Sensitivity analysis is useful in assessing how robust an association is to potential unmeasured or uncontrolled confounding. This article introduces a new measure called the "E-value," which is related to the evidence for causality in observational studies that are potentially subject to confounding. The E-value is defined as the minimum strength of association, on the risk ratio scale, that an unmeasured confounder would need to have with both the treatment and the outcome to fully explain away a specific treatment-outcome association, conditional on the measured covariates. A large E-value implies that considerable unmeasured confounding would be needed to explain away an effect estimate. A small E-value implies little unmeasured confounding would be needed to explain away an effect estimate. The authors propose that in all observational studies intended to produce evidence for causality, the E-value be reported or some other sensitivity analysis be used. They suggest calculating the E-value for both the observed association estimate (after adjustments for measured confounders) and the limit of the confidence interval closest to the null. If this were to become standard practice, the ability of the scientific community to assess evidence from observational studies would improve considerably, and ultimately, science would be strengthened.

Mediation analysis is a useful and widely employed approach to studies in the field of psychology and in the social and biomedical sciences. The contributions of this paper are several-fold. First we seek to bring the developments in mediation analysis for non linear models within the counterfactual framework to the psychology audience in an accessible format and compare the sorts of inferences about mediation that are possible in the presence of exposure-mediator interaction when using a counterfactual versus the standard statistical approach. Second, the work by VanderWeele and Vansteelandt (2009, 2010) is extended here to allow for dichotomous mediators and count outcomes. Third, we provide SAS and SPSS macros to implement all of these mediation analysis techniques automatically and we compare the types of inferences about mediation that are allowed by a variety of software macros.

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This article provides an overview of recent developments in mediation analysis, that is, analyses used to assess the relative magnitude of different pathways and mechanisms by which an exposure may affect an outcome. Traditional approaches to mediation in the biomedical and social sciences are described. Attention is given to the confounding assumptions required for a causal interpretation of direct and indirect effect estimates. Methods from the causal inference literature to conduct mediation in the presence of exposuremediator interactions, binary outcomes, binary mediators, and case-control study designs are presented. Sensitivity analysis techniques for unmeasured confounding and measurement error are introduced. Discussion is given to extensions to time-to-event outcomes and multiple mediators. Further flexible modeling strategies arising from the precise counterfactual definitions of direct and indirect effects are also described. The focus throughout is on methodology that is easily implementable in practice across a broad range of potential applications.

In this tutorial, we provide a broad introduction to the topic of interaction between the effects of exposures. We discuss interaction on both additive and multiplicative scales using risks, and we discuss their relation to statistical models (e.g. linear, log-linear, and logistic models). We discuss and evaluate arguments that have been made for using additive or multiplicative scales to assess interaction. We further discuss approaches to presenting interaction analyses, different mechanistic forms of interaction, when interaction is robust to unmeasured confounding, interaction for continuous outcomes, qualitative or “crossover” interactions, methods for attributing effects to interactions, case-only estimators of interaction, and power and sample size calculations for additive and multiplicative interaction.

For dichotomous outcomes, the authors discuss when the standard approaches to mediation analysis used in epidemiology and the social sciences are valid, and they provide alternative mediation analysis techniques when the standard approaches will not work. They extend definitions of controlled direct effects and natural direct and indirect effects from the risk difference scale to the odds ratio scale. A simple technique to estimate direct and indirect effect odds ratios by combining logistic and linear regressions is described that applies when the outcome is rare and the mediator continuous. Further discussion is given as to how this mediation analysis technique can be extended to settings in which data come from a case-control study design. For the standard mediation analysis techniques used in the epidemiologic and social science literatures to be valid, an assumption of no interaction between the effects of the exposure and the mediator on the outcome is needed. The approach presented here, however, will apply even when there are interactions between the effect of the exposure and the mediator on the outcome.

Authors often do not give sufficient information to draw conclusions about the size and statistical significance of interaction on the additive and multiplicative scales. To improve this, we provide four steps, template tables and examples. We distinguish two cases: when the causal effect of intervening on one exposure, across strata of another factor, is of interest ('effect modification'); and when the causal effect of intervening on two exposures is of interest ('interaction'). Assume we study whether X modifies the effect of A on D, where A, X and D are dichotomous. We propose presenting: (i) relative risks (RRs), odds ratios (ORs) or risk differences (RDs) for each (A, X) stratum with a single reference category taken as the stratum with the lowest risk of D; (ii) RRs, ORs or RDs for A within strata of X; (iii) interaction measures on additive and multiplicative scales; (iv) the A-D confounders adjusted for. Assume we study the interaction between A and B on D, where A, B and D are dichotomous. Steps (i) and (iii) are similar to presenting effect modification. (ii) Present RRs, ORs or RDs for A within strata of B and for B within strata of A. (iv) List the A-D and B-D confounders adjusted for. These four pieces of information will provide a reader the information needed to assess effect modification or interaction. The presentation can be further enriched when exposures have multiple categories. Our proposal hopefully encourages researchers to present effect modification and interaction analyses in as informative a manner as possible.

Recent advances in the causal inference literature on mediation have extended traditional approaches to direct and indirect effects to settings that allow for interactions and non-linearities. In this paper, these approaches from causal inference are further extended to settings in which multiple mediators may be of interest. Two analytic approaches, one based on regression and one based on weighting are proposed to estimate the effect mediated through multiple mediators and the effects through other pathways. The approaches proposed here accommodate exposure-mediator interactions and, to a certain extent, mediator-mediator interactions as well. The methods handle binary or continuous mediators and binary, continuous or count outcomes. When the mediators affect one another, the strategy of trying to assess direct and indirect effects one mediator at a time will in general fail; the approach given in this paper can still be used. A characterization is moreover given as to when the sum of the mediated effects for multiple mediators considered separately will be equal to the mediated effect of all of the mediators considered jointly. The approach proposed in this paper is robust to unmeasured common causes of two or more mediators.