Addendum to “a new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect”

Abstract

Viewed abstractly, the data from a longitudinal epidemiologic study consists of a string of numbers. These numbers represent for each study subject a series of empirical measurements (for example, measurement at monthly intervals of a subject's exposure level, smoking rate, employment status and vital status). In the analysis stage, calculations are performed on these strings of numbers. Based on the results of these calculations, causal inferences are drawn. For example, an investigator might conclude that the analysis provides strong evidence for "a direct effect of arsenic on lung cancer mortality controlling for the intermediate variable cigarette smoking". The nature of the relationship between the English sentences expressing the investigator's causal inferences and the computer calculations performed on the strings of numbers has been obscure. Since the numerical strings and the computer algorithm applied to them are well-defined mathematical objects, it would be important to provide formal mathematical definitions for the English sentences expressing the investigator's causal inferences that agree well with our informal intuitive understanding. In the published paper [1], we developed such a formal theory of causal inference in epidemiologic studies. In this theory causal questions that could be asked in English concerning the study population become mathematical conjectures about the population causal parameters of a causally interpreted structured tree graph (CISTG) as defined on p. 1422 of the original paper. In Ref. [1], we show that, for randomized CISTGs (as defined by equation 4.5 of Ref. [1]), a subset of the population causal parameters, i.e. the G-causal parameters, could be empirically estimated. We will show in Sections AD. 1-AD.4 of this addendum that, under certain assumptions, the G-causal parameters of certain CISTGs that are not R CISTGs can also be empirically estimated. These CISTGs differ from R CISTGs in that, in a sense described more fully below, they are randomized with respect to one outcome (e.g., mortality) but not with respect to another outcome [e.g. employment history]. In Section AD.5 we provide a formulation of the competing risks problem that differs slightly from the formulation given in Section 12 of Ref. [1]. This formulation is more in line with the approach we adopted in Ref. [1]. In Section AD.6 we develop other, alternative, comparability assumptions that again allow for the estimation of the G-causal parameters of CISTGs that are not R CISTGs. In Section AD.7, we show that when risk factors determine exposure, non-differential misclassification of exposure can lead to bias even under the null hypothesis of no causal effect of exposure on mortality. In Section AD.8 we extend our methods to studies with continuous exposure, covariate and outcome measures.