The choice of the primary endpoint may have a large impact on the design of a study and a substantive change of method of analysis. Data from clinical trials in which the outcome represents time-to-event produces what is commonly referred to as survival data. Survival data differs from Normally distributed data in three distinct ways:
(a) the survival time is strictly positive unlike normal data that can take any value (in the range−∞,+∞),
(b) survival data are generally not symmetrically distributed and often positively skewed in contrast to normal data which is symmetric and
(c) there exists censoring or partially observed outcomes unlike normal data that often have fully observed outcomes.
2.2.1
Censoring
Censoring is a defining characteristic of survival data and takes different forms: right, left or interval censoring. Right censoring, also referred to as progressive censoring, is the most common case of the three types of censoring (Collet, 2003). Right censoring can arise from numerous causes, for example,
(1) There may be no event reported on a patient by the end of the study, (2) The patient may have been lost to follow-up,
(3) The patients withdraws from the study and
(4) Patient experiences failure from alternative (competing) risk.
Causes 2− 4 can be either informative or non-informative depending on whether the form of censoring is dependent on the unobserved failure time. Censoring, even if non-informative, must be accounted for in the analysis. Otherwise, results might be misleading and/or biased.
Right censoring are of two types, I and II, depending on what is considered a random variable: time to end of study or the number of failures (events) at the end of study. In type I, the researcher pre-specifies the number of failures (r≤ n, where n is the number of subjects) of interest at the beginning of the study and only terminates the study once this number is realized. The process thus sets the time to end of study as a random variable. This type of censoring commonly occurs in industrial or animal experimentation where items or animals are put on test and observed until failure (O’Quigley, 2008). In contrast, in type II right censoring, the researcher pre-specifies the time of terminating the study at which time the number of failures are determined. This number of failures (r≤ n) now becomes the random variable. Right censoring perhaps derives its name from the fact that the times of failure to the right (larger than T ) are missing. Continuing with a study until a specified number of events are realized may lead to an open-ended random tests. While this may be appealing in indicating preference for type I right censoring, the process may not be practical due to limited resources like time and costs. Type II censoring has the significant advantage in situations where it is important to know a priori the number of failures (e.g. to achieve a pre-specified power of study). We however note that the censored (partially observed) information still provides useful information (Lindsey, 2004) so that if we let U be the observed time then we can define U = min(T, C), where T and C denote survival time and underlying censoring time respectively, i.e. U is the minimum (what comes first) between T and C. An example of right censored data can be realized when a patient leaves town before an event of interest occurs, i.e. T > U .
On the other hand, for left censored data a failure time is only known to be before a certain time, e.g. we may know that a particular patient died sometime before the first month but not exactly when this death occurred. Formally, a time to event T for a specific subject in a study is considered to be left censored if it is less than a censoring time Cl (Cl< C), i.e.
the subject has already failed (realized event of interest) before being observed in the study at time Cl. All we know for such subjects is that they have experienced the event (failed)
known if and only if T≥Cl. Thus in contrast to right censoring above, the observed time U is
defined as U = max (T, Cl). An example of left censoring is when a physician discovers cancer
during a routine visit for a different condition altogether hence the actual time of development of cancer is unknown and the patient sustains ‘event’ before initial time of observation.
Finally interval censoring is often used to reflect uncertainty as to the exact times the units failed within an interval. Lack of constant monitoring may be a cause of such type of data, e.g. if we monitor patients after every three months and we realize that a particular patient was alive at the last evaluation but dead by the next evaluation then the only information we have is that she failed in the 3-6 months interval of time. Probably due to this nature of periodic monitoring is what makes engineers in reliability studies refer to data that is interval censored as inspection data. An example of interval censoring is data obtained when a patient after surgery sustains event between two scheduled post-operation visits, i.e. the event is only known to have occurred in between the two visits. In general, right censored data is more prevalent in survival analysis of medical studies than left and interval censoring (Lee and Go, 1997). Data from the Esprit study were right censored (type II).
According to Marubini and Valsecchi (1995), the principal aims of survival analysis in the biomedical field include:
(i) Estimation of failure time distributions, i.e. summarize the distribution of survival times,
(ii) Compare the distributions of survival times among competing treatments so as to find the best treatment and
(iii) Prognostic evaluation of different variables, i.e. explore and understand the relationship between survival time and important covariates.
2.2.2
Survival function and hazard rate
The two functions extensively used to describe survival times and which are of central im- portance in the analysis of survival data are the survival function and the hazard functions
(Everitt and Pickles, 1999; O’Quigley, 2008). The survival function S(t) describes the proba- bility of an individual surviving beyond time t, i.e., the probability of experiencing the event of interest after time t. Mathematically we can define S(t) = Pr(T > t) = 1− F (t).
The time scale t is often set so that t = 0 refers to the beginning of follow-up. S(t) has the three properties:
(a) S(0) = 1 which implies that a patient is presumed alive at the beginning of the study, (b) limt→∞S(t) = 0, i.e. given a sufficiently long study period all subjects studied will
definitely experience the event (fail) or loosely speaking no patient lives forever and (c) S(t) is a monotonic, non-increasing, and nonnegative function implying that the prob-
ability of survival diminishes as the study progresses.
The probability density function f (t) is related to S(t) by
S(t) = 1− F (t) =
∫ ∞
t
f (u) du,
where by definition in statistical theory F (t) = Pr(T ≤ t) =∫0tf (u) du and f (t) = F′(t). A S(t)-time plot produces the survival curve which begins at S(0) = 1 and decreases to 0 as
t increases to infinity.
An important aspect of the survival distribution is the hazard function h(t) which is also known as the hazard rate or intensity or force of mortality and is defined as
h(t) = lim
δt→0
Pr(t≤ T < t + δt|T ≥ t)
δt , (2.1)
i.e. h(t) is the instantaneous (failure) rate of developing the event of interest in an arbitrar- ily short time interval δt, provided the subject is still at risk at time t (has not fallen ill before time t). From the above Equation (2.1), h(t)δt can be thought of as the approximate probability of an individual who has not experienced the event by time t experiencing the event in the next instant following t. However, we note that although h(t)δt is a probability,
technically the hazard rate h(t) is not a probability in the proper sense because it may take on values greater than one, i.e. it has no upper bound.
Alternatively, the hazard rate can be interpreted as the average number of events in a unit interval of time, hence why it is also commonly referred to as intensity. The incidence rate then validly approximates the hazard rate that assumes piecewise constant rate, a scenario that is realistic for short periods. In such applications incidence rates can be regarded as estimates of a limiting (theoretical) hazard rate h(t), which epidemiologists often refer to as the incidence intensity or force of morbidity (Rothman et al., 2008). Strictly speaking however, incidence and hazard rates do not always coincide.
For a continuous random variable T the following relationships between S(t) and h(t) holds: h(t) = f (t) S(t) =− d dtln S(t), so that S(t) = exp [ − ∫ t 0 h(u) du ] .
For all the three aims of survival analysis listed above, interest often lies in estimating the distribution of failure time T , the hazard h(t) and modelling the relationship between them and a subset of relevant covariates X. Inference from survival data analysis often assumes noninformative censoring, i.e. T is independent of any mechanism which causes the individual’s survival time to be censored at C. This assumption is necessary so as to make the distribution of T become identifiable from the distribution of the observables. Practically the assumption implies that the survival experience of censored individuals can be estimated by using data on the uncensored individuals. This implies that if we consider a group of all patients with same values of relevant prognostic factors, then a patient whose survival is censored at time c is assumed representative of all other patients in that group who have survived until that time.