The hazards of the changing hazard of dialysis modalities Christos P. Argyropoulos, Mark L. Unruh  Kidney International  Volume 86, Issue 5, Pages 884-887 (November 2014) DOI: 10.1038/ki.2014.249 Copyright © 2014 International Society of Nephrology Terms and Conditions

Figure 1 Survival analysis under non-proportional hazards. (a) Mathematical interrelationships in survival analysis: The survival function (S(t)) is equal to the negative exponent of the cumulative hazard function (H(t)). The instantaneous hazard function (h(t)) is the negative logarithmic derivative of the survival function. The integral (area under the curve) from t=0 to t=t of the instantaneous hazard function is also equal to the cumulative hazard function. Under non-proportional hazards, the ratio of the cumulative hazard function of two treatments A and B is equal to the cumulative hazard ratio (CHR(t)). The CHR(t) at any given time point may also be used to relate the value of the survival functions for two treatments A and B. (b) Hypothetical analysis of two treatments A and B (thick black and red lines) under non-proportional hazards. The instantaneous hazard ratio (green line) shows what appears to be a rapidly decreasing benefit of A versus B. Its value exceeds parity, at a point in time in which the two survival curves still indicate a benefit of therapy A versus B. Hence in the case of non-proportional hazards, the instantaneous hazard ratio may lead to misleading conclusions about the benefit of an intervention. On the other hand, the cumulative hazard ratio (CHR, blue line) provides an estimate of the cumulative treatment effect that is consistent with that afforded by the inspection of the survival curves. The value of the CHR remains below 1, until the point at which the two survival curves cross (circle, dashed lines). In the case of non-proportional hazards, one may be tempted to fit Cox models by censoring the follow-up time at progressively increasing intervals. Such an approach will yield average values of the hazard ratio (purple line) that will closely track the value of the CHR. Nevertheless, this approach will overestimate the protective effect of therapy A and will not be consistent with the survival curves. (c) Frailty and interindividual variability in treatment effects. The conventional assumption behind survival analysis is that all individuals have the same (adjusted) survival function. A more realistic approach would be to assume that each individual has his or her own survival curve (dashed lines) as a result either of specific characteristics or of treatment responsiveness. Hence, the population survival curve (or hazard ratio) may not be representative for the individual survival curve. Modeling outcome heterogeneity should be strongly considered in order to establish whether the population response is representative of the outcomes at the individual patient level. Kidney International 2014 86, 884-887DOI: (10.1038/ki.2014.249) Copyright © 2014 International Society of Nephrology Terms and Conditions