Min Lu dissertation defense slides

https://scholarlyrepository.miami.edu/oa_dissertations/2102/ 令 σ | 晶 5ecure I https://scholarlyreposíto叩.míamí.edu/oa_dis自由tí0l1s/21 02/ l J LIBRARIES , MIA~1l lìbra"y.m ami .edu 〈 生旦型坚 旦旦旦旦 >ETDS> QA D1 SSERT.町1 0NS > 旦旦 OPEN ACCESS DISSERTATIONS ]1 Search r Advanced 5 earch Notify me via email or .R.S.S ...而'军3………………………· FAQ Disclaimer Collectíons Disci plines Au thors ·监!lìi画画画画画 Estimating Individual Treat ment E仔ect in Observationa l Data Using Random Forest Methods 旦坠且主，.JlD.些笠MU芷旦旦旦 5 ubmit Research E圆 -o!.圆nm Available ior dow时oad on Thursday, November 14, publication Date 2018-05-23 2019 Avai lability Embargoed SHARE 回 E 田 。-I3 D Enlbargo Period 2019-11-14 Degree Type Dissertation Degree Name Doctor 01 Philosophy (PHD) Departm凹，t Public Health Author Help 旦旦! > 5口 ences (~1edici ne) Dat e of Defense 2018-05-03 First Committee Meluber Hen、ant Ish、;\' aran Second Committee J . 5unil Rao M 凹nber Third CO ll1 mittee Member Daniel Feaster Fourth Committee Member Wei 5 un Abstr act Estimation 01 individual tre.tment e忏ect in observational data is complicated due t。由e challeno目。1 confoundino and selection bias_ A usel ul lnferentlal Iramework to address this is the counterfactual model which takes the hypothetical stance 01 asking what if an individual had received both treatments. 问akino use of random forests ( RF) within the counterfactual Iramew。巾， I estlmate Indlvldual tre.tment e何ects by directly modelino the response. This thesis consists 01 five Chapters. Chapter 1 reviews 白e methodology in causal inferenζe and provide mathematical notations_ Major approaches reviewed include potential outcome approach , oraphical approach and counterfactual approach. Chapter 2 discusses assumptìons for counterfactual approach. P-values are useful in causal inlerence, but whenever it is used, caution must be taken. 5ection 2.3 and 5ection 2.4 propose machine leamino methods as alternatives to p-values and checkino proportional hazards assumption In survlval analysls. These two sectJons are more general in content even beyond the scope 01 counterfactual approach. Chapter 3 describes six random 10re5t methods for estimating individual treatment effects under counterfactual approach I ramework and discusses model conslstency and converoence 01 random lorest in Sectlon 3.6. Chapter 4 demonstrates the performance 01 these methods in ∞mplex simulations and how the most appropriate method is used in a real dataset lor ζontinuous outcome. Chapter 5 addresses causal inlerence in survival anal ysis of ischem ic 日 rdiomγopathγ. Treatm ent effect is viewed as a dynamic causal pro臼du re. New random forest m ethods are proposed in th is chapter to assess individual therapy ove.rlalP. These m ethods possess the uni que f eature of being able to incorporate external expert knowledge either in a fully supe阿ised way ( i.e., we have a strong belief that knowledge is correct) , or in a minimally-supervised fashion (i. e., knowledge is not considered goldstandard) . Keywords Causal l nference; Random Forests; Machine Learning; Survi val; l ndividual Treatment Effect; Observational Data Recommended Citat ion Lu , f'1i n， 仿Estimating lndividual Treatm ent Effect in Observational Data Us[ng Random Forest r-1 ethods" (20 18). Opefl Acc白5 Disset臼tions . 2102. https: llscho larl yreposlt。町.miaml .ed uJ oð_dissertatio nsJ2 1 02 | 出血|约旦[IIj国|旦U皿皿 E虫~田血h l 缸里旦挝且S监皿且 丁 DIGITALCOM… •l ζ一 lW'\'f1ië'm申明 • Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Estimating Individual Treatment Effect in Observational Data Using Random Forest Methods Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Min Lu Dissertation Defense on May 3, 2018 Division of Biostatistics University of Miami Other Papers 1 / 58 Overview Random Forest Individual Causal Inference Min Lu 1 2 Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application 3 Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference 4 5 Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers Other Papers 2 / 58 Introduction Random Forest Individual Causal Inference Min Lu • Comparison of potential outcomes under alternative treatments • We only observe half of the potential outcomes • A non-experimental study is subject to selection bias/confounding Introduction Continuous and Binary Outcome Units Observed Potential Outcomes Treatment Effects Treatment Control Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion 1 .. . Y1 .. . Y1 (1) .. . Y1 (0) .. . Y1 (1) − Y1 (0) .. . i .. . Yi .. . Yi (1) .. . Yi (0) .. . Yi (1) − Yi (0) .. . YN (1) YN (0) YN (1) − YN (0) Ȳ(1) Ȳ(0) Ȳ(1) − Ȳ(0) YN N P T n1 Yi − P C n0 Yi Reference Other Papers 3 / 58 Introduction Random Forest Individual Causal Inference Min Lu Units Observed Potential Outcomes Treatment Effects Treatment Control Outcome Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application 1 .. . Y1 .. . Y1 (1) .. . Y1 (0) .. . Y1 (1) − Y1 (0) .. . i .. . Yi .. . Yi (1) .. . Yi (0) .. . Yi (1) − Yi (0) .. . YN (1) YN (0) YN (1) − YN (0) Ȳ(1) Ȳ(0) Ȳ(1) − Ȳ(0) Discussion Survival Outcome Definitions & notations YN N P T n1 Yi − P C n0 Yi Treatment overlap oj (x) Treatment effect estimation Results Discussion Propensity score approaches (Rosenbaum and Rubin, 1983) can get an average treatment effect Reference Other Papers 4 / 58 Introduction Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Units Potential Outcomes 1 .. . Y1 (1) .. . Y1 (0) .. . Y1 (1) − Y1 (0) .. . i .. . Yi (1) .. . Yi (0) .. . Yi (1) − Yi (0) .. . N YN (1) YN (0) YN (1) − YN (0) Ȳ(1) Ȳ(0) Ȳ(1) − Ȳ(0) Treatment overlap oj (x) Treatment effect estimation Treatment Effects Treatment Control We estimate individual treatment effect (ITE) Results Discussion Reference Other Papers 5 / 58 Definitions and notations Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation • Let {(X1 , T1 , Y1 ), . . . , (Xn , Tn , Yn )} denote the data where Xi is the covariate vector for individual i • Let Ti = 0 represent the control group, and Ti = 1 the intervention group • Let Yi (0) and Yi (1) denote the potential outcome for i under the two treatments • Given Xi = x, the individual treatment effect (ITE) for i is defined as τ (x) = E [Yi (1)|Xi = x] − E [Yi (0)|Xi = x] The average treatment effect (ATE) is defined as Results Discussion Reference τ0 = E [Yi (1)] − E [Yi (0)] = E [τ (X)] Other Papers 6 / 58 ITE estimation Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome As to ITE, τ (x) = E [Yi (1)|Xi = x] − E [Yi (0)|Xi = x], the basis for our counterfactual approaches rests on the assumption of strongly ignorable treatment assignment (SITA). This assumes that P(T = 1|x) > 0 for all x and treatment assignment is conditionally independent of the potential outcomes given the variables; i.e., T ⊥ {Y(0), Y(1)} | X . Under the assumption of SITA, we have Definitions and notations Method Simulation Real Data Application τ (x) = E [Y|T = 1, X = x] − E [Y|T = 0, X = x] Assuming that the outcome Y satisfies Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Y = f (X, T) + ε where E(ε) = 0 and f is the unknown regression function, and assuming that SITA holds, we have τ (x) = f (x, 1) − f (x, 0) Discussion Reference Other Papers 7 / 58 ITE estimation Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome As to ITE, τ (x) = E [Yi (1)|Xi = x] − E [Yi (0)|Xi = x] , the basis for our counterfactual approaches rests on the assumption of strongly ignorable treatment assignment (SITA). This assumes that P(T = 1|x) > 0 for all x and treatment assignment is conditionally independent of the potential outcomes given the variables; i.e., T ⊥ {Y(0), Y(1)} | X. Under the assumption of SITA, we have Definitions and notations Method Simulation Real Data Application τ (x) = E [Y|T = 1, X = x] − E [Y|T = 0, X = x] Assuming that the outcome Y satisfies Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Y = f (X, T) + ε where E(ε) = 0 and f is the unknown regression function, and assuming that SITA holds, we have τ (x) = f (x, 1) − f (x, 0) Discussion Reference Other Papers 8 / 58 ITE estimation Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome As to ITE, τ (x) = E [Yi (1)|Xi = x] − E [Yi (0)|Xi = x], the basis for our counterfactual approaches rests on the assumption of strongly ignorable treatment assignment (SITA). This assumes that P(T = 1|x) > 0 for all x and treatment assignment is conditionally independent of the potential outcomes given the variables; i.e., T ⊥ {Y(0), Y(1)} | X. Under the assumption of SITA, we have Definitions and notations Method Simulation Real Data Application τ (x) = E [Y|T = 1, X = x] − E [Y|T = 0, X = x] Assuming that the outcome Y satisfies Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Y = f (X, T) + ε where E(ε) = 0 and f is the unknown regression function, and assuming that SITA holds, we have τ (x) = f (x, 1) − f (x, 0) Discussion Reference Other Papers 9 / 58 Method Random Forest Individual Causal Inference Min Lu The methods for estimating the ITE are as follows: Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) 1. Virtual twins (VT) 2. Virtual twins interaction (VT-I) 3. Counterfactual RF (CF) 4. Counterfactual synthetic RF (synCF) 5. Bivariate RF (bivariate) 6. Honest RF (honest RF) 7. Bayesian Additive Regression Trees (BART) Treatment effect estimation Results Discussion Reference Other Papers 10 / 58 Virtual twins (VT) Random Forest Individual Causal Inference Min Lu Foster et al. (2011) proposed Virtual Twins (VT) for estimating counterfactual outcomes. In this approach, RF is used to regress Yi against (Xi , Ti ). Given an individual i with Ti = 1, to obtain i’s counterfactual estimate, one runs the altered (Xi , 1 − Ti ) = (Xi , 0) down the forest to obtain the counterfactual estimate Ŷi (0), denoted as ŶVT (x, 0). The VT counterfactual estimate for τ (x) is τ̂VT (x) = ŶVT (x, 1) − ŶVT (x, 0) Introduction Continuous and Binary Outcome Units Potential Outcomes Treatment Effects Treatment Control Definitions and notations Method 1 Yb1 (1) Yb1 (0) Yb1 (1) − Yb1 (0) . . . . . . . . . . . . Survival Outcome i Ybi (1) Ybi (0) Ybi (1) − Ybi (0) Definitions & notations . . . . . . . . . . . . N b YN (1) b YN (0) b YN (1) − b YN (0) Simulation Real Data Application Discussion Treatment overlap oj (x) Treatment effect estimation Training data Units Y T 1 . . . . . . . . . X1 . . . XP . . . N R ANDOM F OREST Results Discussion Reference Other Papers 11 / 58 Virtual twins interaction (VT-I) Random Forest Individual Causal Inference Min Lu Introduction As noted in Foster et al. (2011), the VT approach can be improved by manually including treatment interactions in the design matrix. Thus, one runs a RF regression with Yi regressed against (Xi , Ti , Xi Ti ) Units Continuous and Binary Outcome Definitions and notations Potential Outcomes Treatment Effects Treatment Control 1 Yb1 (1) Yb1 (0) Yb1 (1) − Yb1 (0) . . . . . . . . . . . . Discussion i Ybi (1) Ybi (0) Ybi (1) − Ybi (0) Survival Outcome . . . . . . . . . . . . N b YN (1) b YN (0) Method Simulation Real Data Application Training data Units Y T 1 . . . . . . . . . X1 . . . XP TX10 . . . TXP0 . . . . . . N R ANDOM F OREST Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 12 / 58 Out-of-bag (OOB) estimates Random Forest Individual Causal Inference Min Lu Each tree in a forest is constructed from a bootstrap sample which uses approximately 63% of the data. The remaining 37% of the data is called OOB and are used to calculate an OOB predicted value for a case Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 13 / 58 Out-of-bag (OOB) estimates Random Forest Individual Causal Inference Min Lu For example, if 1000 trees are grown, approximately 370 will be used in calculating the OOB estimate for the case. The