Generalizing inferences about failure-time outcomes from randomized individuals to a target population Sarah Robertson Brown University sarah robertson@brown.edu June 2, 2019 1 / 34
Acknowledgments PCORI awards ME-1306-03758 and ME-1502-27794, NIH grant R37 AI102634, AHRQ T32AGHS00001 Joint work with JA Steingrimsson, MA Hern´ an, IJ Dahabreh 2 / 34
Problem- Clinical trials may have limited applicability Even if trial is run perfectly, trial results often do not apply/extend to a target population Target population = trial-eligible individuals Want estimates of treatment effects in the target population, not just the trial Need methods to extend trial results by combining randomized and non-randomized data 3 / 34
Using a nested study design to extend trial results Nested study design : Trial embedded in a cohort of trial-eligible individuals (e.g., health-care system) ∗ ∗ Image from Choudhry, 2017 NEJM 4 / 34
Nested study design Cohort of eligible patients Trial nested within cohort X S A Y X 1 y 1 Yes Hypothetical 0 y Sample Target Participate Population in trial? No 0 S=indicator for trial participation; A=treatment indicator; Y=outcome; X=baseline covariates 5 / 34
Applied example of a nested study design Coronary Artery Surgery Study (CASS) August 1975 to December 1996 Study Design Outcome All-cause mortality Met trial- eligibility criteria (N=2099) Median follow-up time was 14 years Randomized individuals Non-randomized individuals (N=780) (N=1310) Some outcomes censored in last intervals of the study Surgery Medical (N=390) Therapy (N=390) 6 / 34
CASS trial-only analysis: survival curves (Kaplan-Meier) Trial-only Survival probability 1.0 Surgery Medical 0.9 0.8 0.7 0.6 0 4 8 12 16 Time (years) IPW 7 / 34
CASS trial-only analysis: survival probabilities 16-year survival probability, % Estimator Surgery Medical Trial-only 66.3 (61.5, 71.1) 65.8 (61.5, 71.1) Counterfactual survival analysis : survival status, Y a , c =0 , at time t interval t , under treatment a and no censoring c = 0 Want to compare counterfactual survival curves in the target population with followup over time intervals j = 1 , ..., t (e..g, follow-up in yearly intervals) Use methods that adjust for informative censoring/drop-out 8 / 34
Observed and missing data in the clinical trial Y j : indicator for having the event by interval j C j : indicator for being censored by interval j Consider one trial participant (id 1), followed for j=1,2 years id S j X A Y j C j Y 0,c=0 Y 1,c=0 0 1 0 1 1 1 1 2 Goal : Use observed data (and assumptions) to identify counterfactual Y 0 , c =0 and Y 1 , c =0 Assume trial no treatment switching, no measurement error Drop out occurs in trial 9 / 34
Identifying the causal effect in the trial S j A Y j C j Y 0,c=0 id X Y 1,c=0 1 0 0 x 1 1 1 1 2 Consistency Observed outcome=counterfactual outcome under the assigned treatment and no censoring in an interval 10 / 34
Identifying the causal effect in the trial S j A Y j C j Y 0,c=0 id X Y 1,c=0 1 0 0 x 1 1 1 1 2 Consistency Observed outcome=counterfactual outcome under the assigned treatment and no censoring in an interval Positivity of treatment assignment Nonzero probability of being randomized to either treatment 11 / 34
Identifying the causal effect in the trial S j A Y j C j Y 0,c=0 id X Y 1,c=0 1 0 0 x 1 1 1 1 2 Consistency Observed outcome=counterfactual outcome under the assigned treatment and no censoring in an interval Positivity of treatment assignment Nonzero probability of being randomized to either treatment Positivity of being observed during follow-up Nonzero probability of being observed 12 / 34
Identifying the causal effect in the trial S j A Y j C j Y 0,c=0 id X Y 1,c=0 1 0 0 x 1 1 1 1 2 Consistency Observed outcome=counterfactual outcome under the assigned treatment and no censoring in an interval Positivity of treatment assignment Nonzero probability of being randomized to either treatment Positivity of being observed during follow-up Nonzero probability of being observed Non-informative censoring Counterfactual outcome is independent of censoring status given covariates 13 / 34
Identifying the causal effect in the trial S j Y j C j Y 0,c=0 id X A Y 1,c=0 1 0 0 x 1 1 1 1 2 Consistency Observed outcome=counterfactual outcome under the assigned treatment and no censoring in an interval Positivity of treatment assignment Nonzero probability of being randomized to either treatment Positivity of being observed during follow-up Nonzero probability of being observed Non-informative censoring Counterfactual outcome is independent of censoring status given covariates Exchangeability over treatment Counterfactual outcome is independent of treatment given covariates 14 / 34
