Non-proportionality of hazards in the competing risks framework

The simplest means of determining the effect of an exposure on the frequency and timing of two competing events is to contrast the cumulative incidences between the exposed and unexposed groups for each event type.Methods and software are widely available to semi-parametrically model the sub-hazards of the cumulative incidences as proportional and to test whether the constant relative sub-hazards (a₁ and a₂) are different from 1. In this chapter, we show that a₁ and a₂ are tethered by a strong relationship which is independent of the timing of the competing events; the relationship is fully determined by the overall frequencies of events, and a₁ and a₂ must be on opposite sides of 1. When violations of proportionality occur, separate analyses for the two competing events often yield an inadmissible result in which the estimates of a₁ and a₂ are on the same side of 1, and may even exhibit statistical significance. We further characterize the compatibility of concurrent proportionality of cause-specific hazards and sub-hazards, and show that strong tethering also occurs among these quantities; and that, of the sub-hazards and cause-specific hazards, at most two of the four can be proportional, but without restriction on which two. Because proportionality rarely holds in practice, the default analytical approach should allow the relative hazards to depend on time, which can be easily carried out with widely available software. However, the statistical power of this approach is limited in the case of large numbers of eventfree observations. An application using data from a North American cohort study of children with kidney disease is presented.