inbag predicted value, on the other hand, uses all 1000 trees Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 14 / 58 Out-of-bag (OOB) estimates Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome OOB estimates are generally much more accurate than insample (inbag) estimates (Breiman, 1996). To ∗ illustrate how OOB estimation applies to VT, suppose that case x is assigned treatment T = 1. Let ŶVT (x, T) denote the OOB predicted value for (x, T). The OOB counterfactual estimate for τ (x) is ∗ τ̂VT (x) = ŶVT (x, 1) − ŶVT (x, 0) Note that ŶVT (x, 0) is not OOB. This is because (x, 0) is a new data point and technically speaking cannot have an OOB predicted value as the observation is not even in the training data Units Definitions and notations Potential Outcomes Treatment Effects Treatment Control Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results 1 Yb1 (1) Yb1 (0) Yb1 (1) − Yb1 (0) . . . . . . . . . . . . i Ybi (1) Ybi (0) Ybi (1) − Ybi (0) . . . . . . . . . . . . N b YN (1) b YN (0) Training data Units Y T 1 . . . . . . . . . b YN (1) − b YN (0) X1 . . . XP . . . N Use all data for training RF R ANDOM F OREST Only use OOB for prediction Discussion Reference Other Papers 15 / 58 Counterfactual random forests (CF) Random Forest Individual Causal Inference Min Lu Forests CF1 and CF0 are fit separately to data {(Xi , Yi ) : Ti = 1} and {(Xi , Yi ) : Ti = 0}, respectively. To obtain a counterfactual ITE estimate, each data point is run down its natural forest, as well as its counterfactual forest. If ŶCF,j (x, T) denotes the predicted value for (x, T) from CFj , for j = 0, 1, and ∗ ŶCF,1 (x, 1) denotes the OOB predicted value for (x, 1), the counterfactual OOB ITE estimate is ∗ τ̂CF (x) = ŶCF,1 (x, 1) − ŶCF,0 (x, 0) Introduction Continuous and Binary Outcome Units Definitions and notations Potential Outcomes Treatment Effects Training data: T=1 X1 . . . XP . . . Units Y T . . 1 . . . . 1 Treatment Control Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation 1 Yb1 (1) Yb1 (0) Yb1 (1) − Yb1 (0) . . . . . . . . . . . . i Ybi (1) Ybi (0) Ybi (1) − Ybi (0) . . . . . . . . . . . . N b YN (1) b YN (0) b YN (1) − b YN (0) R ANDOM F OREST (1) Training data: T=0 X1 . . . XP . . . Units Y T . . 0 . . . . 0 Results Discussion Reference R ANDOM F OREST (0) Other Papers 16 / 58 Counterfactual synthetic random forests (synCF) Random Forest Individual Causal Inference Min Lu Using a collection of Breiman forests (called base learners) grown under different tuning parameters (mtry and nodesize), each generating a predicted value called a synthetic feature, a synthetic forest (Ishwaran and Malley, 2014) is defined as a secondary forest calculated using the new input synthetic features, along with all the original features. We denote its ITE estimate by τ̂synCF (x): τ̂synCF (x) = ŶsynCF,1 (x, 1) − ŶsynCF,0 (x, 0) Introduction Continuous and Binary Outcome where ŶsynCF,j (x, T) is the predicted value for (x, T). As before, OOB estimation is used whenever possible Units Definitions and notations Potential Outcomes Treatment Effects Training data: T=1 X1 . . . XP Ŷ 1 . . .Ŷ M . . . . . . Units Y T . . 1 . . . . 1 Treatment Control Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation 1 Yb1 (1) Yb1 (0) Yb1 (1) − Yb1 (0) . . . . . . . . . . . . i Ybi (1) Ybi (0) Ybi (1) − Ybi (0) . . . . . . . . . . . . N b YN (1) b YN (0) b YN (1) − b YN (0) R ANDOM F OREST (1) Synthetic features Training data: T=0 X1 . . . XP Ŷ 1 . . .Ŷ M . . . . . . Units Y T . . 0 . . . . 0 Results Discussion Reference R ANDOM F OREST (0) Other Papers 17 / 58 Bivariate imputation approach Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Causal Inference as Missing Data Problem Y(0) Y(1) X1 ... Xp ? y2 ? y4 .. . y1 ? y3 ? .. . x11 x21 x31 x41 .. . ... ... ... ... .. . x1p x2p x3p x4p .. . ? yn xn1 ... xnp Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference To impute these missing outcomes, a bivariate splitting rule is used (Tang and Ishwaran, 2015). The complete bivariate values are used to calculate the bivariate counterfactual estimate τ̂bivariate (x) = Ŷbivariate,1 (x) − Ŷbivariate,0 (x) Other Papers 18 / 58 Honest RF Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations In this method, a RF is run by regressing Yi on (Xi , Ti ), but using only a randomly selected 50% subset of the data. When fitting RF to this training data, a modified regression splitting rule is used. Rather than splitting tree nodes by maximizing the node variance, honest RF instead uses a splitting rule which maximizes the treatment difference within a node (see Procedure 1 and Remark 1 in Wager and Athey, 2015). We denote the honest RF estimate by τ̂honestRF (x) Units Potential Outcomes Treatment Effects Treatment Control Method Simulation Real Data Application Discussion 1 Y1 (1) Y1 (0) Y1 (1) − Y1 (0) Survival Outcome . . . . . . . . . . . . Definitions & notations Treatment overlap oj (x) Treatment effect estimation i Yi (1) Yi (0) Yi (1) − Yi (0) . . . . . . . . . . . . N YN (1) YN (0) YN (1) − YN (0) Results Discussion Training data (50%) Units Y T X1 . . . XP 1 . . . . . . . . . . . . N/2 H ONEST R ANDOM F OREST Use the remaining 50% for prediciton Reference Other Papers 19 / 58 Bayesian Additive Regression Trees (BART) Random Forest Individual Causal Inference Min Lu Hill (2011) proposed using BART to directly model the regression surface to estimate potential outcomes. Therefore, this is similar to VT, but where RF is replaced with BART. The BART ITE estimate is defined as τ̂BART (x) = ŶBART (x, 1) − ŶBART (x, 0) where ŶBART (x, T) denotes the predicted value for (x, T) from BART. Note that due to the highly adaptive nature of BART, no forced interactions are included in the design matrix Introduction Continuous and Binary Outcome Units Potential Outcomes Treatment Effects Treatment Control Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Training data Units Y T 1 . . . . . . . . . 