Abstract
In competing risks setting, we account for death according to a specific cause and the quantities of interest are usually the cause‐specific hazards (CSHs) and the cause‐specific cumulative probabilities. A cause‐specific cumulative probability can be obtained with a combination of the CSHs or via the subdistribution hazard. Here, we modeled the CSH with flexible hazard‐based regression models using B‐splines for the baseline hazard and time‐dependent (TD) effects. We derived the variance of the cause‐specific cumulative probabilities at the population level using the multivariate delta method and showed how we could easily quantify the impact of a covariate on the cumulative probability scale using covariate‐adjusted cause‐specific cumulative probabilities and their difference. We conducted a simulation study to evaluate the performance of this approach in its ability to estimate the cumulative probabilities using different functions for the cause‐specific log baseline hazard and with or without a TD effect. In the scenario with TD effect, we tested both well‐specified and misspecified models. We showed that the flexible regression models perform nearly as well as the nonparametric method, if we allow enough flexibility for the baseline hazards. Moreover, neglecting the TD effect hardly affects the cumulative probabilities estimates of the whole population but impacts them in the various subgroups. We illustrated our approach using data from people diagnosed with monoclonal gammopathy of undetermined significance and provided the R‐code to derive those quantities, as an extension of the R‐package mexhaz.
Highlights
IntroductionThe one-to-one relationship between the risk of an event (probability scale) and the rate at which the event occurs (hazard scale) is well known when studying a single event/cause
In survival analysis, the one-to-one relationship between the risk of an event and the rate at which the event occurs is well known when studying a single event/cause
The relative bias and the RMSEs were very low, the model standard errors (ModSE) were quite close to the empirical ones, and the majority of the coverage probabilities were within the acceptable coverage probability range ([0.931, 0.969])[30]
Summary
The one-to-one relationship between the risk of an event (probability scale) and the rate at which the event occurs (hazard scale) is well known when studying a single event/cause. This is a key feature in hazard regression models in order to examine how covariates affect the survival probability.[1] assuming that the survival time t of an Statistics in Medicine. Individual can be described by a positive random variable T with probability density function f, the cumulative distribution function F is defined as F(t) = P(T ≤ t) = ∫0t f (u)du = ∫0t λ(u)S(u)du, where S(t) = P(T > t) = 1 − F(t) is the survival probability and λ(t) is the hazard function. In the competing cause-specific hazards (CSHs) rλi(stk)s=se∑ttiJjn=g1 where more λj(t), where than one the cause cause are acting, the (total) hazard is j-specific hazard λj represents the rate the sum of failure of all from cause j per time unit for individuals who are still at risk.[2,3] The cumulative probability of dying from a particular cause until time t in the presence of all other causes ( called the cumulative incidence function) depends on all the CSHs t
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