The paper relates estimation and testing for additive interaction in proportional hazards models to causal interactions within the counterfactual framework. A definition of a causal interaction for time-to-event outcomes is given that generalizes existing definitions for dichotomous outcomes. Conditions are given concerning the relative excess risk due to interaction in proportional hazards models that imply the presence of a causal interaction at some point in time. Further results are given that allow for assessing the range of times and baseline survival probabilities for which parameter estimates indicate that a causal interaction is present, and for deriving lower bounds on the prevalence of such causal interactions. An interesting feature of the time-to-event setting is that causal interactions can disappear as time progresses, ie, whether a causal interaction is present depends on the follow-up time. The results are illustrated by hypothetical and data analysis examples.
From the Departments of aEpidemiology and bBiostatistics, Harvard School of Public Health, Boston, MA.
Submitted 23 November 2010; accepted 16 February 2011; posted 10 May 2011.
Supported by National Institutes of Health (grant ES017876).
Correspondence: Tyler J. VanderWeele, Departments of Epidemiology and Biostatistics, Harvard School of Public Health, 677 Huntington Ave, Boston, MA 02115. E-mail: email@example.com.