Causal Inference Basics

Transcription

1 Causal Inference Basics Sam Lendle October 09, 2013

2 Observed data, question, counterfactuals Observed data: n i.i.d copies of baseline covariates W, treatment A {0, 1}, and outcome Y. O i = (W i, A i, Y i ) P 0 for i = 1,..., n. Question: What is the effect of treatment A on outcome Y. E(Y 1 Y 0 ) where Y a is the counterfactual outcome that would have happened if a patient had received treatment a.

3 Identifiability How can we write E(Y 1 Y 0 ) as a function of the distribution of the observed data? The no unmeasured confounders assumption, A (Y 1, Y 0 ) W and the counterfactual consistency assumption: Y = Y A. E(Y a ) =E[E(Y a W )] =E[E(Y a A = a, W )] by A Y a W =E[E(Y A = a, W )] by consist. for a {0, 1}. Also need to assume 0 < P(A = a W ) < 1 a.e.

4 Statistical model and parameter The causal assumptions we have made so far have put no restriction on the distribution of the observed data, so the model M is nonparametric. We will focus on the parameter Ψ(P) = E[E(Y A = 1, W )] and ψ 0 = Ψ(P 0 ). Let Q(a, w) = E[Y A = a, W = w)] and g(w) = P(A = 1 W = w)

5 Estimation Plug in estimator A plug in estimator takes an estimate of (relavent parts of) P 0, and plugs it in to the parameter mapping Ψ. For ψ 0, estimate the outcome regression with Q n (1, w), and average w.r.t. the empirical distribution of W : n n 1 Q n (1, W i ) i=1 If W is discrete, then we can use Q n (1, w) = n i=1 I (A i = 1, W i = w)y i n i=1 I (A i = 1, W i = w) Otherwise need to use some sort of smoothing for Q n. The plug in estimator is consistent if Q n is consistent for Q 0.

6 Estimation Propensity score based estimators Can write E ( ) AY g(w ) [ ( )] AY =E E g(w ) A, W [ =E E ( Y g(w ) A = 1, W =E[E(Y A = 1, W )] ) ] g(w ) Inverse probability of treatment weighted (IPTW) estimator: take weighted average of outcomes where weights are one over estimate propensity score: n 1 n i=1 A i Y i g n (W i ) Consistent if g n is consistent for g 0. Not efficient.

7 Estimation Propensity score based estimators II Propensity score matching estimators match untreated observations with the treated observations most similar in estimated propensity score g n. n 1 n i=1 Ŷ i where Ŷ i = { Y i if A i = 1 Y Mi if A i = 0 and M i = arg min g n (W i ) g n (W j ) j:a j =1 There are many other variants to this particular matching estimator. Matching estimators are consistent if g n is consistent for g 0, or under some less strict conditions. Not efficient.

8 Estimation Doubly robust estimators Combine estimates Q n and g n to build an estimator that is consistent if either initial estimate is consistent, and efficient if both are consistent. Examples: augmented IPTW, targeted maximum likelihood/minimum loss based estimation (TMLE). TMLE: Estimate ɛ in the model Q 0 (a, w) = Q 0 a n(a, w) + ɛ g n (w) with maximum likelihood/least squares and calculate Q n(a, w) = Q 0 a n(a, w) + ɛ n g n (w) Estimate ψ 0 by plugging Q n into parameter mapping along with empirical distribution of W.

9 Balancing scores Rosenbaum and Rubin [1983] define a balancing score b(w ) as a function of W such that A W b(w ) Examples: W, g 0 (W ), a monotone transformation of g 0 (W ). The no unmeasured confounders assumption implies A (Y 1, Y 0 ) b(w ) so E(Y 1 ) = E[E(Y A = 1, b(w ))] This is not really surprising, because we already know IPTW and PS matching work, and those only rely on the propensity score.

10 The balancing score property Propensity score matching is essentially estimating E(Y A = 1, g n (W )) nonparametrically with nearest neighbor or kernel regression. Such estimators are consistent as long as g n converges to some balancing score, not necessarily the true propensity score. Say that an estimator that is robust to this sort of propensity score model misspecification has the balancing score property. Can we construct a doubly robust, locally efficient estimator that also has the balancing score property?

11 Estimator with balancing score property Idea: Estimate initial Q 0 n. Use initial estimate as an offset in the model E(Y a, w) = Q 0 n(a, w) + θ(a, g n (w)) and update to Q 1 n(a, w). If g n (w) is discrete, use saturated parametric model (no TMLE update step needed) Otherwise, stratify on categories of g n (W ), do matching on Y i Q 0 n(a i, W i ), or use some nonparametric smoothing: GAM, loess, etc. Perform TMLE update step on Q 1 n(a, w) to remove additional bias.

12 P.R. Rosenbaum and D.B. Rubin. The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1):41, April 1983.