The era of big data has witnessed an increasing availability of multiple data sources for statistical analyses. We consider estimation of causal effects combining big main data with unmeasured confounders and smaller validation data with supplementary information on these confounders. Under the unconfoundedness assumption with completely observed confounders, the smaller validation data allow for constructing consistent estimators for causal effects, but the big main data can only give error-prone estimators in general. However, by leveraging the information in the big main data in a principled way, we can improve the estimation efficiencies yet preserve the consistencies of the initial estimators based solely on the validation data. Our framework applies to asymptotically normal estimators, including the commonly used regression imputation, weighting, and matching estimators, and does not require a correct specification of the model relating the unmeasured confounders to the observed variables. We also propose appropriate bootstrap procedures, which makes our method straightforward to implement using software routines for existing estimators. Supplementary materials for this article are available online.

Randomization is a basis for the statistical inference of treatment effects without strong assumptions on the outcome-generating process. Appropriately using covariates further yields more precise estimators in randomized experiments. R. A. Fisher suggested blocking on discrete covariates in the design stage or conducting analysis of covariance in the analysis stage. We can embed blocking in a wider class of experimental design called rerandomization, and extend the classical analysis of covariance to more general regression adjustment. Rerandomization trumps complete randomization in the design stage, and regression adjustment trumps the simple difference-in-means estimator in the analysis stage. It is then intuitive to use both rerandomization and regression adjustment. Under the randomization inference framework, we establish a unified theory allowing the designer and analyser to have access to different sets of covariates. We find that asymptotically, for any given estimator with or without regression adjustment, rerandomization never hurts either the sampling precision or the estimated precision, and, for any given design with or without rerandomization, our regression-adjusted estimator never hurts the estimated precision. Therefore, combining rerandomization and regression adjustment yields better coverage properties and thus improves statistical inference. To quantify these statements theoretically, we discuss optimal regression-adjusted estimators in terms of the sampling precision and the estimated precision, and then measure the additional gains of the designer and the analyser. We finally suggest the use of rerandomization in the design and regression adjustment in the analysis followed by the Huber–White robust standard error.

Instrumental variable methods can identify causal effects even when the treatment and outcome are confounded. We study the problem of imperfect measurements of the binary instrumental variable, treatment and outcome.We first consider nondifferential measurement errors, that is, the mismeasured variable does not depend on other variables given its true value.We show that the measurement error of the instrumental variable does not bias the estimate, that the measurement error of the treatment biases the estimate away from zero, and that the measurement error of the outcome biases the estimate toward zero. Moreover, we derive sharp bounds on the causal effects without additional assumptions. These bounds are informative because they exclude zero. We then consider differential measurement errors, and focus on sensitivity analyses in those settings.

Difference-in-differences is a widely used evaluation strategy that draws causal inference from observational panel data. Its causal identification relies on the assumption of parallel trends, which is scale-dependent and may be questionable in some applications. A common alternative is a regression model that adjusts for the lagged dependent variable, which rests on the assumption of ignorability conditional on past outcomes. In the context of linear models, Angrist and Pischke (2009) show that the difference-in-differences and lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming parallel trends will overestimate the effect; in contrast, if the parallel trends assumption is correct, then mistakenly assuming ignorability will underestimate the effect. We show that the same bracketing relationship holds in general nonparametric (model-free) settings. We also extend the result to semiparametric estimation based on inverse probability weighting. We provide three examples to illustrate the theoretical results with replication files in Ding and Li (2019).

