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Pre-treatment selection or censoring (`selection on treatment') can occur when two treatment levels are compared ignoring the third option of neither treatment, in `censoring by death' settings where treatment is only defined for those who survive long enough to receive it, or in general in studies where the treatment is only defined for a subset of the population. Unfortunately, the standard instrumental variable (IV) estimand is not defined in the presence of such selection, so we consider estimating a new survivor-complier causal effect. Although this effect is generally not identified under standard IV assumptions, it is possible to construct sharp bounds. We derive these bounds and give a corresponding data-driven sensitivity analysis, along with nonparametric yet efficient estimation methods. Importantly, our approach allows for high-dimensional confounding adjustment, and valid inference even after employing machine learning. Incorporating covariates can tighten bounds dramatically, especially when they are strong predictors of the selection process. We apply the methods in a UK cohort study of critical care patients to examine the mortality effects of prompt admission to the intensive care unit, using ICU bed availability as an instrument.

The method of instrumental variables (IV) provides a framework to study causal effects in both randomized experiments with noncompliance and in observational studies where natural circumstances produce as-if random nudges to accept treatment. Traditionally, inference for IV relied on asymptotic approximations of the distribution of the Wald estimator or two-stage least squares, often with structural modeling assumptions and/or moment conditions. In this paper, we utilize the randomization inference approach to IV inference. First, we outline the exact method, which uses the randomized assignment of treatment in experiments as a basis for inference, but lacks a closed-form solution and may be computationally infeasible in many applications. We then provide an alternative to the exact method, the almost exact method, which is computationally feasible but retains the advantages of the exact method. We also review asymptotic methods of inference, including those associated with two-stage least squares, and analytically compare them to randomization inference methods. We also perform additional comparisons using a set of simulations. We conclude with three different applications from the social sciences.

In parameter determination for the heteroskedastic probit model, both in simulated data and in actual data, we observe a failure of traditional local search methods to converge consistently to a single parameter vector, in contrast to the typical situation for the regular probit model. We identify features of the heteroskedastic probit log likelihood function that we argue tend to lead to this failure, and suggest ways to amend the local search methods to remedy the problem.

In observational studies of treatment effects, matched samples are created so treated and control groups are similar in terms of observable covariates. Traditionally such matched samples consist of matched pairs. If a pair match fails to make treated and control units sufficiently comparable, alternative forms of matching may be necessary. One general strategy to improve balance is to match a variable number of control units to each treated unit. A more tailored strategy is to adopt a fine balance constraint. Under a fine balance constraint, a nominal covariate is exactly balanced, but it does not require individually matched treated and control subjects for this variable. In the example, we seek to construct a matched sample for an ongoing evaluation of Peer Health Exchange, an intervention in schools designed to decrease risky health behaviors among youth. We find that an optimal pair match that minimizes distances between pairs creates a matched sample where balance is poor. Here we propose a method to allow for fine balance constraints when each treated unit is matched to a variable number of control units, which is not currently possible using existing matching algorithms. Our approach uses the entire number to first determine the optimal number of controls for each treated unit. For each strata of matched treated units, we can then apply a fine balance constraint. We then demonstrate that a matched sample for the evaluation of the Peer Health Exchange based on a variable number of controls and fine balance constraint is superior to simply using a variable-ratio match., Comment: 30 pages, 2 figures

A distinctive feature of a clustered observational study is its multilevel or nested data structure arising from the assignment of treatment, in a non-random manner, to groups or clusters of units or individuals. Examples are ubiquitous in the health and social sciences including patients in hospitals, employees in firms, and students in schools. What is the optimal matching strategy in a clustered observational study? At first thought, one might start by matching clusters of individuals and then, within matched clusters, continue by matching individuals. But as we discuss in this paper, the optimal strategy is the opposite: in typical applications, where the intracluster correlation is not perfect, it is best to first match individuals and, once all possible combinations of matched individuals are known, then match clusters. In this paper we use dynamic and integer programming to implement this strategy and extend optimal matching methods to hierarchical and multilevel settings. Among other matched designs, our strategy can approximate a paired clustered randomized study by finding the largest sample of matched pairs of treated and control individuals within matched pairs of treated and control clusters that is balanced according to specifications given by the investigator. This strategy directly balances covariates both at the cluster and individual levels and does not require estimating the propensity score, although the propensity score can be balanced as an additional covariate. We illustrate our results with a case study of the comparative effectiveness of public versus private voucher schools in Chile, a question of intense policy debate in the country at the present.

Causal mediation analysis is routinely conducted by applied researchers in a variety of disciplines. The goal of such an analysis is to investigate alternative causal mechanisms by examining the roles of intermediate variables that lie in the causal paths between the treatment and outcome variables. In this paper we first prove that under a particular version of sequential ignorability assumption, the average causal mediation effect (ACME) is nonparametrically identified. We compare our identification assumption with those proposed in the literature. Some practical implications of our identification result are also discussed. In particular, the popular estimator based on the linear structural equation model (LSEM) can be interpreted as an ACME estimator once additional parametric assumptions are made. We show that these assumptions can easily be relaxed within and outside of the LSEM framework and propose simple nonparametric estimation strategies. Second, and perhaps most importantly, we propose a new sensitivity analysis that can be easily implemented by applied researchers within the LSEM framework. Like the existing identifying assumptions, the proposed sequential ignorability assumption may be too strong in many applied settings. Thus, sensitivity analysis is essential in order to examine the robustness of empirical findings to the possible existence of an unmeasured confounder. Finally, we apply the proposed methods to a randomized experiment from political psychology. We also make easy-to-use software available to implement the proposed methods., Comment: Published in at http://dx.doi.org/10.1214/10-STS321 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)