Nonparametric Assessment of Differences Between Competing Risk Hazard Ratios: Application to Racial Differences in Pediatric Chronic Kidney Disease Progression

Abstract

Associations between an exposure and multiple competing events are typically described by cause-specific hazard ratios (csHR) or subdistribution hazard ratios (sHR). However, diagnostic tools to assess differences between them have not been described. Under the proportionality assumption for both, it can be shown mathematically that the sHR and csHR must be equal, so reporting different time-constant sHR and csHR implies non-proportionality for at least one. We propose a simple, intuitive approach using the ratio of sHR/csHR to nonparametrically compare these metrics. In general, for the non-null case, there must be at least one event type for which the sHR and csHR differ, and the proposed diagnostic will be useful to identify these cases. Furthermore, once standard methods are used to estimate the csHR, multiplying it with our nonparametric estimate for the sHR/csHR ratio will yield estimates of sHR which fulfill intrinsic linkages of the subhazards that separate analysis may violate. In addition, for non-null cases, at least one must be time dependent (i.e., non-proportional), and thus our tool serves as an indirect test of the proportionality assumption. We applied this proposed diagnostic tool to data from a cohort of children with congenital kidney disease to describe racial differences in the time to first dialysis or first transplant and extend methods to include adjustment for socioeconomic factors.

Keywords: cause-specific hazard ratios; chronic kidney disease; competing risk analysis; nonparametric methods; sub-distribution hazard ratios; survival analysis.

Conflict of interest statement

The authors report no conflicts of interest in this work.

© 2020 Ng et al.

Figures

Figure 1

Extended nonparametric (solid) cumulative incidence functions of composite RRT (A) and first occurrence of dialysis (grey) or transplant (black) as competing events (B), by African-American (bold), and non-African-American (unbolded) participants with a pediatric diagnosis of kidney disease.

Figure 2

Results from cause-specific hazard (i.e., Cox regression) and subhazard (i.e., Fine and Gray regression) models with first dialysis (A) and first transplant (B) as competing events presenting the time-varying hazard ratios comparing African-American to non-African-American participants with a pediatric onset of kidney disease.

Figure 3

(A and B) Ratios of subhazard ratio to cause-specific hazard ratio for dialysis and transplant, respectively. (C and D) Subhazard ratios via nonparametric and classical semiparametric cause-specific hazard ratios for dialysis and transplant, respectively. For (A and B), solid lines depict the nonparametric estimator with bootstrapped 95% confidence intervals and dashed lines depict the semiparametric estimator based on the models presented in Figure 2.

Figure 4

Adjusted nonparametric competing risk analyses based on inverse probability weights controlling for socioeconomic status variables associated with race. (A) presents the extended nonparametric (solid) cumulative incidence functions of first occurrence of dialysis (grey) or transplant (black) as competing events by race. Adjusted nonparametric ratios of subhazard ratio to cause-specific hazard ratio for dialysis (grey, B) and transplant (black, C) with adjusted bootstrapped 95% confidence intervals.