Frequently, empirical studies are plagued with missing data. When the data are missing not at random, the parameter of interest is not identifiable in general. Without additional assumptions, we can derive bounds of the parameters of interest, which, unfortunately, are often too wide to be informative. Therefore, it is of great importance to sharpen these worst-case bounds by exploiting additional information. Traditional missing data analysis uses only the information of the binary missing data indicator, that is, a certain data point is either missing or not. Nevertheless, real data often provide more information than a binary missing data indicator, and they often record different types of missingness. In a motivating HIV status survey, missing data may be due to the units’ unwillingness to respond to the survey items or their hospitalization during the visit, and may also be due to the units’ temporarily absence or relocation. It is apparent that some missing types are more likely to be missing not at random, but other missing types are more likely to be missing at random. We show that making full use of the missing types results in narrower bounds of the parameters of interest. In a real-life example, we demonstrate substantial improvement of more than 50% reduction in bound widths for estimating the prevalence of HIV in rural Malawi. As we illustrate using the HIV study, our strategy is also useful for conducting sensitivity analysis by gradually increasing or decreasing the set of types that are missing at random. In addition, we propose an easy-to-implement method to construct confidence intervals for partially identified parameters with bounds expressed as the minimums and maximums of finite parameters, which is useful for not only our problem but also many other problems involving bounds.

Unmeasured confounding may undermine the validity of causal inference with observational studies. Sensitivity analysis provides an attractive way to partially circumvent this issue by assessing the potential influence of unmeasured confounding on causal conclusions. However, previous sensitivity analysis approaches often make strong and untestable assumptions such as having an unmeasured confounder that is binary, or having no interaction between the effects of the exposure and the confounder on the outcome, or having only one unmeasured confounder. Without imposing any assumptions on the unmeasured confounder or confounders, we derive a bounding factor and a sharp inequality such that the sensitivity analysis parameters must satisfy the inequality if an unmeasured confounder is to explain away the observed effect estimate or reduce it to a particular level. Our approach is easy to implement and involves only two sensitivity parameters. Surprisingly, our bounding factor, which makes no simplifying assumptions, is no more conservative than a number of previous sensitivity analysis techniques that do make assumptions. Our new bounding factor implies not only the traditional Cornfield conditions that both the relative risk of the exposure on the confounder and that of the confounder on the outcome must satisfy but also a high threshold that the maximum of these relative risks must satisfy. Furthermore, this new bounding factor can be viewed as a measure of the strength of confounding between the exposure and the outcome induced by a confounder.

For binary experimental data, we discuss randomization-based inferential procedures that do not need to invoke any modeling assumptions. In addition to the classical method of moments, we also introduce model-free likelihood and Bayesian methods based solely on the physical randomization without any hypothetical super population assumptions about the potential outcomes. These estimators have some properties superior to moment-based ones such as only giving estimates in regions of feasible support. Due to the lack of identification of the causal model, we also propose a sensitivity analysis approach that allows for the characterization of the impact of the association between the potential outcomes on statistical inference.

It is often of interest to decompose the total effect of an exposure into a component that acts on the outcome through some mediator and a component that acts independently through other pathways. Said another way, we are interested in the direct and indirect effects of the exposure on the outcome. Even if the exposure is randomly assigned, it is often infeasible to randomize the mediator, leaving the mediator-outcome confounding not fully controlled. We develop a sensitivity analysis technique that can bound the direct and indirect effects without parametric assumptions about the unmeasured mediator-outcome confounding.

Recent terrorist attacks by first- and second-generation immigrants in the United States and Europe indicate that radicalization may result from the failure of ethnic integration-or the rise of intergroup prejudice in communities where home-grown extremists are raised. Yet, these community-level drivers are notoriously difficult to study because public opinion surveys provide biased measures of both prejudice and radicalization. We examine the relationship between anti-Muslim and pro-ISIS (Islamic State of Iraq and Syria) Internet searches in 3099 U.S. counties between 2014 and 2016 using instrumental variable models that control for various community-level factors associated with radicalization. We find that anti-Muslim searches are strongly associated with pro-ISIS searches-particularly in communities with high levels of poverty and ethnic homogeneity. Although more research is needed to verify the causal nature of this relationship, this finding suggests that minority groups may be more susceptible to radicalization if they experience discrimination in settings where they are isolated and therefore highly visible-or in communities where they compete with majority groups for limited financial resources. We evaluate the validity of our findings using several other data sources and discuss the implications of our findings for the study of terrorism and intergroup relations, as well as immigration and counterterrorism policies.