When there is interference, a subject's outcome depends on the treatment of others and treatment effects may take on several different forms. This situation arises often, particularly in vaccine evaluation. In settings where interference is likely, two-stage cluster randomized trials have been suggested as a means of estimating some of the causal contrast of interest. Working in the finite population setting to investigate rare and unplanned subgroup analyses using some of the estimators that have been suggested in the literature, include Horvitz-Thompson, Hajek, and what might be called the natural extension of the marginal estimators suggested in Hudgens and Halloran 2008. I define the estimands of interest conditional on individual, group and both individual and group baseline variables, giving unbiased Horvitz-Thompson style estimates for each. I also provide variance estimators for several estimators. I show that the Horvitz-Thompson (HT) type estimators are always unbiased provided at least one subject within the group or population, whatever the level of interest for the estimator, is in the subgroup of interest. This is not true of the "natural" or the Hajek style estimators, which will often be undefined for rare subgroups.
We elaborate on a deconvolution method, used to estimate the empirical distribution of unknown parameters, as suggested recently by Efron (2013). It is applied to estimating the empirical distribution of the 'sampling probabilities' of m sampled items. The estimated empirical distribution is used to modify the Horvitz-Thompson estimator. The performance of the modified Horvitz-Thompson estimator is studied in two examples. In one example the sampling probabilities are estimated based on the number of visits until a response was obtained. The other example is based on real data from panel sampl…
When collections of functional data are too large to be exhaustively observed, survey sampling techniques provide an effective way to estimate global quantities such as the population mean function. Assuming functional data are collected from a finite population according to a probabilistic sampling scheme, with the measurements being discrete in time and noisy, we propose to first smooth the sampled trajectories with local polynomials and then estimate the mean function with a Horvitz-Thompson estimator. Under mild conditions on the population size, observation times, regularity of the trajec…
Inverse probability weighting (IPW) is a general tool in survey sampling and causal inference, used both in Horvitz-Thompson estimators, which normalize by the sample size, and Hájek/self-normalized estimators, which normalize by the sum of the inverse probability weights. In this work we study a family of IPW estimators, first proposed by Trotter and Tukey in the context of Monte Carlo problems, that are normalized by an affine combination of these two terms. We show how selecting an estimator from this family in a data-dependent way to minimize asymptotic variance leads to an iterative proce…
We systematically investigate issues due to mis-specification that arise in estimating causal effects when (treatment) interference is informed by a network available pre-intervention, i.e., in situations where the outcome of a unit may depend on the treatment assigned to other units. We develop theory for several forms of interference through the concept of exposure neighborhood, and develop the corresponding semi-parametric representation for potential outcomes as a function of the exposure neighborhood. Using this representation, we extend the definition of two popular classes of causal est…
We study conditions under which treatment effect estimators constructed under the no-interference assumption in randomized experiments are asymptotically normal in the presence of interference. We prove that the standard Horvitz-Thompson estimator is asymptotically normal under a restricted interference condition characterized by limiting the degree of the dependency graph. The amount of interference is allowed to grow with the population size. We then provide a central limit theorem for the difference-in-means estimator that can handle interference that exists between all pairs of units, prov…
In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz-Thompson empirical process arising from complex sampling designs. Global theorems such as Glivenko-Cantelli and Donsker theorems, and local theorems such as local asymptotic modulus and related ratio-type limit theorems are proved for both the Horvitz-Thompson empirical process, and its calibrated version. Limit theorems of other variants and their conditional versions are also established. Our approach reveals an interesting feature: the problem of deriving uniform limit theorems fo…
Under the Neyman causal model, it is well-known that OLS with treatment-by-covariate interactions cannot harm asymptotic precision of estimated treatment effects in completely randomized experiments. But do such guarantees extend to experiments with more complex designs? This paper proposes a general framework for addressing this question and defines a class of generalized regression estimators that are applicable to experiments of any design. The class subsumes common estimators (e.g., OLS). Within that class, two novel estimators are proposed that are applicable to arbitrary designs and asym…
The generalization ability of minimizers of the empirical risk in the context of binary classification has been investigated under a wide variety of complexity assumptions for the collection of classifiers over which optimization is performed. In contrast, the vast majority of the works dedicated to this issue stipulate that the training dataset used to compute the empirical risk functional is composed of i.i.d. observations. Beyond the cases where training data are drawn uniformly without replacement among a large i.i.d. sample or modelled as a realization of a weakly dependent sequence of r.…
We consider a potential outcomes model in which interference may be present between any two units but the extent of interference diminishes with spatial distance. The causal estimand is the global average treatment effect, which compares outcomes under the counterfactuals that all or no units are treated. We study a class of designs in which space is partitioned into clusters that are randomized into treatment and control. For each design, we estimate the treatment effect using a Horvitz-Thompson estimator that compares the average outcomes of units with all or no neighbors treated, where the …
The global average treatment effect (GATE) is a primary quantity of interest in the study of causal inference under network interference. With a correctly specified exposure model of the interference, the Horvitz-Thompson (HT) and Hájek estimators of the GATE are unbiased and consistent, respectively, yet known to exhibit extreme variance under many designs and in many settings of interest. With a fixed clustering of the interference graph, graph cluster randomization (GCR) designs have been shown to greatly reduce variance compared to node-level random assignment, but even so the variance is …
A/B testing is a standard approach for evaluating the effect of online experiments; the goal is to estimate the `average treatment effect' of a new feature or condition by exposing a sample of the overall population to it. A drawback with A/B testing is that it is poorly suited for experiments involving social interference, when the treatment of individuals spills over to neighboring individuals along an underlying social network. In this work, we propose a novel methodology using graph clustering to analyze average treatment effects under social interference. To begin, we characterize graph-t…