1 Yb1 (1) Yb1 (0) Yb1 (1) − Yb1 (0) . . . . . . . . . . . . i Ybi (1) Ybi (0) Ybi (1) − Ybi (0) . . . . . . . . . . . . N b YN (1) b YN (0) b YN (1) − b YN (0) X1 . . . XP . . . N BAYESIAN A DDITIVE R EGRESSION T REES (BART) Results Discussion Reference Other Papers 20 / 58 Simulation study to assess different methods Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application • 20 covariates: 10 continuous variables, 10 binary variables • A logistic regression model was used to simulate T in which the linear predictor F(X) defined on the logit scale was F(X) = −2 + .028X1 − .374X2 − .03X3 + .118X4 − 0.394X11 + 0.875X12 + 0.9X13 • Three different models were used for continuous outcome Y, which was assumed to be Yi = fj (Xi , Ti ) + εi , where εi were independent N(0, σ 2 ). The mean functions fj for the three simulations were Discussion Survival Outcome f1 (X, T) = 2.455 − 1{T=0} × (.4X1 + .154X2 − .152X11 − .126X12 ) − 1{T=1, g(X)>0} f2 (X, T) = 2.455 − 1{T=0} × sin(.4X1 + .154X2 − .152X11 − .126X12 ) − 1{T=1, g(X)>0} Definitions & notations f3 (X, T) = 2.455 − 1{T=0} × sin(.4X1 + .154X2 − .152X11 − .126X12 ) − 1{T=1, h(X)>0} Treatment overlap oj (x) Treatment effect estimation 2 where g(X) = .254X22 − .152X11 − .4X11 − .126X12 and h(X) = .254X32 − .152X4 − .126X5 − .4X52 Results Discussion Reference Other Papers 21 / 58 Performance measures Random Forest Individual Causal Inference Min Lu Given an estimator τ̂ of τ , the bias for group Gm was defined as h i h i Bias(m) = E τ̂ (X)|X ∈ Gm − E τ (X)|X ∈ Gm , m = 1, . . . , M Introduction Continuous and Binary Outcome Definitions and notations Method Our simulation experiments were replicated independently B times. Let Gm,b denote those x values that lie within the qm quantile of the propensity score from realization b. Let τ̂b be the ITE estimator from realization b. The conditional bias was estimated by Simulation Real Data Application B B 1X 1X d τ̂m,b − τm,b Bias(m) = B b=1 B b=1 Discussion Survival Outcome Definitions & notations where Treatment overlap oj (x) Treatment effect estimation Results τ̂m,b = X 1 τ̂b (xi ), #Gm,b xi ∈Gm,b τm,b = X 1 τ (xi ) #Gm,b xi ∈Gm,b Discussion Reference Other Papers 22 / 58 Performance measures Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Similarly, we define the conditional root mean squared error (RMSE) of τ̂ by s    2 X ∈ Gm , m = 1, . . . , M RMSE(m) = E τ̂ (X) − τ (X) Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations which we estimated using v u X i2 X h u1 B 1 \ RMSE(m) =t τ̂b (xi ) − τb (xi ) B b=1 #Gm,b xi ∈Gm,b Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 23 / 58 Simulation result: Bias and Root-Mean-Squared-Error Discussion 0.4 n = 500 n = 5000 0.5 RMSE 0.5 RMSE RMSE 0.5 Simulation 3 0.6 n = 500 n = 5000 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART Results Simulation 2 0.6 n = 500 n = 5000 VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART Simulation 1 0.6 VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART −0.05 VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART −0.05 Discussion Treatment effect estimation Bias 0.00 −0.05 Real Data Application Treatment overlap oj (x) 0.05 0.00 Simulation Definitions & notations 0.10 0.05 0.00 Method n = 500 n = 5000 0.15 0.10 0.05 Definitions and notations Survival Outcome Simulation 3 n = 500 n = 5000 0.15 0.10 VT VT−I honest RF bivariate CF synCF BART VT VT−I honest RF bivariate CF synCF BART Continuous and Binary Outcome Simulation 2 n = 500 n = 5000 0.15 Bias Introduction Simulation 1 Bias Random Forest Individual Causal Inference Min Lu Reference Other Papers 24 / 58 Project Aware: a counterfactual approach to understand the role of drug use in sexual risk Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion • Aim: how substance use plays a role in sexual risk (n = 2813, p = 99) • Treatment (exposure) variable T: 0 = no substance use in the prior 6 months, 1 = any substance use in the prior 6 months leading to the study • Outcome: number of unprotected sex acts within the last six months as reported by the individual • Method: a synthetic forest was fit separately to each exposure group. This yielded estimated causal effects {τ̂synCF (xi ), i = 1, . . . , n} for τ (x) defined as the mean difference in number of unprotected sex acts for drug versus non-drug users Reference Other Papers 25 / 58 RF estimated causal effect of drug use Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 26 / 58 RF estimated causal effect of drug use Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome The true model for the outcome (number of unprotected sex acts) is Y = f (X, T) + ε, where f (X, T) = α0 T + h(X, T) and h is some unknown function. Under the assumption of SITA, Definitions and notations Method Simulation Real Data Application τ (x) = f (x, 1) − f (x, 0) = α0 + h(x, 1) − h(x, 0) Discussion Survival Outcome Definitions & notations Now since we canPassume a linear model α + pj=1 βj xj for the ITE, we have Treatment overlap oj (x) Treatment effect estimation Results Discussion α0 + h(x, 1) − h(x, 0) = α + p X βj xj j=1 Reference Other Papers 27 / 58 Explore causal effect of drug use Random Forest Individual Causal Inference Min Lu Now we assume a linear model for the ITE. Standard errors and significance of linear model coefficients were determined using p subsampling. We have X τ (x) = α + βj xj Introduction j=1 Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Intercept (drug use) CESD Condom change 2 Condom change 3 Condom change 4 Condom change 5 Usual care 3 Marriage No health insurance SU treatment last 6 months 2 Frequency of injection Estimate Std. Error Z 16.97 0.60 -19.38 -23.33 -21.46 -24.02 4.91 -1.61 2.72 6.38 3.59 9.36 0.13 2.96 3.23 3.39 3.41 2.11 0.73 0.99 3.20 1.77 1.81 4.54 -6.56 -7.22 -6.32 -7.04 2.33 -2.21 2.75 2.00 2.02 Other Papers 28 / 58 Discussion Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results • In observational data with complex heterogeneity of treatment effect, individual estimates of treatment effect can be obtained in a principled way by directly modeling the response surface • Successful estimation mandates highly adaptive and accurate regression methodology and for this we relied on RF, a machine learning method with well known properties for accurate estimation in complex nonparametric regression settings • We encourage the use of out-of-bag estimation, a simple but underappreciated out-of-sample technique for improving accuracy • The success of counterfactual synthetic RF can be attributed to three separate effects: (a) fitting separate forests to each treatment group, which improves adaptivity to confounding; (b) replacing Breiman forests with synthetic forests, which reduces bias; and (c) utilizing OOB estimation, which improves accuracy Discussion Reference Other Papers 29 / 58 Research Questions Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome We base our analysis on data from 1468 patients who were treated for ischemic cardiomyopathy at Cleveland Clinic from 1997 to 2007. Treatments include · Coronary artery bypass grafting alone (CABG) · CABG plus mitral valve anuloplasty (MVA) · CABG plus surgical ventricular reconstruction (SVR) · Listing for cardiac transplantation (LCTx) Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome • What is the average treatment effect (ATE) and the individual treatment effect (ITE)? Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 30 / 58 Research Questions Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations We base our analysis on data from 1468 patients who were treated for ischemic cardiomyopathy at Cleveland Clinic from 1997 to 2007. Treatments include · Coronary artery bypass grafting alone (CABG) · CABG plus mitral valve anuloplasty (MVA) · CABG plus surgical ventricular reconstruction (SVR) · Listing for cardiac transplantation (LCTx) Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion MVA LCTx 31 81 CABG 43 126 231 90 SVR 65 37 77 110 124 52 121 109 171 • What is the treatment effect? • How to check overlap and how to utilize all data points when lack of overlap often results in sample size reduction? • Have patients received optimal treatments? Reference Other Papers 31 / 58 Research Questions Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations We base our analysis on data from 1468 patients who were treated for ischemic cardiomyopathy at Cleveland Clinic from 1997 to 2007. Treatments include · Coronary artery bypass grafting alone (CABG) · CABG plus mitral valve anuloplasty (MVA) · CABG plus surgical ventricular reconstruction (SVR) · Listing for cardiac transplantation (LCTx) Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion MVA LCTx 31 81 CABG 43 126 231 90 SVR 65 37 77 110 124 52 121 109 171 • What is the treatment effect? • How to check overlap and how to utilize all data points when lack of overlap often results in sample size reduction? • Have patients received optimal treatments? Reference Other Papers 31 / 58 Treatment effect on survival outcome Random Forest Individual Causal Inference Min Lu Introduction Let {(X1 , Z1 , T1 , δ1 ), . . . , (Xn , Zn , Tn , δn )} denote the data. The observed survival time Ti = min(Tio , Cio ), where Tio is the true event time and Cio is the true censoring time. We assume Tio ⊥ Cio |(Xi , Zi ). Let T o (j) denote the potential outcome (event time) under treatment Z = j Continuous and Binary Outcome Definitions and notations Units Observed Potential Outcomes Treatment Effects Method Simulation Outcome Treatment Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) 1 . . . i δ1 T1 Z1 . . . Ti δi Zi Therapy 1 . . . Therapy M for j over k T1o (1) T1o (M) Sj (t|x1 ) − Sk (t|x1 ) . . . . . . . . . Tio (1) Tio (M) Sj (t|xi ) − Sk (t|xi ) Treatment effect estimation Results Discussion Reference Other Papers 32 / 58 The individual treatment effect (ITE) τj,k (t, x) 1.0 Random Forest Individual Causal Inference Min Lu 0.8 Survival under CABG Survival under MVA Treatment effect of CABG over MVA 0.6 Introduction 0.4 Continuous and Binary Outcome 0.2 Definitions and notations Method Simulation 0.0 Real Data Application −0.2 Discussion Survival Outcome 0 Definitions & notations 2 4 6 Time (Year) 8 Treatment overlap oj (x) Treatment effect estimation Results Discussion Definition The individual treatment effect
(ITE) at time t
for covariate x for treatment j over treatment k is defined as
τj,k (t, x) = Sj t|x − Sk t|x , where Sl t|x = P{T o (l) > t|X = x} is the survival function Reference Other Papers 33 / 58 0.8 Survival under CABG Survival under MVA Treatment effect of CABG over MVA Method Simulation Real Data Application 0.2 Overlap Assumption 0.0 Overlap between treatments j and k is said to hold for x if P(Z = j|x) > 0 and P(Z = k|x) > 0. We define the overlap function as follows oj (x) = 1{P(Z=j|x)>0} −0.2 Definitions and notations 0.4 Introduction Continuous and Binary Outcome Weak Unconfoundedness Assumption We say that weak unconfoundedness holds, if for all j ∈ {1, . . . , M},
1{Z=j} ⊥ T o (j), Co (j) | X 0.6 Random Forest Individual Causal Inference Min Lu 1.0 The individual treatment effect (ITE) τj,k (t, x) 0 2 4 6 Time (Year) 8 Discussion Survival Outcome Definition Definitions & notations The individual treatment effect (ITE) at time t for covariate x for treatment j over treatment k is defined as


τj,k (t, x) = Sj t|x − Sk t|x , where Sl t|x = P{T o (l) > t|X = x} is the survival function. Under weak Treatment overlap oj (x) unconfoundedness, Treatment effect estimation Results Discussion Reference τj,k (t, x) = = = o o P{T (j) > t|X = x} − P{T (k) > t|X = x} o o P{T > t|X = x, Z = j} − P{T > t|X = x, Z = k}

