A causal framework for classical statistical estimands in failure-time settings with competing events

Abstract

In failure-time settings, a competing event is any event that makes it impossible for the event of interest to occur. For example, cardiovascular disease death is a competing event for prostate cancer death because an individual cannot die of prostate cancer once he has died of cardiovascular disease. Various statistical estimands have been defined as possible targets of inference in the classical competing risks literature. Many reviews have described these statistical estimands and their estimating procedures with recommendations about their use. However, this previous work has not used a formal framework for characterizing causal effects and their identifying conditions, which makes it difficult to interpret effect estimates and assess recommendations regarding analytic choices. Here we use a counterfactual framework to explicitly define each of these classical estimands. We clarify that, depending on whether competing events are defined as censoring events, contrasts of risks can define a total effect of the treatment on the event of interest or a direct effect of the treatment on the event of interest not mediated by the competing event. In contrast, regardless of whether competing events are defined as censoring events, counterfactual hazard contrasts cannot generally be interpreted as causal effects. We illustrate how identifying assumptions for all of these counterfactual estimands can be represented in causal diagrams, in which competing events are depicted as time-varying covariates. We present an application of these ideas to data from a randomized trial designed to estimate the effect of estrogen therapy on prostate cancer mortality.

Keywords: causal inference; competing risks; g-formula; inverse probability weighting; longitudinal data; survival analysis.

Figure 8:

A SWIG template $\mathcal{G}(a,\overline{c},\overline{d})$ derived from the causal DAG in Figure 1 which implies exchangeability (20) holds.

Figure 9:

A SWIG template $\mathcal{G}(a,\overline{c},\overline{d})$ derived from the causal DAG in Figure 2 which implies exchangeability (20) fails.

Figure 10:

A SWIG template $\mathcal{G}(a,\overline{c})$ derived from the causal DAG in Figure 2 which implies exchangeability (26) holds.

Figure 1:

A causal DAG representing observed data generating assumptions under which (i) competing events may mediate the effect of treatment A on the event of interest and (ii) exchangeability (20) holds such that the direct effect (6) may be identified.

Figure 2:

A causal DAG representing observed data generating assumptions under which exchangeability (26) holds and the total effect of treatment A on the event of interest (8) may be identified.

Figure 3:

Graphical illustration of why contrasts in counterfactual hazard ratios may not have a causal interpretation even under conditions that give identification of contrasts in any estimand in Table 1 under different levels of a and $\overline{c}=0$.

Figure 4:

Parametric g-formula estimates (38), IPW estimates (39) (IPW sub) and IPW estimates (40) (IPWcs) of the the risk of prostate cancer death without elimination of competing events by follow-up month k + 1 = 1,…60 under high-dose estrogen therapy (DES) versus placebo.

Figure 5:

Parametric g-formula estimates (61), IPW estimates (62) (IPWcs) and IPW estimates of (39) but replacing the event of interest with the competing event (IPW sub) of the risk of other causes of death by follow-up month k+1 = 1,…60 under high-dose estrogen therapy (DES) versus placebo.

Figure 6:

Parametric g-formula and IPW estimates of the risk of the composite outcome (based on sums of corresponding estimates of the risk of prostate cancer death without elimination of competing events and risk of other causes of death) by follow-up month k + 1 = 1,…60 under high-dose estrogen therapy (DES) versus placebo.

Figure 7:

Parametric g-formula estimates (36) and IPW estimates (37) of the risk of prostate cancer death under elimination of competing events by follow-up month k + 1 = 1,…60 under high-dose estrogen therapy (DES) versus placebo.