1. Introduction

The survival analysis is a collection of statistical methods for analyzing time-to-event data. The commencement of the survival analysis dates back to the 18th century when analyses of mortality experience of human populations started. During the World War II, the survival analysis focused on engineering – reliability of military equipment was being analyzed. After the World War II the interest turned towards economics and medicine. In 1960s, after the fundamental article of E. L. Kaplan and P. Meier [9] had been published, medical applications of the survival analysis shifted to the center of statistical focus.

2. Basic concepts in the survival analysis

2.1 The survival and hazard function

•  The probability density of X: f(x), x≥0.
  

•  The survival function:

•   The cumulative hazard function:

Thus

the hazard x₁ at is

where S(x^-)=lim_t→x^-S(t). The survival function and probability mass function can be also written as (see [8])

and the cumulative hazard function is

Let dΛ(x) be a differential increment of the cumulative hazard over the interval [x,x+dx):

The survival function in the discrete, continuous, or mixed cases can then be written as

is the product integral [8].

2.2 Censoring and truncation

Survival data possess a special feature of censoring, compared to other statistical data. Censoring is used when the survival time is not known exactly, the event is only known to have occurred within some time interval. There are several types of censoring: right, left and interval. In biomedical applications, right censoring is the most common type of censoring. It occurs when the survival time is incomplete on the right-hand side of the follow-up period, i.e. the study ends before all patients experience the event or a patient is lost to follow-up (dies due to reasons other than the event of interest, withdraws from the study, moves to another city, etc.).

Let X₁,X₂,...,X_n be independent and identically distributed (i.i.d.) survival times and C₁,C₂,...C_n be i.i.d. censoring times. The lifetime X_i of the i-th individual will be known if, and only if, X_i<C_i. If C_i<X_i the event time will be censored at C_i. Thus it is convenient to represent the survival experience of a group of patients by the pairs of random variables  (T_i, δ_i), where T_i=min(X_i,C_i), δ_i=I(X_i<C_i), and I is an indicator of the event's occurring, having value one if the event occurs, and zero otherwise.

Another feature, common in survival data, is a truncation. Truncation occurs when only those individuals whose event time lies within a certain time interval (T_L, T_R) are observed. For left truncation, T_R=∞, in case of right truncation, T_L=0. Individuals, whose event time is not in this interval, are not observed and no information on these subjects is available. This is in contrast to censoring where there is at least partial information available on each patient. When data are truncated, a conditional distribution has to be used in constructing the likelihood (see [10]).

where Y(t)=1 indicates that the individual is at risk of failure at time t (has neither failed nor been censored prior to t).

2.3 Counting processes and martingales

N_i(t) is a counting process, while Y_i(t) is a predictable process, i.e. a process whose value at time t is known infinitesimally before t, at time t^-. This process has left-continuous sample paths. Right-censored survival data are included in this formulation as a special case: N_i(t)=I(T_i≤t,δ_i=1) and Y_i(t)=I(T_i≥t).

where Y_i(s⁺)=lim_u→s+Y_i(u). Thus F_t contains the information up to and including time t. The information in F_t increases with increasing time on study, i.e. for s≤t [4]. Let dN_i(t) denote the increment of N_i over the time interval [t,t+dt).

For each t>0, let

denote the full history of the processes N_i(s),Y_i(s),i=1,...,n, up to but not including t. Then (see [4]):

where λ_i(t) is the hazard function. The process

is called the intensity process. At each fixed t, this process is a random variable which approximates the number of jumps by N_i over (0,t]. In fact, EN_i(t)=EΛ_i(t) and thus

For any given i, define the process

(1)

where dM_i(t)=dN_i(t)-Y_i(t)λ_i(t)dt.

It can be seen that

for all t and

(2)

The approach using martingale methods is very useful in yielding results for censored and truncated data, especially for calculating and verifying asymptotic properties of test statistics and estimators.

3. Non-parametric and semi-parametric models

when a continuous distribution is estimated by a discrete one. For an uncensored sample of n distinct failure times, the empirical survival function is then estimated by S_n(t)=1–F_n(t). The only problem with this approach is the censoring – it is not taken into account in standard statistical methods. Important steps in the development of appropriate methods were done by Kaplan and Meier [9] and Cox [3].

3.1 Kaplan-Meier and Nelson-Aalen estimators

The product-limit estimator can also be used to estimate the cumulative hazard function:

 

Based on the Nelson-Aalen estimator of the cumulative hazard function, an alternative estimator of the survival function becomes

 

A common interest is to compare two or more samples, i.e. to test whether there is a significant difference in survival experience of distinct groups of patients. Several generalizations of standard non-parametric tests have been developed to deal with censored and truncated data. The most common tests are the log-rank test, Gehan-Wilcoxon test and Peto-Peto test. For more information, see e.g. [10].

3.2 The Cox model

where t₁<...<t_k are the uncensored failure times of the study group, R(t_j) is the set of subjects at risk of failure at time t^-_j (just prior to time t_j), and X_j denotes the covariate vector for an individual failing at tj. The partial likelihood function is treated as a standard likelihood, and inference is carried out by usual means.

