Assess Assumptions and Sensitivity Analysis. Fan Li March 26, 2014

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1 Assess Assumptions and Sensitivity Analysis Fan Li March 26, 2014

2 Two Key Assumptions 1. Overlap: 0<Pr(W i =1 X i )<1, for all i 2. Nonconfoundedness: Y i (0),Y i (1) W i X i Overlap means that each unit has non-zero probabilities to be assigned to either treatment condition. Nonconfoundedness means that no unmeasured confounders, equivalently, it means that for units with the same distributions of measured covariates, the treatment assignment can be viewed as random.

3 Assessing Assumptions In randomized experiments, both assumptions are valid. In observational studies, they are not guaranteed, in fact, most time they are at the best only approximately true. It is important to assess the plausibility of these assumptions; when this is not feasible, one should check how sensitive (or robust) the results are to violation to these assumptions.

4 Assessing Overlap The overlap assumption is on the distribution of the observed quantities (assignment, covariates) testable. The first method to detect lack of overlap is to plot distributions of covariates by treatment groups. In the case with one or two covariates one can do this directly. In high dimensional cases, however, this becomes more difficult.

5 Assessing Overlap One can inspect pairs of marginal distributions by treatment status, but these are not necessarily informative about lack of overlap. It is possible that for each covariate the distribution for the treatment and control groups are identical, even though there are areas where the propensity score is zero or one. A more direct method is to inspect the distribution of the propensity score in both treatment groups, which can reveal lack of overlap in the multivariate covariate distributions. Area (subpopulation) with no or limited overlap, if detected, should be removed or down weighted.

6 Example: The LaLonde Data

7 Assessing Unconfoundedness The unconfoundedness assumption involves unobserved potential outcomes generally untestable in practice. But in some settings, it has testable implications. One setting is when multiple control groups exists. Another setting is to use pseudo outcomes. Not often done in practice due to the restriction of data, sensitivity analysis is much more common.

8 Sensitivity Analysis When unconfoundedness can be tested and found to be plausible we are happy, go on with our analysis, and present the results with confidence. When unconfoundedness is not testable, we can still check how sensitivity the analysis results are to the violation of unconfoundedness. Another approach is to obtain the bounds of the treatment effect with minimum assumptions.

9 Rosenbaum-Rubin type of Sensitivity Analysis Sensitivity analysis aims at assessing the bias of causal effect estimates when the uncounfoundedness assumption is assumed to fail in some specific and meaningful ways Sensitivity is different from testing because the identifying assumption is intrinsically non-testable: the data are uninformative about the distribution of Y (0) for treated units and Y (1) for control units Rosenbaum and Rubin (1983, JRSSB) propose assessing the robustness of the estimated causal effects with respect to assumptions about an unobserved binary covariate that is associated with both the treatment and the response

10 Rosenbaum-Rubin type of Sensitivity Analysis Central assumption: the assignment to treatment is not unconfounded given the set of observable variables X, Pr(W Y(0),Y(1),X) Pr(W X), but uncounfoundedness holds given X and an unobserved binary covariate U: Pr(W Y(0),Y(1),X,U) = Pr(W X,U). Given these assumptions, Rosenbaum and Rubin suggest specifying a set of parameters characterizing the distribution of U and the association of U with W, Y (1) and Y (0) given observed covariates Then parameters are called sensitivity parameters, encoding the degree of violation to unconfoundedness.

11 Rosenbaum-Rubin type of Sensitivity Analysis Specify a model for W, Y (0),Y (1), U given X, estimate the model parameters, holding the sensitivity parameters as fixed known values It is then possible to judge the sensitivity of inferential conclusions with respect to certain plausible variations of the association of U with W, Y (0) and Y (1) If conclusions are relatively insensitive over a range of plausible assumptions about U, causal inference is more defensible

12 Sensitivity Analysis: Example Rosenbaum and Rubin (1983, JRSSB) Goal: comparing different treatments in relieving symptoms of coronary artery disease Treatment: 1 surgery, 0 medical Outcome: improvement after 6 months (1 improve; 0 no improve) 74 covariates, stratified by propensity scores.

13 Sensitivity Analysis: Example Assume there is one unmeasured confounder, u. Specify (1) proportion of u in the population; (2) effect of u on the treatment assignment; (3) effect of u on the outcome in different groups. Compare the results (estimated treatment effects) under a range of values of the above three quantities (known as sensitivity parameter). If the results are similar, the conclusion is not sensitive to the unmeasured confounder.

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