Statistical Analysis of Randomized Experiments with Nonignorable Missing Binary Outcomes

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1 Statistical Analysis of Randomized Experiments with Nonignorable Missing Binary Outcomes Kosuke Imai Department of Politics Princeton University July Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 1 / 13 Overview Introduction Overview Missing outcomes in randomized experiments. A growing literature on the topic: Method of bounds (e.g. Horowitz and Manksi 2000). Semiparametric methods (e.g. Scharfstein et al. 1999). Ignorability (e.g. Yau and Little 2001). Latent ignorability (e.g. Frangakis and Rubin 1999). Nonignorable missing outcomes: Political science: self-reported voting behavior. Economics: self-reported income. Medicine: self-reported health status. The paper offers (with and without noncompliance): 1 Alternative identification and estimation strategies. 2 New sensitivity analyses. 3 Applications in political science psychology and public health. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 2 / 13

2 Standard Randomized Experiments Framework for Standard Randomized Experiments Setup Causal inference via potential outcomes (e.g. Holland 1986). Experimental unit: i = n. Binary treatments: T i {0 1}. Potential outcomes: Y i (T i ). Observed outcome: Y i = T i Y i (1) + (1 T i )Y i (0). Potential response indicators: R i (T i ). Observed response indicator: R i = T i R i (1) + (1 T i )R i (0). Pre-treatment covariates: X i. No interference among units (Cox 1958; Rubin 1990). Randomized treatment: (Y i (1) Y i (0) R i (1) R i (0)) T i for all i. Estimands: Average Treatment Effect (ATE): τ ATE E[Y i (1) Y i (0)] = E[Y i T i = 1] E[Y i T i = 0]. Conditional Average Treatment Effect (CATE): τ CATE 1 n n i=1 E[Y i(1) Y i (0) X i ]. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 3 / 13 Standard Randomized Experiments Identification Problem in the Binary Case Assume Y i (0) Y i (1) {0 1}. Define p jk Pr(Y i = 1 T i = j R i = k) π jk Pr(T i = j R i = k) Then the ATE can be written as τ ATE = p 10π 10 + p 11 π 11 π 10 + π 11 p 00π 00 + p 01 π 01 π 00 + π 01 where p 00 and p 10 are not identifiable from the data. Since p j0 [0 1] the sharp bounds (Horowitz & Manski 2000) are given by [ p11 π 11 (π 00 + π 01 ) (π 00 + p 01 π 01 )(π 10 + π 11 ) τ ATE (π 10 + π 11 )(π 00 + π 01 ) (π 10 + p 11 π 11 )(π 00 + π 01 ) p 01 π 01 (π 10 + π 11 ) (π 10 + π 11 )(π 00 + π 01 ) Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 4 / 13 ].

3 Standard Randomized Experiments Identification Strategies Ignorability Assumption (Little & Rubin 1987): For j {0 1} Pr(R i (j) = 1 T i = j Y i (j) = 1 X i = x) = Pr(R i (j) = 1 T i = j Y i (j) = 0 X i = x) Nonignorability (NI) Assumption: For k {0 1} and x X Pr(R i (j) = 1 T i = 0 Y i (0) = k X i = x) = Pr(R i (j) = 1 T i = 1 Y i (1) = k X i = x). Missing-data mechanism directly depends on the realized value of the outcome variable itself but is conditionally independent of the treatment status. Identification of the ATE is established via Bayes rule (PROPOSITION 1). Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 5 / 13 Standard Randomized Experiments Inference under the Nonignorability Assumption 1 Without covariates (or within strata defined by covariates): the ML estimator is in a closed form (PROPOSITION 2). 2 With covariates: Modeling approach (e.g. logistic regression): q j (x) = Pr(Y i = 1 T i = j X i = x) r jk (x) = Pr(R i = 1 T i = j Y i = k X i = x) Complete-data likelihood function: n [ r 1 (X i ) R i {1 r 1 (X i )} 1 R ] Yi [ i r 0 (X i ) R i {1 r 0 (X i )} 1 R ] 1 Yi i i=1 [ q 1 (X i ) Y i {1 q 1 (X i )} 1 Y i ] Ti [ q0 (X i ) Y i {1 q 0 (X i )} 1 Y i ] 1 Ti where r k (x) = r 1k (x) = r 0k (x) for x X under the NI assumption. Computation: EM algorithm Gibbs sampler with prior distributions. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 6 / 13

