Structural equation modeling (SEM) has dominated causal analysis in the social and behavioral sciences since the 1960s. Currently, many SEM practitioners are having difficulty articulating the causal content of SEM and are seeking foundational answers.

This paper studies the causal interpretation of counterfactual sentences using a modifiable structural equation model. It is shown that two properties of counterfactuals, namely, composition and effectiveness, are sound and complete relative to this interpretation, when recursive (i.e., feedback-less) models are considered. Composition and effectiveness also hold in Lewis's closest-world semantics, which implies that for recursive models the causal interpretation imposes no restrictions beyond those embodied in Lewis's framework. A third property, called reversibility, holds in nonrecursive causal models but not in Lewis's closest-world semantics, which implies that Lewis's axioms do not capture some properties of systems with feedback. Causal inferences based on counterfactual analysis are exemplified and compared to those based on graphical models.

Simpson's Paradox occurs when an observed association is spurious – reversed after taking into account a confounding factor. At best, Simpson's Paradox is used to argue that association is not causation. At worst, Simpson's Paradox is used to argue that induction is impossible in observational studies (that all arguments from association to causation are equally suspect) since any association could possibly be reversed by some yet unknown confounding factor. This paper reviews Cornfield's conditions – the necessary conditions for Simpson's Paradox – and argues that a simple-difference form of these conditions can be used to establish a minimum effect size for any potential confounder. Cornfield's minimum effect size is asserted to be a key element in statistical literacy. In order to teach this important concept, a graphical technique was developed to illustrate percentage-point difference comparisons. Some preliminary results of teaching these ideas in an introductory statistics course are presented.

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The paper provides a simple test for deciding, from a given causal diagram, whether two sets of variables have the same bias-reducing potential under adjustment. The test requires that one of the following two conditions holds: either (1) both sets are admissible (i.e., satisfy the back-door criterion) or (2) the Markov boundaries surrounding the manipulated variable(s) are identical in both sets. Applications to covariate selection and model testing are discussed.

Abstract: The defining conditions for a binary confounder to nullify an association have been identified for a non-interactive model involving a binary predictor and a binary outcome. When the association involves a relative risk or prevalence, three values are required to specify the nature of the confounder. One goal of this paper is to identify a meaningful single-value that can specify the numerical properties of a binary confounder that would nullify a given association. Associations that can withstand a certain size confounder without being nullified are considered confounder resistant. A second goal is to identify conditions under which the influence of a confounder can be shown as confounderintervals for an observed ratio and a given size confounder. Formulas for the upper and lower limits of confounder intervals are determined for relative prevalences. In order to highlight the influence of potential confounders, data analysts using relative risks or prevalences from observational data should be accompany these with some measure of their susceptibility to confounding using either the size confounder that would nullify the association or the interval for a given size confounder.

Abstract: Confounding is present in most observational studies. Yet by its nature, confounding is not generally present in the data. In order to use statistical associations as evidence for causal connections, one must try to take into account the influence of confounding. This paper reviews the role of confounding in the epic debate between Cornfield and Fisher on the statistical association between smoking and lung cancer and Cornfield’s measure of the influence of an unobserved confounder in terms of a necessary condition. This paper extends the approach of Cornfield and Gastwirth to obtain defining conditions under which a binary confounder will nullify – render spurious – an association between binary variables when using a noninteractive (NI) linear OLS regression model. These defining conditions are used to derive necessary conditions for NI spuriosity and reversal. From these necessary conditions, simple tests are obtained to infer whether an association will be increased, decreased or reversed after controlling for a confounder. Using this non-interactive linear model, families of confounders are identified as mathematical objects based on their ability to nullify an observed relative prevalence. This paper also identifies the numerical properties of a binary confounder that would nullify a given association. Associations that can withstand a certain size confounder without being nullified are considered confounder resistant. This paper also identifies conditions under which the influence of a confounder can be shown as confounder intervals for an observed ratio and a given size confounder. Formulas for the upper and lower limits of confounder intervals are determined. In order to highlight the influence of potential confounders on relative risks or prevalences in observational studies, data analysts should accompany these measures with some measure of their susceptibility to confounding using either the size confounder that would nullify the association or the interval for a given size confounder. Keywords: Epidemiology, Simpson’s Paradox, nullify.

Abstract: The defining conditions for a binary confounder to nullify an association have been identified for a non-interactive model involving a binary predictor and a binary outcome. When the association involves a relative risk or prevalence, three values are required to specify the nature of the confounder. One goal of this paper is to identify a meaningful single-value that can specify the numerical properties of a binary confounder that would nullify a given association. Associations that can withstand a certain size confounder without being nullified are considered confounder resistant. A second goal is to identify conditions under which the influence of a confounder can be shown as confounderintervals for an observed ratio and a given size confounder. Formulas for the upper and lower limits of confounder intervals are determined for relative prevalences. In order to highlight the influence of potential confounders, data analysts using relative risks or prevalences from observational data should be accompany these with some measure of their susceptibility to confounding using either the size confounder that would nullify the association or the interval for a given size confounder.

The paper provides a simple test for deciding, from a given causal diagram, whether two sets of variables have the same bias-reducing potential under adjustment. The test requires that one of the following two conditions holds: either (1) both sets are admissible (i.e., satisfy the back-door criterion) or (2) the Markov boundaries surrounding the manipulated variable(s) are identical in both sets. Applications to covariate selection and model testing are discussed. 1