01.12.2010 | Research article | Ausgabe 1/2010 Open Access

# Methods of competing risks analysis of end-stage renal disease and mortality among people with diabetes

- Zeitschrift:
- BMC Medical Research Methodology > Ausgabe 1/2010

## Electronic supplementary material

## Competing interests

## Authors' contributions

## 1. Background

_{ KM }(t)}, has been used in some research. However, studies have demonstrated that {1 - S

_{ KM }(t)} is inappropriate because it overestimates the probability of occurrence of the event of interest [7–12]. The bias is especially great when the hazard of the competing events is large [13]. An alternative method to the inappropriate cumulative hazard function is Cox cause-specific hazard [14] and cumulative incidence functions (CIF), which are the most important approaches to analyse competing risks data [12]. The cause-specific hazard measures the instantaneous failure rate due to one risk at a time. It is routinely estimated by constructing the Cox models on cause-specific hazards and treating time to event from the other competing risks as censored [11, 12]. For each risk, the effects of prognostic factors are assessed as constant hazards ratios on the instantaneous failure rate of this risk. The CIF is an important quantity related to one risk in the context of competing risks. The CIF curve provides a better incidence curve associated with one risk than {1 - S

_{ KM }(t)}. It also provides a meaningful interpretation in terms of failure due to one risk regardless of whether competing risks are independent. Comparing the CIF curves is analogous to the log-rank test and is identical to the log-rank test in the absence of competing risks [15]. Gray considered a class of K-sample tests for the cumulative incidence based on weighted averages of subdistribution hazard functions [15]. Such tests do not require the independence assumption and does not adjust for other covariates.

## 2. Study Description

### 2.1. Clinical Background

### 2.2 Study Population

## 3. Models

### 3.1. Standard single event time model

_{ 0 }(t) is an unspecified baseline hazard function and gives the shape of the hazard function. If all explanatory covariates are zero, the hazard function will be the baseline hazard h

_{ 0 }(t). If two individuals have identical values of the measured covariates, they will have identical hazard functions. The cumulative hazard function given zis defined by $\text{\Lambda}(t;\mathit{z})\phantom{\rule{0.25em}{0ex}}={\text{\Lambda}}_{0}(t)\phantom{\rule{0.50em}{0ex}}{e}^{{\beta}^{\text{'}}\phantom{\rule{0.25em}{0ex}}Z}$, where Λ

_{0}(t) is the cumulative baseline hazard and ${\text{\Lambda}}_{0}(t)={\displaystyle \underset{0}{\overset{t}{\int}}{h}_{0}\text{(u)}du}$. The survival function is then obtained from the cumulative hazard function such that S(t) = exp{- Λ(t; z ) }.

### 3.2. Models on cause-specific hazards

_{ k }(t; z ) = h

_{ 0k }(t) e

^{ β Z }.

_{ k }(t; z ) = exp{- Λ

_{ k }(t; z )}. Although we can estimate S

_{ k }(t; z ) from the cause-k specific cumulative hazard, exp{- Λ

_{ k }(t; z )} is not interpretable as the marginal survival function for cause-k specific alone [12]. Instead S

_{ k }(t; z ) is the survival probability for the k

^{th}risk if all other risks were hypothetically removed.

_{ k }(t), is defined by the probability of failing from cause k,

_{ k }(t; z ) are the adjusted overall survival and cumulative hazard based on certain types of cause-specific hazard regression models [12]. This expression shows that the cumulative incidence of a specific cause k is a function of both the probability of not having the event prior to another event first (S(u)) up to time t and the cause-specific hazard (h

_{ k }(u)) for the event of interest at that time [7, 8, 12]. Estimation of the CIF can be obtained by using the cause-specific hazard.

### 3.3. Model on a subdistribution hazards

_{ k }(t; z), which expresses the effect of covariates directly on the CIF. This is done via the subdistribution hazard function h*

_{ k }(t; z):

_{ k }(t; z) is not the cause-specific hazard. The CIF for cause k not only depends on the hazard of cause k, but also on the hazards of all other causes. For this approach, the subdistribution hazard is also defined as

_{ k }(t; z), the risk set decreases at each time point at which there is an event of another cause. For the subdistribution hazard, h*

_{ k }(t; z), a person who has an event from another cause remains in the risk set [16]. In our study, we have applied the Cox models on the cumulative incidences of ESRD and death without ESRD, and have determined the subdistribution hazards ratios.

## 4. Results

Competing Risk | Model | Covariate | H.R | 95% C.I | p-value |
---|---|---|---|---|---|

ESRD | Male | 1.513 | 1.146 - 1.998 | 0.004 | |

Cox | |||||

cause-specific | Age < 40 | - | - | - | |

hazards model | 40 < Age < 6 | 1.149 | 0.849 - 1.554 | 0.368 | |

60 < Age | 1.405 | 0.891 - 2.215 | 0.144 | ||

----------- | ----------- | ------ | ----------- | ----- | |

Cox | Male | 1.323 | 1.006 - 1.742 | 0.045 | |

subdistribution | |||||

hazards model | Age < 40 | - | - | - | |

40 < Age < 60 | 0.923 | 0.685 - 1.243 | 0.6 | ||

60 < Age | 0.53 | 0.335 - 0.838 | 0.007 | ||

Death without ESRD | |||||

Cox | Male | 1.377 | 1.243 - 1.525 | < 0.0001 | |

cause-specific | |||||

hazards model | Age < 40 | - | - | - | |

40 < Age < 60 | 2.68 | 2.267 - 3.158 | < 0.0001 | ||

60 < Age | 10.23 | 8.653 - 12.09 | < 0.0001 | ||

------------ | ------------ | ------ | ------------ | ------ | |

Cox | Male | 1.36 | 1.226 - 1.498 | < 0.0001 | |

subdistribution | |||||

hazards model | Age < 40 | - | - | - | |

40 < Age < 60 | 2.65 | 2.248 - 3.126 | < 0.0001 | ||

60 < Age | 9.96 | 8.453 - 11.74 | < 0.0001 |

Competing Risk | Covariate | H.R | 95% C.I | p-value |
---|---|---|---|---|

ESRD | Male _{ESRD}
| 1.395 | 1.057 - 1.84 | 0.0186 |

Age _{(ESRD)} < 40 | - | - | - | |

40 < Age _{(ESRD)} < 60 | 1.078 | 0.797 - 1.457 | 0.627 | |

60 < Age _{(ESRD)}
| 1.004 | 0.641 - 1.574 | 0.985 | |

------------------------- | ------------ | ----------------- | ---------------- | |

Death without ESRD | Risk type * | 2.44 | 1.788 - 3.328 | < 0.0001 |

Male _{(death)}
| 1.40 | 1.264 - 1.55 | < 0.0001 | |

Age _{(death)} < 40 | - | - | - | |

40 < Age _{(death)} < 60 | 2.708 | 2.291 - 3.202 | < 0.0001 | |

60 < Age _{(death)}
| 10.73 | 9.078 - 12.68 | < 0.0001 |

## 5. Discussion

_{ k }(u)) for the event of interest at that time [7, 8, 12].

^{ β }represents the increase of the hazard of the subdistribution due to one unit increase of z. However, the cause-specific models

_{ KM }(t)} or by plotting residuals (Cox-Snell , Martingale, or Deviance residuals) or by adding time-dependent covariates in the model [12, 35, 46]. For the Cox cause-specific hazard model, the statistical software is available in many commercial statistical software packages and makes it easy to fit the models. However, for the sub distribution hazard models, currently standard procedure is not available in SAS, but SAS macros [47], STATA with compet.adoor R-package cmprskare available.