S t|x, Z = j − S t|x, Z = k Other Papers 34 / 58 Potential survival function under CABG Potential survival function under SVR Potential survival function under MVA Potential survival function under LCTx 0.8 Random Forest Individual Causal Inference Min Lu 1.0 Individualized Treatment Rules Definitions and notations 0.4 Continuous and Binary Outcome 0.6 Introduction Method Real Data Application 0.2 Simulation Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference 0.0 Discussion Survival Outcome 0 2 4 6 Time (Year) 8 The optimal individualized treatment rule (ITR) selects the best treatment for each patient. For example, LCTx is the optimal treatment in the above figure. Other Papers 35 / 58 Individualized Treatment Rules Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method The optimal decision rule for x, denoted as dopt (x), maximizes the restricted mean survival time (RMST) Z t0 
opt d (x) = argmax S t|x, Z = l dt l∈{l:ol (x)=1} 0 and is uniquely determined by the ITE Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Integrating over t ∈ [0, t0 ], we define the ITE before time t0 as τj,k ([0, t0 ], x) = Z t 0 0 τj,k (t, x) dt which can be interpreted as the difference in the number of years alive before time t0 for treatment j over k. Typically, t0 is chosen to equal the maximum observed follow-up time Results Discussion Reference Other Papers 36 / 58 Continuous and Binary Outcome Definitions and notations 0.10 0.05 0.00 Introduction Individual Treatment effect of CABG over MVA Average Treatment effect of CABG over MVA −0.05 Random Forest Individual Causal Inference Min Lu 0.15 The average treatment effect (ATE) τj,k (t) Simulation Real Data Application −0.15 Method 0 Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results 2 4 6 Time (Year) 8 Definition Define the average treatment effect (ATE) at time t for treatment j over treatment k, as i h τj,k (t) = EX τj,k (t, X) oj (X) = 1, ok (X) = 1 . Discussion Reference Rt We define the ATE before time t0 as τj,k ([0, t0 ]) = 0 0 τj,k (t) dt Other Papers 37 / 58 The average treatment effect (ATE) τj,k (t) Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Definition Define the average treatment effect (ATE) at time t for treatment j over treatment k, as h i τj,k (t) = EX τj,k (t, X) oj (X) = 1, ok (X) = 1 . Results Discussion Reference Rt We define the ATE before time t0 as τj,k ([0, t0 ]) = 0 0 τj,k (t) dt Other Papers 37 / 58 The average treatment effect (ATE) τj,k (t) Random Forest Individual Causal Inference Min Lu Overlap Assumption Overlap between treatments j and k is said to hold for x if P(Z = j|x) > 0 and P(Z = j|x) > 0. We formally define the overlap function as follows Introduction Continuous and Binary Outcome oj (x) = 1{P(Z=j|x)>0} Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Definition Define the average treatment effect (ATE) at time t for treatment j over treatment k, as h i τj,k (t) = EX τj,k (t, X) oj (X) = 1, ok (X) = 1 . {z } | {z } | ITE Treatment Overlap Results Discussion Reference R t0 We define the ATE before time t0 as τj,k ([0, t0 ]) = 0 τj,k (t) dt Other Papers 38 / 58 Treatment overlap oj (x) = 1{P(Z=j|x)>0} Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome • Strategy 1. The overlap function can be estimated using ôj (x; C) = 1{P̂(Z=j|x)>C} where 0 < C < 1 is a selected cutoff value Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome • Strategy 2. A unique feature of our study was the availability of expert knowledge for defining treatment eligibility Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 39 / 58 Treatment overlap oj (x) = 1{P(Z=j|x)>0} Random Forest Individual Causal Inference Min Lu • Strategy 1. The overlap function can be estimated using ôj (x; C) = 1{P̂(Z=j|x)>C} Introduction Continuous and Binary Outcome where 0 < C < 1 is a selected cutoff value Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference • Strategy 2. A unique feature of our study was the availability of expert knowledge for defining treatment eligibility Table: Expert knowledge used for determining treatment eligibility Treatment CABG SVR∗ MVA LCTx∗ ∗ Expert Knowledge Eligibility Criteria (a) Ischemic symptoms (angina); viable myocardium with diseased but by-passable coronary arteries. If (a) was not available, eligibility was determined using: (b) ACC/AHA guidelines for CABG based on angina and coronary artery disease Anterior wall akinesia/dyskinesia; left ventricular end-diastolic diameter>6 cm 3+/4+ mitral regurgitation (MR) present Age<70 years; NYHA functional class III/IV; creatinine level<1.7 mg·dL−1 Treatments where expert knowledge is considered less accurate for determining eligibility Other Papers 39 / 58 Treatment overlap oj (x) = 1{P(Z=j|x)>0} Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome • Strategy 1. The overlap function can be estimated using ôj (x; C) = 1{P̂(Z=j|x)>C} where 0 < C < 1 is a selected cutoff value Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation • Strategy 2. A unique feature of our study was the availability of expert knowledge for defining treatment eligibility Let Ej ∈ {0, 1} denote eligibility for treatment j. The overlap function can also be estimated using Results Discussion Reference ôj (x; C) = 1{P̂(Ej =1|x)>C} Other Papers 39 / 58 Random forest approaches addressing ôj (x, C) Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) • Approach I. Random forest classification (RF-C) approach. Our first approach uses the treatment received Zi as the outcome and Xi as features and fits a random forest classification (RF-C) model to estimate P{Z = j|x} — strategy 1 • Approach II. Random forest distance (RF-D) approach. The general idea is to assign patient i’s eligibility for treatment j by using the “random forest distance” of i to treatment j patients — strategy 1 • Approach III. Multivariate random forest (MRF). We directly model expert knowledge by using the expert data as M-multivariate outcomes in a multi-label classfication analysis — strategy 2 Treatment effect estimation Results Discussion Reference Other Papers 40 / 58 Approach II. Random forest distance approach Random Forest Individual Causal Inference Min Lu Let diA be the count of the edges from i to the closest common ancestor of i and i0 . Similarly, let diA0 count the edges from i0 to the closest (i, i0 ) common ancestor. Define DA = diA + diA0 . Let diR and diR0 be the count of i,i0 the edges from i and i0 to the root node and define DR = diR + diR0 . The distance is defined as i,i0 DA i,i0 . Introduction di,i0 = Continuous and Binary Outcome The forest distance is defined as the forest averaged distance, which we denote by di,i0 . We define the probability of assigning i to treatment j by the closeness of i to treatment j patients, Definitions and notations Method Simulation Real Data Application P P̂{Z = j|Xi } = DR 0 i,i i0 :Z 0 =j (1 i P i0 (1 − di,i0 ) − di,i0 ) . Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference The distance between Xi and Xi0 is the ratio of the number of edges connecting the red nodes to the ancestor, NA , to the number of edges connecting the red nodes to the root node, NR . Thus di,i0 = (2 + 1)/(4 + 3) = 3/7 Other Papers 41 / 58 Cutoff criteria C in ôj (x, C) Random Forest Individual Causal Inference Min Lu ôj (x; C) = 1{P̂(E =1|x)>C} j Random Forest Classification (RF−C) Random Forest Distance (RF−D) Multivariate Random Forest (MRF) Simulation Real Data Application 0.26 Definitions & notations Treatment overlap oj (x) Treatment effect estimation   n  1 X  X 1{E 0 6=ô 0 (X ,c)} i  ij j 0<c<1  2n i=1 0 j ∈M 0 = arg min Table: Cutoff values for estimating treatment eligibility C=0.08 0.18 C=0.12 Discussion Method 0.04 Cutoff Value C=0.61 0.0 Survival Outcome ∗ 0.4 C 0.2 Method Misclassification Error Definitions and notations 0.6 Introduction Continuous and Binary Outcome • Let M 0 = {j1 , j2 } denote the subset of treatment groups corresponding to CABG and MVA. We define the CABG and MVA cutoff as follows: • Treatment overlap is determined either ôj (x; C) = 1{P̂(Z=j|x)>C} or 0.0 0.2 0.4 0.6 0.8 1.0 Cutoff, c RF-C RF-D MRF 0.08 0.12 0.61 Misclassification Error CABG MVA 0.26 0.18 0.04 All four treatments 0.32 0.35 0.13 Fig: Misclassification error of CABG and MVA as a function of the cutoff value c Results Discussion Reference Other Papers 42 / 58 Counterfactual survival analysis using random survival forests Random Forest Individual Causal Inference Min Lu We utilize ALL the data to estimate S(t|x, Z) We estimate the survival function S(t|x, Z) using virtual twin random survival forest interactions, denoted as RSF-VT-I where we add all possible interactions between the treatment variable Z and covariates X to the design matrix to grow random survival forest. The counterfactual ITE estimate is defined as Introduction Continuous and Binary Outcome Definitions and notations τ̂j,k (t, Xi ) = Ŝ∗ (t|Xi , Z = Zi = j) − Ŝ(t|Xi , Z = k), where Ŝ∗ (t|Xi , Zi = j) is the Out-of-Bag estimated survival value based on i’s original (unaltered) data Units Potential Outcomes Treatment Effects Treatment j Simulation Treatment k Real Data Application Discussion Survival Outcome Definitions & notations 1 . . . Treatment overlap oj (x) i Treatment effect estimation . . . Results Discussion N b S1 (t|Z = j) . . . b Si (t|Z = j) . . . b SN (t|Z = j) b S1 (t|Z = k) . . . b Si (t|Z = k) . . . b SN (t|Z = k) Training data Units T δ 1 . . . . . . Method τ bj,k (t|X1 ) Z . . . X1 . . . XP ZX10 . . . ZXP0 . . . . . . N . . . τ bj,k (t|Xi ) . . . R ANDOM S URVIVAL F OREST τ bj,k (t|XP ) Reference Other Papers 43 / 58 Counterfactual survival analysis using random survival forests Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations We utilize ALL the data to estimate S(t|x, Z) We estimate the survival function S(t|x, Z) using virtual twin random survival forest interactions, denoted as RSF-VT-I where we add all possible interactions between the treatment variable Z and covariates X to the design matrix to grow random survival forest. The counterfactual ITE estimate is defined as τ̂j,k (t, Xi ) = Ŝ∗ (t|Xi , Z = Zi = j) − Ŝ(t|Xi , Z = k), where Ŝ∗ (t|Xi , Zi = j) is the Out-of-Bag estimated survival value based on i’s original (unaltered) data Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 43 / 58 Average treatment effect on the treated (ATT) Random Forest Individual Causal Inference Min Lu Treatment assignment is not only adjusted, but also judged Definition Introduction Define the average treatment effect on the treated (ATT) at time t for the treated j, for treatment j over treatment k, as h i τ j k (t) = EX τj,k (t, X) Z = j, oj (X) = 1, ok (X) = 1 Continuous and Binary Outcome Likewise, the ATT for the treated k, for treatment j over k, is h i τj k (t) = EX τj,k (t, X) Z = k, oj (X) = 1, ok (X) = 1 Definitions and notations Method 0.15 0.10 0.05 0.00 0.00 Treatment overlap oj (x) Overlap for CABG and MVA Received CABG and overlap for MVA Received MVA and overlap for CABG −0.05 Definitions & notations −0.05 Survival Outcome 0.10 Discussion 0.05 Real Data Application 0.15 Simulation Average Treatment effect of CABG over MVA Individual Treatment effect of patient received CABG Individual Treatment effect of patient received MVA Discussion Reference −0.15 Results 0 2 4 6 Time (Year) 8 −0.15 Treatment effect estimation 0 2 4 6 Time (Year) 8 Other Papers 44 / 58 Results Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 45 / 58 Results Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome The areas under the black, blue, and red lines of previous figure equal the ATE and ATT before t0 (the maximum observed follow-up time), and thus represent the difference in number of years alive before t0 o