A distinctive feature of survival data is the concept of censoring. And an implicit concept in the definition of censoring is that if the study had been prolonged (or if subjects had not dropped out), eventually the outcome of interest should be observed to occur, for all the subjects. Conventional statistical methods for the analysis of survival data make the important assumption of independent or non-informative censoring. This means that at a given point in time, subjects who remain under follow-up have the same future risk for the occurrence of the event as those subjects will no longer be followed (either because of censoring or study dropout), as if losses to follow-up were random and thus non-informative.
A competing risk is an event whose occurrence precludes, the occurrence of the primary event of interest. For instance, in a study in which the primary outcome was time to death due to a cardiovascular cause, a death due to a non-cardiovascular serves as a competing risk.
Conventional statistical methods for the analysis of survival data assume that competing risk are absent. Two competing risks are said to be independent if information about a subject’s risk of experiencing one type of event provides no information about the subject’s risk of experiencing the other type of event. The methods that will be described later on will involve impeding risks which are independent of one another and further also in which competing risks are not independent of one another.
In biomedical applications, the biology often suggests at least some dependence between competing risks, which in many cases may be quite strong. Accordingly, independent competing risks may be relatively rare in biomedical applications.
When analyzing survival data in which competing risks are present, analysts frequently censor subjects when a competing event occurs. Thus, when the outcome is time to death attributable to cardiovascular causes, an analyst may consider a subject as censored once that subject dies of noncardiovascular causes. However, censoring subjects at the time of death attributable to noncardiovascular causes may be problematic.
First, it may violate the assumption of noninformative censoring: it may be unreasonable to assume that subjects who died of noncardiovascular causes (and were thus treated as censored) can be represented by those subjects who remained alive and had not yet died of any cause.
Second, even when the competing events are independent, censoring subjects at the time of the occurrence of a competing event may lead to incorrect conclusions because the event probability being estimated is interpreted as occurring in a setting where the censoring (eg, the competing events) does not occur.
In the cardiovascular example described above, this corresponds to a setting where death from noncardiovascular causes is not a possibility. Although such probabilities may be of theoretical interest, they are of questionable relevance in many practical applications, and generally lead to overestimation of the cumulative incidence of an event in the presence of the competing events.
Suppose, the baseline time in the cohort is well defined and that T denotes the time from baseline time until the occurrence of the event of interest. In the absence of competing risks, the survival function,
\[ S(t) = Pr[T>t] \]
describes the distribution of event times. One minus the survival function (ie, the complement of the survival function),
\[F(t)= 1 -S(t) = Pr[T\leq t]\] describes the incidence of the event over the duration of follow-up.
Two key properties of the survival function being
\[S(0) = 1\]
, i.e. at the beginning of the study, the event has not yet occurred for any subjects,
\[ \underset{t \rightarrow \infty}{lim} S(t) = 0 \]
, i.e. eventually the event of interest occurs for all subjects
In the absence of competing risks, F(t) may be a good measure when describing the incidence of occurrence of event(deaths). However, in the presence of competing risks, the deaths due to competing risks are treated as censored observations(most of the times, loss during to follow up) , as a result an absurd assumption is taken into consideration that the event (death) occurs ultimately due to the primary cause. This results to a heavy bias in estimation.
To do away with this demerit, the Cumulative Incidence Function which allows for estimation of the incidence of the occurrence of an event while taking competing risk into account, is a much better measure. The cumulative incidence function for the kth cause is defined as:
\[ CIF_k(t) = Pr[T\leq t , D=k] \] ; where D is a random variable denoting the type of event that occured. A key point is that, in the competing risks setting, only 1 event type can occur, such that the occurrence of 1 event precludes the subsequent occurrence of other event types. The function \(CIFk(t)\) denotes the probability of experiencing the kth event before time t and before the occurrence of a different type of event. The CIF has the desirable property that the sum of the CIF estimates of the incidence of each of the individual outcomes will equal the CIF estimates of the incidence of the composite outcome consisting of all of the competing events. Unlike the survival function in the absence of competing risks, CIFk(t) will not necessarily approach unity as time becomes large, because of the occurrence of competing events that preclude the occurrence of events of type k.
The hazard function, describes the instantaneous rate of occurrence of the event of interest in subjects who are still at risk of the event. In a setting in which the outcome was all-cause mortality, the hazard function at a given point in time would describe the instantaneous rate of death in subjects who were alive at that point in time.
The Cox proportional hazards regression model relates the hazard function to a set of covariates. In the absence of competing events, the Cox proportional hazards regression model can be written as:
\[ \begin{eqnarray*} h(t)=h_0(t)e^{X\beta} \end{eqnarray*} \]
;where \(h_0(t)\) denotes the baseline hazard function
The Cox model relates the covariates to the hazard function of the outcome of interest (and not directly to the survival times themselves). The covariates have a relative effect on the hazard function because of the use of the logarithmic transformation. The regression coefficients are interpreted as log-hazard ratios. The hazard ratio is equal to the exponential of the associated regression coefficient. The hazard ratio denotes the relative change in the hazard function associated with a 1-unit increase in the predictor variable. Although the regression coefficients from the Cox model describe the relative effect of the covariates on the hazard of the occurrence of the outcome, the following relationship also holds in the absence of competing risks:
\[ S(t)=S_0(t)e^{X\beta} \]
where \(S_0(t)\) denotes the baseline survival function.