4. Multivariate survival analysis

In most clinical applications the univariate survival analysis assumes that the observed survival times are mutually independent (i.i.d. failure times). In practice, however, dependence can occur for very different kinds of data, e.g. survival of twins or other several individuals, similar organs, recurrent events or multi-state events. The multivariate survival analysis covers the field where independence between survival times cannot be assumed. According to [7], the various approaches to analyzing multivariate survival data fall into four main categories: multi-state models, frailty models, marginal modeling and non-parametric methods. The data structure should be considered as well. The data can be parallel (where the number of failures is fixed by the design of the study) or longitudinal (where the number of failures is random for each object under study). The data sets are classified into six types: several individuals, similar organs, recurrent events, repeated measurements, different events and competing risks. Relation of the data types to the two main approaches of analysis (multi-state and frailty models) is described in Table 1. Only these two approaches to analyzing multivariate survival data are presented in this paper. For more information on marginal and non-parametric methods, see e.g. [7], [16], or [8].

Tab. 1. Overview of data types and approaches; x means relevant, blank not relevant. Adopted from [7].

4.1 Competing risk and multi-state models

Multi-state models are commonly used for describing the development of longitudinal data. They model stochastic processes, which at any time point occupy one of a set of discrete states. In medicine, the states can be e.g. healthy, diseased, and dead. A change of state is called a transition. The competing risk model is an example of multi-state modeling. In competing risks, various causes of death "compete" in the life of patient, and occurrence of one event precludes occurrence of the other events. There are generally three areas of interest in the analysis of competing risks [8]:

1. Studying the relationship between a vector of covariates and the rate of occurrence of specific types of failure.

2. Analyzing whether patients at high risk of one type of failure are also at high risk for others.

3. Estimating the risk of one type of failure after removing others.

To model competing risks, a cause-specific hazard function is considered:

It is possible to calculate the Kaplan-Meier estimator for each type of failure separately, but it is difficult to give this a survival function interpretation and therefore this is not recommended [7]. Instead, generalizations of the Kaplan-Meier and Nelson-Aalen estimators can be made (see e.g. [8]). The generalized estimator includes all causes of failure and is usually denoted the Aalen-Johanson estimator.

Both the baseline hazards λ_0j and the regression coefficients β_j vary arbitrarily over the m failure types. Estimation and comparison of the coefficients  β_j can be conducted by applying asymptotic likelihood techniques individually to the m factors.

which holds for all A(u),0≤u<t with A(t^-)=i and i,j{1,...,m}, j≠i. The process is memoryless in that only the current state occupied is relevant in specifying the transition rates [8]. In the continuous case, dΛ_ij(t)=λ_ij(t)dt for all i,j=1,...,m, so that λ_ij(t),i≠j is the continuous-time intensity function for i-to-j transitions. Estimation of the cumulative intensity functions Λ_ij(t) proceeds as follows [8]: consider a possibly right-censored sample of n individuals. For k=1,...,n, let N_ijk(t) be the right continuous process that counts the number of observed direct i-to-j transitions for k-th individual, i,j=1,...m,i≠j. Let Y_ik(t) be the corresponding at-risk process. Define the filtration process as

for k=1,...n; i,j=1,...m and suppose that censoring is independent, so that

which must hold for all i, j, k and t>0. The Nelson-Aalen estimator of Λ_ij(t) is then given by

for all i≠j.

Parametric and semi-parametric models for λ_ijk are obtained analogously as earlier and may be found in [8].

4.2 Frailty models

Frailty models represent an extension of the Cox proportional hazards model. The concept of frailty provides a way to introduce random effects into the model to account for association (correlation) and unobserved heterogeneity. This heterogeneity may be difficult to assess but is nevertheless of a great importance. The frailty is an unobserved random factor that modifies multiplicatively the hazard function of an individual or a group of individuals. The key idea of these models is that individuals most "frail" die earlier than the others [16]. The frailty models are relevant to lifetimes of several individuals, similar organs and repeated measurements. They are not generally relevant for the case of different events [7].

First, bivariate models will be considered. Let

The marginal survival functions are then

If T₁ and T₂ are independent, S₁₂(t₁,t₂)=S₁(t₁)S₂(t₂). The joint hazard function is

and the marginal hazards are

Usually, the frailty is assumed to act multiplicatively on the hazard, so that

where g(z) is the probability density of Z. For the bivariate survival function thus

where the Laplace transform of g(z) is evaluated at s=Λ₀₁(t₁)+Λ₀₂(t₂).

with

The gamma function in the denominator of the probability density function is defined as

It satisfies Γ(k+1)=kΓ(k). The gamma distribution fits very well to failure data and is also convenient from computational and analytical point of views [19].

 

 

where

5. Conclusion

The survival analysis is a collection of specific statistical methods. In this paper, a short overview of these methods was presented. The standard univariate models were extended to multivariate models dealing with parallel and longitudinal data. The two major multivariate concepts were introduced: multi-state and frailty models.

Acknowledgement

The paper has been supported by the project SVV-2010-265513.

References

Jana Fürstová
Faculty of Medicine and Dentristy
Palacky University
Tř. Svobody 8
771 26  Olomouc
Czech Republic
e-mail:jana.furstova@email.cz