ATE before t0 (black line) o ATT before t0 where j is the treated (blue line) o ATT before t0 where k is the treated (red line) ATEjk = τj,k ([0, t0 ]) ATTjk = τ j k ([0, t0 ]) ATTkj = τj k ([0, t0 ]) Definitions and notations Method Simulation Real Data Application Table: Difference in number of months alive before maximum follow-up time, t0 = 9.36 years Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion ATEojk Treatment j vs. k (a) CABG vs. SVR (b) CABG vs. MVA (c) CABG vs. LCTx (d) SVR vs. MVA (e) SVR vs. LCTx (f) MVA vs. LCTx ATTojk ATTokj MRF RF-C RF-D Mean SE Mean SE 0.31 4.88 0.85 5.95 -1.40 -11.80 0.29 5.06 3.67 5.49 -0.55 -6.08 0.60 5.21 3.50 5.47 -1.08 -6.81 -2.67 4.20 5.85 5.97 2.57 -0.84 3.74 2.89 2.26 1.41 1.52 2.62 0.70 5.02 -0.74 5.70 -4.81 -14.97 0.93 1.55 1.11 5.61 1.53 1.36 Reference Other Papers 46 / 58 Results Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Confidence intervals for individual treatment effects τ̂j,k (t, x) at t = 5 years. Each subfigure indicates a pairwise comparison for treatment j versus k. Red and blue indicate patients with significant treatment effect (p-value < .05), where blue are from treatment j group and red are from treatment group k. Thus, blue and red boxes correspond to some of the patients from blue and red lines in previous figure Reference Other Papers 47 / 58 Treatment effect heterogeneity test Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 48 / 58 Subgroup analysis Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations We fit a bump hunting model (Friedman and Fisher, 1999; Duong, 2015) for subgroup analysis. To improve efficiency of the algorithm, we only used variables found important by using random forest variable selection. The estimated ITE was used for the outcome and all pre-treatment covariates as independent variables Table: Subgroup detection using bump hunting after variable selection. CATEojk equals the conditional ATE before t0 , conditioned on subgroup criteria Method Treatment j vs. k Subgroup Discussion CABG vs. SVR CABG vs. SVR Survival Outcome CABG vs. LCTx BSA>2.23 Regurgitation Grade>0 Blood Urea Nitrogen<30 Creatinine<1.8 BMI>27.04 GFR>44.75 Blood Urea Nitrogen<25 LDL<133.31 BSA>1.83 BMI>27.77 55.29<GFR<120.80 Simulation Real Data Application Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results SVR vs. LCTx CATEojk /ATEojk Size/Total % in j % in k -4.08/0.31 -7.26/0.31 44/246 31/246 28.57 10.71 16.51 12.84 5.31/0.85 125/406 59.18 21.75 7.66/-1.40 60/292 30.37 12.10 Discussion Reference BSA=body surface aera (m2 ); BMI=body mass index; GFR=glomerular filtration rate; LDL=low-density lipoprotein cholesterol Other Papers 49 / 58 Results Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Confidence intervals for individual treatment effects τ̂j,k (t, x) at t = 5 years. Each subfigure indicates a pairwise comparison for treatment j versus k. Red and blue indicate patients with significant treatment effect (p-value < .05), where blue are from treatment j group and red are from treatment group k. Thus, blue and red boxes correspond to some of the patients from blue and red lines in previous figure Reference Other Papers 50 / 58 Results Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Fig. Identifying patients who received optimal treatment and those who did not. Optimal therapy is defined as eligible treatment maximizing restricted mean survival time (RMST). Pie charts display gain in months for alternative optimized therapies and their respective sample sizes. If optimized treatment is the assigned treatment, gain is defined as zero. Reference Other Papers 51 / 58 Gain in months for patients who received SVR but where optimal therapy was CABG Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 52 / 58 Gain in months for patients who received SVR but where optimal therapy was CABG Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 53 / 58 Treatment decisions Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 54 / 58 Concluding remarks Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion • One contribution of this paper is to offer estimation methods for assessing treatment overlap under the scenario that some treatments may have either gold standard expert knowledge, or controversial knowledge for judging eligibility • For personalized treatment decision and dynamic causal procedure of treatment effect, we develop a virtual twin random survival forest, extended to include interactions between treatment variables and all pre-treatment covariates • A key insight of this paper is to judge current treatment decisions using pairwise ATT comparisons Reference Other Papers 55 / 58 Reference Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 56 / 58 Other papers Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Method Simulation Real Data Application Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) • Lu M. and Ishwaran H. (2017). A Prediction-Based Alternative to P-values in Regression Models. To appear The Journal of Thoracic and Cardiovascular Surgery https://arxiv.org/abs/1701.04944 • Ishwaran H. and Lu M. (2018). Standard Errors and Confidence Intervals for Variable Importance in Random Forest Regression, Classification, and Survival. To appear Statistics in Medicine • Rice T.W., Lu M., Ishwaran H., and Blackstone, E.H. (2017). Precision Surgical Therapy for Adenocarcinoma of the Esophagus and Esophagogastric Junction Treatment effect estimation Results Discussion Reference Other Papers 57 / 58 Random Forest Individual Causal Inference Min Lu Introduction Continuous and Binary Outcome Definitions and notations Thank you all very much! Method Simulation Real Data Application You can reach me at [email protected] Discussion Survival Outcome Definitions & notations Treatment overlap oj (x) Treatment effect estimation Results Discussion Reference Other Papers 58 / 58