Additional assumptions for generalizing trial results id S j X A Y j C j Y 0,c=0 Y 1,c=0 1 0 0 x 1 1 1 1 2 Positivity of trial participation Nonzero probability of participating 15 / 34
Additional assumptions for generalizing trial results id S j X A Y j C j Y 0,c=0 Y 1,c=0 1 0 0 x 1 1 1 1 2 1 0 x 2 2 Positivity of trial participation Nonzero probability of participating Exchangeability over trial participation Know enough factors that determine the outcome so that trial participation itself is unimportant 16 / 34
Methods for extrapolating trial results Missing data problem in the non-randomized Estimate the effect of the intervention had it been applied to the target population 3 classes of estimators ∗ : Outcome model-based estimator , g-formula computation Probability of trial participation estimator , IPW Doubly robust estimator , DR ∗ Dahabreh et al., Biometrics 2018, and https://arxiv.org/abs/1805.00550 17 / 34
Outcome model-based estimator (g-formula) Regression-based extrapolation using pooled logistic regression Model the hazard of the outcome (dying) in each interval among those remaining in the risk set, conditional on baseline covariates, in each treatment arm of the trial Represent time flexibly (e.g, squared term), include time and treatment interactions (allow non-proportionality) Predict over all trial-eligible individuals and marginalize to get estimated counterfactual survival at different intervals in the target population Consistent when outcome model is correctly specified 18 / 34
Baseline covariates in CASS Non Trial participants Variable Level participants Surgery Medical N 955 368 363 Age, years 50.9 (7.7) 51.4 (7.2) 50.9 (7.4) Angina None 195 (20.4) 83 (22.6) 81 (22.3) Present 760 (79.6) 285 (77.4) 282 (77.7) History of MI No 406 (42.5) 159 (43.2) 135 (37.2) Yes 549 (57.5) 209 (56.8) 228 (62.8) LAD % obstruction 39.1 (38.7) 36.4 (38.0) 34.9 (37.0) Left ventricular score 7.1 (2.7) 7.4 (2.9) 7.3 (2.8) Diseased vessels 0 347 (36.3) 146 (39.7) 133 (36.6) ≥ 1 608 (63.7) 222 (60.3) 230 (63.4) Ejection fraction, % 60.2 (12.3) 60.9 (13.1) 59.8 (12.8) Results presented as mean (standard deviation) for continuous variables and count (%) for discrete variables. CASS = Coronary Artery Surgery Study; LAD = left anterior descending coronary artery; MI = myocardial infarction. 19 / 34
CASS re-analysis: survival curves (OM) Survival curves in the trial vs survival curves in the target population Trial-only OM Survival probability Survival probability 1.0 Surgery 1.0 Medical 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0 4 8 12 16 0 4 8 12 16 Time (years) Time (years) IPW AIPW 20 / 34
CASS re-analysis: survival probabilities (OM) 16-year survival probability, % Estimator Surgery Medical Trial-only 66.3 (61.5, 71.1) 65.8 (61.5, 71.1) OM 63.9 (58.6, 69.0) 64.0 (58.9, 69.0) 21 / 34
Inverse participation weighting estimator (IPW) Weighted Kaplan-Meier estimator Weights depend on correctly specifying a model for the probability of participation and probability of being censored in each interval Estimate participation model conditional on baseline covariates at baseline Estimate censoring model among trial participants conditional on remaining in the risk set and baseline covariates in each treatment arm 22 / 34
CASS re-analysis: survival curves (OM, IPW) Trial-only OM Survival probability Survival probability 1.0 Surgery 1.0 Medical 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0 4 8 12 16 0 4 8 12 16 Time (years) Time (years) IPW AIPW Survival probability Survival probability 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0 4 8 12 16 0 4 8 12 16 Time (years) Time (years) 23 / 34
CASS re-analysis: survival probabilities (OM, IPW) 16-year survival probability, % Estimator Surgery Medical Trial-only 66.3 (61.5, 71.1) 65.8 (61.5, 71.1) OM 63.9 (58.6, 69.0) 64.0 (58.9, 69.0) IPW 63.5 (58.3, 68.8) 63.6 (58.5, 68.8) 24 / 34
Doubly robust/ Augmented inverse probability weighting estimator (AIPW) Combines working models from the outcome-based estimator and IPW for two opportunities for valid inference At least as efficient as IPW, in large samples, when all models are correctly specified Necessary for machine learning 25 / 34
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