4 Standard Randomized Experiments Sensitivity Analysis Sensitivity Analysis Neither MAR nor NI assumptions are testable. Sensitivity analysis based on the following parameter θ NI k Pr(R i(1) = 1 T i = 1 Y i (1) = k) Pr(R i (0) = 1 T i = 0 Y i (0) = k) for k = 0 1 where the range of the parameter is given by (1 p 11 )π 11 θ0 NI (1 p 01)π 01 + π 00 (1 p 11 )π 11 + π 10 (1 p 01 )π 01 p 11 π 11 θ1 NI p 01π 01 + π 00. p 11 π 11 + π 10 p 01 π 01 τ ATE is now a function of θk NI and identifiable parameters. See how τ ATE varies along with the value of θ k. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 7 / 13 Setup Randomized encouragement design: Binary encouragement: Z i {0 1}. Potential binary treatments: T i (Z i ) {0 1}. Observed treatment: T i = Z i T i (1) + (1 Z i )T i (0). Potential outcomes: Y i (Z i ). Observed outcome: Y i = Z i Y i (1) + (1 Z i )Y i (0). Potential response indicators: R i (Z i ). Observed response indicator: R i = Z i R i (1) + (1 Z i )R i (0). Randomization of encouragement: (Y i (1) Y i (0) T i (1) T i (0) R i (1) R i (0)) Z i Intention-To-Treat (ITT) effect: τ ITT E[Y i (T i (1) 1) Y i (T i (0) 0)]. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 8 / 13

5 Setup Instrumental Variables (Angrist Imbens & Rubin 1996) Noncompliance Complier: T i (1) = 1 and T i (0) = 0. Noncomplier: 1 Always-taker (C i = c): T i (1) = T i (0) = 1. 2 Never-taker (C i = n): T i (1) = T i (0) = 0. 3 Defier (C i = d): T i (1) = 0 and T i (0) = 1. Assumptions: 1 Monotonicity (no defier): T i (1) T i (0). 2 Exclusion restriction for noncompliers: Y i (1) = Y i (0) for C i = a n (i.e. zero ITT effect for always-takers and never-takers). Complier Average Causal Effect (IV estimand): τ CACE E[Y i (1) Y i (0) C i = c] = E[Y i(1) Y i (0)] E[T i (1) T i (0)]. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 9 / 13 Identification Strategies Ignorability (Yau & Little 2001): For j = 0 1 and l = 0 1 Pr(R i (l) = 1 Y i (l) = 1 T i (l) = j Z i = l X i = x) = Pr(R i (l) = 1 Y i (l) = 0 T i (l) = j Z i = l X i = x). Latent Ignorability (Frangakis & Rubin 1999): 1 Latent ignorability: For l = 0 1 and t {c n a} Pr(R i (l) = 1 Y i (l) = 1 Z i = l C i = t X i = x) = Pr(R i (l) = 1 Y i (l) = 0 Z i = l C i = t X i = x). 2 Compound exclusion restriction for noncompliers: Y i (0) = Y i (1) and R i (1) = R i (0) for C i = n a. Nonignorability: For j = 0 1 and k = 0 1 Pr(R i (1) = 1 T i (1) = j Y i (1) = k Z i = 1 X i = x) = Pr(R i (0) = 1 T i (0) = j Y i (0) = k Z i = 0 X i = x). Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 10 / 13

6 Theoretical Results in the Binary Case Apply the same analytical strategy as before. Define p jkl Pr(Y i = 1 T i = j R i = k Z i = l) π jkl Pr(T i = j R i = k Z i = l). Rewrite the ITT effect as τ ITT = k=0 p jk1π jk1 k=0 π jk1 k=0 p jk0π jk0 k=0 π jk0 where π jkl and p j1l are identifiable but p j0l is not. Thus the identification of τ ITT requires four constraints (PROPOSITION 3). Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 11 / 13 Inference and Sensitivity Analysis With no covariate: ML estimator and its asymptotic variance are in a closed-form. Sensitivity analysis parameters: ψ NI jk Pr(R i(1) = 1 T i (1) = j Y i (1) = k Z i = 1) Pr(R i (0) = 1 T i (0) = j Y i (0) = k Z i = 0) Modeling approach: p jl (x) Pr(Y i = 1 T i = j Z i = l X i = x) q l (x) Pr(T i = 1 Z i = l X i = x) r jk (x) Pr(R i = 1 T i = j Y i = k X i = x). τ ITT (x) = [p 11 (x)q 1 (x) + p 01 (x){1 q 1 (x)}] [p 10 (x)q 0 (x) + p 00 (x){1 q 0 (x)}] Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 12 / 13

7 Concluding Remarks Concluding Remarks Nonignorable missing data in randomized experiments. Identification and estimation strategies for randomized experiments with and without noncompliance. Sensitivity analyses to examine robustness of conclusiosns. Kosuke Imai (Princeton University) Nonignorable Missing Outcomes 13 / 13