Binary Confounders as Mathematical Objects: Confounder Influence and Confounder Intervals

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.