Prentice et al. Respond to “Studying Co-Occurrence of Multiple Outcomes”

We thank Haneuse et al. (1) for their generous and insightful commentary on our article (2). Here, we offer additional perspective on 3 topics: namely, 1) interpretation of single- and dual-outcome hazard rates, 2) the value added by including dual-outcome analyses for mechanistic studies and clinical decision-making, and 3) multivariate failure-time outcome data displays.

First, a couple of statistical modeling points: The dual-outcome hazard rates that we consider condition only on lack of prior occurrence for the outcome pair under investigation (and covariates), giving hazard rate estimators that have a marginal population-averaged interpretation. This is in contrast to long-standing counting process intensity models that condition on the subject’s entire preceding failure-time history (3). The new methods allow censoring rates to differ among the multiple outcomes under analysis. In addition, the single- and dual-outcome hazard rates we consider implicitly condition on the continued survival of the study subject and therefore differ from those typically targeted in the semicompeting risk literature mentioned by Haneuse et al. (1) for the joint analysis of a nonterminal outcome and death. The latter literature primarily targets the marginal distribution of the nonterminal outcome under the strong additional assumption that dying individuals would never develop the nonterminal outcome had their deaths somehow been averted (4). In comparison, our marginal methods allow joint survival functions to be estimated for the 2 outcomes without further assumption, and they do so with positive dual-outcome hazard rates only when the follow-up time for death is equal to or greater than that for the nonterminal event.

The potential value of dual-outcome hazard rate analyses, beyond single-outcome analyses, includes mechanistic studies to elucidate treatment or exposure effects. For example, our dual-outcome analysis of gallbladder disease and breast cancer (2), which includes higher dual-outcome hazard ratios when the gallbladder disease precedes the breast cancer diagnosis, raises the hypothesis that poor estrogen metabolizers may be at elevated risk for gallbladder disease and may therefore not realize the breast cancer benefits that would otherwise accrue from conjugated equine estrogens alone. As noted by Haneuse et al. (1), these analyses may also contribute to clinical decision-making, which routinely requires consideration of multiple outcomes simultaneously. For example, our analyses of the dual outcome of hypertension and death (2) may suggest evaluating continuation of conjugated equine estrogens plus medroxyprogesterone acetate among patients undergoing treatment who develop hypertension. Similarly, dual-outcome analyses allowed us to conclude, using intention-to-treat analyses between randomized groups, that a previously noted hazard ratio reduction in a low-fat diet intervention group (6) for the dual outcome of breast cancer followed by death was primarily due to survival improvement following breast cancer diagnosis (5).

With regard to multivariate outcome data displays: Kaplan-Meier curves, cumulative hazard rate estimators, and Cox regression hazard rate and ratio estimators provide major tools for the display of univariate failure-time regression data. These same tools can be considered for bivariate failure-time regression data. For example, single- and dual-outcome empirical hazard rates combine to estimate bivariate survival function estimators (5, 7), and cumulative single- and dual-outcome hazard rate estimators are readily obtained. These hazard rate estimators are appropriate even when the modeled covariate includes “internal” time-dependent covariates. Single- and dual-outcome hazard ratio displays include the usual single failure-time hazard ratios across treatments/covariates, as well as dual-outcome “platter plots,” as shown in our paper (2). It will be valuable to develop additional displays that incorporate estimated absolute hazard rate comparisons. In addition, in some contexts patients may frequently experience several outcomes, which supports the value of hazard ratio modeling for trivariate and higher-dimensional time-to-response outcomes (5, 7).