Thus, the relative effect of a given covariate on the hazard of the outcome is equal to the relative effect of that covariate on the logarithm of the survival function. Therefore, in the absence of competing risks, making inferences about the effect of a covariate on the hazard function permits one to make equivalent inferences about the effect of that covariate on prognosis or survival.
This direct correspondence between the hazard function and incidence in the absence of competing risks may be used to conclude that a given risk factor or variable increased the risk of an event, without specifying whether risk denoted the hazard of an event (ie, the rate of the occurrence of the event in those still at risk of the event) or the incidence of the event (ie, the probability of the occurrence of the event).
Competing risks implies that a subject can experience one of a set of different events or outcomes. In this case, two different types of hazard functions are of interest:
the cause-specific hazard function and
the subdistribution hazard function.
The cause-specific hazard function denotes the instantaneous rate of occurrence of the kth event in subjects who are currently event free (ie, in subjects who have not yet experienced any of the different types of events). It is defined as:
\[ h^{CS}_k(t) = \underset{\Delta t \rightarrow 0}{lim} \frac{Pr \left[ t \leq T \leq t+\Delta t , D = k | T \geq t \right]}{\Delta t} \]
It denotes the instantaneous risk of failure from the kth event in subjects who have not yet experienced an event of type k. Note that this risk set includes those who are currently event free as well as those who have previously experienced a competing event. This differs from the risk set for the cause-specific hazard function, which only includes those who are currently event free. It is defined as:
\[ h^{SD}_k(t) = \underset{\Delta t \rightarrow 0}{lim} \frac{Pr \left[ t \leq T \leq t+\Delta t , D = k | T > t \cup (T< t \cap K \neq k) \right]}{\Delta t} \]
In settings in which competing risks are present, two different hazard regression models are available: modeling the cause-specific hazard and modeling the subdistribution hazard function. Both models account for competing risks, but do so by modeling the effect of covariates on different hazard functions. Consequently, each model has its unique interpretation. We refer to these two models as cause-specific hazard models and subdistribution hazard models. The second model has also been described as a CIF regression model. The latter name makes explicit the link between the subdistribution hazard and the effect on the incidence of an event. That is, one may directly predict the cumulative incidence for an event of interest using the usual relationship between the hazard and the incidence function under the proportional hazards model. Thus, the subdistribution hazard model allows one to estimate the effect of covariates on the cumulative incidence function for the event of interest.
If \(x_1,x_2,\dots,x_n\) is a random sample of size ‘n’ drawn from a population with density function $f(x;\theta)$. The likelihood function of the sample values \(x_1,\dots,x_n\) , id given by:
\[ L(\theta) = L(\theta;x_1,\dots,x_n) = \prod_{i=1}^n f(x_i ; \theta) \]
The principle of Maximum Likelihood consists in finding an estimator for the unknown parameters $\theta = (\theta_1,\dots,\theta_k)’$ which maximizes the likelihod function $L(\theta)$ for variation in parameters.
That is, if
\[ \hat{\theta} = (\hat{\theta}_1,\dots,\hat{\theta}_n)\] is an MLE of $\theta$ then,
\[ L(\hat{\theta}) = \underset{\theta}{min} \quad L(\theta) \]
The method of minimum chi-square makes the use of the Pearsonian chi-squre statistic.
Suppose, $f_1,\dots,f_k$ be the observed frequencies in the groups or classes and \(p_1,\dots,p_k\) are the unknown probabilities the \(f_1,\dots,f_k\) elements belong to the respective groups or classes. We assume that \(p_i\)’s are the functions of the unknown parameters \((\theta_1,\dots,\theta_n)\) . Thus,
\[ p_i = p_i(\theta) \]
;where \(\theta = (\theta_1,\dots,\theta_n)\)
Suppose the total sample size is \(N = \sum_i f_i\) , the expected frequencies are \(Np_i, Np_2, \dots , Np_k\) . We know, the Pearsonian chi-square statistic,
\[ \begin{eqnarray*} \chi^2 &=& \sum_{i=1}^{k} \frac{(f_i - Np_i)^2}{Np_i} \\ &=& \sum_{i=1}^{k} \frac{f_i^2}{Np_i} - N \end{eqnarray*} \]
If \(\hat{\theta}\) is an estimator obtained by the method of minimum chi-square estimator of \(\theta\) , then
\[ \chi^2 (\hat{\theta}) = \underset{\theta}{min } \quad \chi^2 (\theta) \]
In the method of minimum chi-square estiimation, both the numerator and the denominator in the chi-square statistic involves the parameter \(\theta\), due to which it mostly takes a complicated form. The modified method introduces the \(f_i\) in place of the expected value in the denominator, i.e
\[ \chi'^2(\theta) = \sum_{i=1}^{k} \frac{\left(f_i - Np_i \right)^2}{f_i} = \sum_{i=1}^{k} \frac{N^2p_i^2}{f_i} - N \]
If \(\hat{\theta}\) is the estimator obtained by the modified method of minimum chi-square estimator of \(\theta\) , then
\[ \chi'^2 (\hat{\theta}) = \underset{\theta}{min } \quad \chi'^2 (\theta) \]
Introduction to the Analysis of Survival Data in the Presence of Competing Risks – Austin, Lee & Fine
Statistical Inference - Theory of Estimation - Srivastava, Khan and Srivastava.