竞争风险模型competing risk model

生存时间相关研究样本的几种结局

1. censoring
2. uncensoring: (1.interesting event; 2. competing event)
   比如肿瘤相关死亡是你关心的，则为interesting event
   非肿瘤相关死亡则为competing event

什么是competing risk

in some situations more than one type of event can happen.
The occurrence of one type of event can hinder the observation or change the probability of other types of events being observed.

• the event whose occurrence either precludes the occurrence of another event under investigation or fundamentally alters the probability of occurrence of this other event.

为什么要使用competing risk model

• 如果我要研究乳腺癌相关死亡，那么非乳腺癌相关死亡就是competing risk
• 我们使用KM法分析乳腺癌相关死亡，是将非乳腺癌相关死亡认为是censor，这个就不合适！！
• 生存分析常用的方法是KM，logrank，cox，但是KM没有考虑competing risk，所以需要改进！！

• KM method can be considered an extension for calculating the probability of event in the presence of censoring.
• CIF estimation is an extension of the KM method for calculating the probability of event in the presence of competing risks.
  两种假设
• If competing risks are not present, the CIF is identical to 1-KM.
• If competing risks exist but there is no censoring, the CIF is identical to the ratio of the number of events of interest to the number of subjects.

competing risk model一些统计学知识

• density
• distribution， the survivor funtion

• the hazard function

  
  
  image.png

• there are only two types of events and ev = 1 and ev = 2 refer to events of types 1 and 2, respectively
• F(t) is an increasing function ranging between 0 and 1.
• each of the probabilities for a specific event can reach only a value p < 1. Thus the probability of one event in the presence of another event ranges between 0 and a value p <1
• It follows that F1 and F2 cannot be regarded as true, proper distributions. They are called subdistributions.

对于每个subdistributions，F1和F2， 都有一个subdensity ( f1 and f2)
根据F1和f1，可以计算subhazard和subdistribution hazard

image.png

对于subhazard，有两种情况

• the two events are independent

that the subhazard is the same as the hazard of the marginal distribution

• the two events are not independent

the subhazard is no longer the hazard of the marginal mode
对于subdistribution hazard
the analysis of the subdistribution hazard does not assume independence, and it can be interpreted as reflecting the observable effect

用什么统计学方法

The cumulative incidence function and the Fine and Gray model will be introduced as the main methods to analyze competing risks data.

CIF和KM有什么不同

相较于The cumulative incidence function ，KM会高估累计发生率

image.png

• CIF对应曲线为Nelson-Aalen累计风险曲线，差异性检验对应Gray‘s检验
• CIF包括两个模型：1. cause-specific hazard function; 2. subdistribution hazard function

the Fine and Gray model和the partial likelihood of Cox regression 的不同

• weight

The involvement of the competing risks event is mitigated by the weight: the longer the duration between the current event and the observed competing risks event, the smaller the weight

• risk set

• the set of observations with longer observed time that the current event (in COX);
• also includes all the competing risks events at all time points regardless of the time at which the competing risk was observed (in fine and gray model)
  
  image.png

为什么不用COX

• 只有在interesting event和competing risk event相互独立的情况下，用COX分析生存数据才能解释
• 但多数情况下interesting event和competing risk event并不相互独立 （这也是为什么要用fine-gray model的原因）

competing risk event对样本量的影响

marginal hazards of competing risk event 上升
probability of the event of interest to occur during the study period 下降
总的样本量 上升

• since the competing risks hinder the observation of the event of interest
• The higher the rate of competing risks, the less likely is to observe the event of interest, and therefore a larger initial sample sizes is needed.

怎么做

cmprsk包中的crr函数和cuminc函数

• cuminc--可以计算CIF
  可以得到cumulative incidence plot以及每个事件内部的比较pvalue（如果有分组的话）

• crr--计算Competing Risks Regression
  为了得到HR，95%CI，pvalue