# 第 90 章 競爭風險模型 competing risk

## 90.1 Cause-specific hazard

$h_k(t) = \lim_{\delta t \rightarrow 0} \frac{1}{\delta t}\text{Pr}\{ t\leqslant T < t + \delta t, D = k | T\geqslant t \}$

(Cumulative cause-specific hazard)累積因素別風險方程則可以被定義為：

$H_k(t) = \int_0^t h_k(s)ds$

$S_k(t) = \exp(-H_k(t))$

Overall hazard is the sum of all cause-specific hazards:

\begin{aligned} h(t) & = \sum_{e=1}^K \lim_{\delta t\rightarrow 0} \frac{1}{\delta t}\text{Pr}\{ t\leqslant T < t + \delta t, D = e | T\geqslant t \} \\ & = \lim_{\delta t \rightarrow 0} \frac{1}{\delta t}\text{Pr}\{ t\leqslant T < t + \delta t | T \geqslant t \} \end{aligned}

It follows that the overall survival can be written as useful application of this cause-specific survivor function:

$S(t) = \exp[-\sum_{e=1}^KH_e(t)] = \prod_{e = 1}^K \exp(-H_e(t))$

### 90.1.1 Cause-specific hazards models

$h_k(t|x) = h_{k, 0} (t)e^{\beta_k x}$

• People are censored at the time of any event that is not the event of interest
• We fit a separate Cox model for each event type
• $$\beta_k$$ represents the impact of $$x$$ on the hazard for event type k,** among those at risk of event type k**

## 90.2 Cumulative incidence function

Other names: absolute cause-specific risk/Crude Probabilty of event

$I_k(t) = \int_0^t h_k(s)S(s)ds$

## 90.3 Subdistribution hazard - Fine and Gray model

The approach uses an alternative definition of the hazard, called the subdistribution hazard, which represents the instantaneous risk of dying from cause k given that an individual has not already died from cause k, that is:

$h^s_k(t) = \lim_{\delta t \rightarrow 0} \frac{1}{\delta t} \{ \text{Pr}(t \leqslant T < t + \delta t, K = k | T > t \text{ or } (T \leqslant t, K \neq k)) \}$

This differs from the cause-specific hazard in its risk set; here individuals are not removed from the risk set if they die from another competing cause of death than cause k.

### 90.3.1 Subdistribution hazard model

$h_k^s(t) = -\frac{d}{dt} \log(1 - I_k(t))$

$h_k^s(t|x) = h_{0,k}(t)e^{\beta^T_k x}$

The relationsship between the CIFs in the two treatment groups is given by:

$1 - I_k(t|1) = [1 - I_k(t|0)]^{\exp(\beta_kx)}$

## 90.4 Multi-state models

### 90.4.1 The Markov model

A common assumption for multi-state mode is that upon entering a particular state i, individuals are subject to common trasition rate for movement to state j, irrespective of their history. In other words, we assume that the transition rate does not differ according to the previous states an individual has been in. This is called the Markov assumption, and is often quite a strong assumption to make.

### 90.4.2 Cox proportional hazards model for transition intensities

The transition intensities for transition i to j is given by:

$h_{ij} (t | x) = h_{ij,0} (t)\exp(\beta_{ij}^Tx)$