly
computed from the observed data
I Sample E(g ,g ′)·jk from E·jk – use Monte Carlo integration to
evaluare expectation18 / 33
Estimating π(g ,g ′)ij
(Aij ,E ·jSj
)
ID LAS A Pr(A = 1|O = 1) Pr(O = 1) Pr(A)
763 52.54 0 0.563 1.000 0.437761 ? ? ? 0.437 ?197 43.33 1 0.339 ? ?491 ? ? ? ? ?553 41.60 0 0.302 ? ?66 33.38 0 0.160 ? ?
592 33.31 0 0.158 ? ?
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Estimating π(g ,g ′)ij
(Aij ,E ·jSj
)
Hot Deck Imputation:
I For all transplant recipients, impute missing data assumingnever transplanted by borrowing data from their nearestneighbor, the “lender”
I Simulate the organ allocation when everyone follows g ′,assuming model for accepting organs holds
I Assume low quality organs are accepted with probability 0
We’ve created a hypothetical data set where all patients follow g ′.
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After Imputation
Assume the organ is defined as high quality under g ′
ID LAS A Pr(A = 1|O = 1) Pr(O = 1) Pr(A)
763 52.54 0 0.563 1.000 0.437761 49.48 0 0.367 0.437 0.840197 43.33 1 0.339 0.277 0.094491 42.55 0 0.237 0.183 0.957553 41.60 0 0.302 0.140 0.95866 33.38 0 0.160 0.097 0.984
592 33.31 0 0.158 0.082 0.987
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After Assigning Organ Under g , g ′
Assume the organ is defined as high quality under g ′
ID LAS A A(g ,g ′) Pr(A = 1|O = 1) Pr(O = 1) Pr(A)
763 52.54 0 0 0.563 1.000 0.437761 49.48 0 0.367 0.437 0.840197 43.33 1 1 0.339 0.277 0.094491 42.55 0 0.237 0.183 0.957553 41.60 0 0 0.302 0.140 0.95866 33.38 0 0 0.160 0.097 0.984
592 33.31 0 0 0.158 0.082 0.987
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Simulation Study
Data Generation:
I Patient and organ arrivals: independent Poisson processeswith rate parameters 0.5 and 0.32, respectively
I bi0 ∼ N (−1, 1), bi1 ∼ N(
1365 ,
1(4·365)2
)I Xij = bi0 + bi1 · b j−Li30 c · 30, where b·c is the floor function, Li
is arrival time
I Organs were low-quality with probability 0.5
I Organs were accepted with probability{
1 + e2.5−0.25Xij}−1
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Simulation Study
Estimation:
I Estimated survival for regime g : decline all low- quality organs
I Estimated survival assuming “one follows” and “all follow”
I Hot deck imputation: lender i ′ for patient i selected asarg min
i ′
(|Xij − Xi ′j ′ | : j − Li = j ′ − Li ′
)I Nonparametric bootstrap to estimate standard errors (SEs)
and form 95% confidence intervals (CI)
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Simulation Results
Bias CP
Target t Truth St(g , ∅) St(g , g) St(g , ∅) St(g , g)
St(g , ∅) 180 0.785 -0.006 -0.018 0.954 0.735360 0.636 -0.000 -0.036 0.966 0.519540 0.533 0.003 -0.056 0.954 0.269720 0.463 0.004 -0.073 0.951 0.152
St(g , g) 180 0.770 0.009 -0.003 0.882 0.967360 0.599 0.037 0.001 0.333 0.978540 0.471 0.065 0.006 0.045 0.972720 0.382 0.084 0.007 0.014 0.960
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Application
Data are from an observational registry maintained by the UnitedNetwork for Organ Sharing
I May 4, 2005-Sept 30, 2011
I 9,091 transplants
I 13,039 patients total
Treatment regimes:
I Decline organs below pth percentile of donor quality whileLAS < M; if LAS ≥ M, any organ is acceptable
I p and M can both vary
I Estimated anticipated survival assuming “one follows” theregime, or “all follow” the regime.
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Application
Donor quality:
I Donor quality modeled with Cox regression
I For each transplant, donor quality was estimated for eachpotential recipient
Model for accepting organs:
I Logistic regression model
I Predictors: patient age, LAS, time on waiting list, nativedisease, patient-donor-height difference; and indicators fordonor smoking ≥ 20 pack-years, donor age > 50, and theirinteraction
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Application
Hot deck imputation for “all follow”:
I Lender i ′ for patient i selected asarg min
i ′
(|LASij − LASi ′j ′ | : j − Li = j ′ − Li ′
)I LAS was only imputed variable
Nonparametric bootstrap to estimate SEs and form 95% CIs
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Application Results
0 1 2 3
0.0
0.1
0.2
0.3
0.4
Decline All Organs Until LAS Exceeds 40
Years From Entering Waitlist
Cum
ulat
ive
Pro
babi
lity
of D
eath
Kaplan−Meier SurvivalOne Participant Follows RegimeAll Participants Follow Regime
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Application Results
0 1 2 3
0.0
0.1
0.2
0.3
0.4
All Follow Regime 'Decline All Organs Until LAS Exceeds Cutoff'
Years From Entering Waitlist
Cum
ulat
ive
Pro
babi
lity
of D
eath
Kaplan−Meier SurvivalLAS Cutoff = 35LAS Cutoff = 40LAS Cutoff = 45LAS Cutoff = 50
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Application Results
0 1 2 3
0.0
0.1
0.2
0.3
0.4
One Follows Regime 'Decline Worst p% of Organs Until LAS Exceeds 50'
Years From Entering Waitlist
Cum
ulat
ive
Pro
babi
lity
of D
eath
Kaplan−Meier Survivalp = 25p = 50p = 75p = 100
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Summary
Simulation shows:
I Estimators have little bias for their target
I Causal estimand must be specified with care
Application
I “One follows” and “all follows” are different
I Patients may gain a modest increase in survival probability bydeclining transplantation while LAS is low
I Effect may be greater if the entire population adopts thestrategy
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Summary
Thank you!
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