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S u r v i v a l o f t h e s i c ke s t : S u r v i v a l o f t h e s i c ke s t : t r a c k i n g t h e i r p r o g r e s st r a c k i n g t h e i r p r o g r e s s

In the year that commemorates the 200th anniversary of the birth of Charles Darwin, “survival of the fi ttest” is a phrase that is on

many lips. However, researchers looking for new medicines to treat fatal diseases such as cancer have a more appropriate phrase: “survival of the sickest”. The medicines they seek to discover are those that extend the life of terminally ill patients. This is, in a sense, one of the most fundamental aspects of medicine and, with quality of life, one of the most hoped-for by patients. But how are we to assess the benefi ts of alternative treatments so that one can be recommended over the other?

One important metric is the time a patient remains alive when taking a particular treat-ment. This is the aptly named survival time and to understand it better we need a model for its probability distribution. Using this we can then compare the survival rates of two alternative treatments and give estimates of the precision with which these have been estimated. How-ever, we must fi rst understand patients’ reducing chance of survival as time passes. There is a use-ful plot that can summarise this.

Plotting survival times

To keep things simple for a moment, let us as-sume that there is only one treatment under

consideration and that this has been given to a single group of patients. We assume that all these patients are similar in the type and sever-ity of their disease when they start taking the treatment. If we consider this as a clinical trial the obvious way to proceed would be to wait un-til all the patients have died and then use their data to provide a summary of the effectiveness of the treatment. As we shall see, this is often neither possible nor desirable and a different ap-proach will be needed. However, let us continue with our simple approach.

An important summary of the survival times is provided by the survivor function S(t), which is the probability that a patient will survive to at least time t. Time might be measured in days, weeks, months or years depending on the nature of the disease.

An obvious estimator of this function, based on the actual survival times, is the empirical survivor function SE, where:

SE(t) = (number of patients with survival times ≥ t)/(number of patients in the data set).

The plot of SE(t) versus t provides a useful de-scriptive summary of the survival times.

Byron Jones gives a brief introduction to the analysis of survival data and shows techniques that can model events

such as waiting times and even the length of pregnancies.

© iStockphoto.com/Mike Manzano

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The survivor function might be more recognis-able to statisticians when it is written as S(t) = 1 – F(t), where F(t) is the cumulative distribution function of t. For a continuous distribution, the probability density function (p.d.f.) of survival time t, is f(t) = F’(t), the derivative of F(t) with respect to t. Hence f(t) = –S’(t).

Perhaps the simplest p.d.f. for t is the ex-ponential distribution, f(t) = λexp(–λt), which has survivor function S(t) = exp(–λt). This dis-tribution has the “memoryless” property that the probability of death at time t does not depend on the amount of time survived up to time t. Its mean is 1/λ and its median is log(2)/λ. The parameter λ is the rate of death. If λ = 0.1 months, for example, the mean survival time is 10 months, i.e. an average of one death every 10 months. The median survival time is about 7 months. Often more complicated survivor func-tions are needed to explain actual survival times and statistical modelling techniques that take account of important patient characteristics are used. We shall mention one of these below.

However, let us return to the estimation of the survivor function. As we hinted earlier, the rather obvious approach of waiting until all patients have died has limitations in practice because we often do not know the survival times of all the patients in our clinical trial. How can this be, when all we have to do is wait? There are several reasons, the most common of which is that some patients survive beyond the appointed time at which the trial will be stopped and the data analysed. (Some fortunate few, of course, may outlive the analyst, which means that the analysis would never be completed.) This is good for those patients who survive, but not for esti-mating the survivor function! If T is the time at which the recording of survival times is stopped, and a patient is still alive at time T, then all we know is that the eventual survival time of that patient is greater than T. We refer to such times as censored (i.e. t ≥ T). Our simple approach cannot handle censored survival times. Patients with such times would need to be removed from the estimation of the survivor function, which is something we want to avoid.

One alternative that does not discard the censored times is the Kaplan–Meier estimate of the survivor function, which we describe next.

Kaplan–Meier estimate of the survivor function

The basic idea is to consider the time period during which patients are treated as divided into a series of consecutive intervals of time. During each of these intervals a patient is at risk of death and we are interested in the prob-ability that a patient dies within that interval. However, to die during any particular interval a patient must have survived all of the previous

time intervals. Assuming that the time intervals are independent, as far as the chance of a death is concerned, the probability of surviving up to the start of a particular interval is the product, for all the previous intervals, of the probabilities of not dying during each of the intervals.

How do we estimate these probabilities? As we have data for only those deaths that have occurred, the time intervals we use are the intervals of time between these successive deaths. Suppose there are r distinct death times t1, t2, …, tr. Consider the interval (tk, tk+1), and let nk denote the number of patients that are still alive at the start of this interval. This is referred to as the risk-set for interval (tk, tk+1) and we see that it includes any patients whose times are censored during the time interval as well as those who might have died during the interval. If dk is the number that die during this interval, then nk–dk is the number that are alive at the end of the interval. An estimate of the

probability of not dying during the interval is then (nk–dk)/nk. Hence an estimate of the prob-ability of being alive at the start of interval (tk, tk+1) is

(n1–d1)/ n1 x(n2–d2)/ n2x ... x(nk–1–dk–1)/ nk–1

An estimate of the probability of not dying during interval (tk, tk+1) is, as already noted, (nk–dk)/nk, and so an estimate of the probability that a patient of survives until the end of interval (tk, tk+1) is

SKM(tk) = (n1 – d1)/ n1 x(n2 – d2)/ n2 x ... x(nk–1 – dk–1)/ nk–1 x(nk – dk)/nk

A plot of SKM(tk) versus tk is the well-known Kaplan–Meier plot. If no patients are censored during the trial, then (nk – dk) = nk+1, and the Kaplan–Meier plot is the same as the empirical survivor function SE(t).

Table 1. Calculation of the survivor function

Time interval nj dj (nj –dj)/(nj) SKM(tj)

0–9 20 0 20/20 1.0010–11 20 2 18/20 0.9012–19 18 4 14/18 0.7020–29 12 2 10/12 0.5830–46 10 1 8/10 0.4747–55 7 1 6/7 0.4056–59 6 3 3/6 0.2060 2 1 1/2 0.10

Survival Time in Months

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Figure 1. Example of a Kaplan–Meier plot

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To illustrate the calculations needed for this plot, consider an invented set of survival times from 20 patients who took part in a clinical trial to evaluate a treatment for cancer: 10, 10, 12, 12, 12, 12, 12*,12*, 20, 20, 30, 30, 30*, 47, 56, 56, 56, 56*, 60, 60*. The times are in months and an asterisk indicates that a time was censored. We might suppose the trial lasted 5 years, which accounts for the last censored time. The other censored times are the result of patients dropping out of the trial for vari-ous reasons. Among these reasons could be that patients withdrew themselves from the trial or perhaps moved away and could not continue to visit the local medical centre where the trial was being conducted. The calculation of the esti-mated survivor function using the Kaplan–Meier method is illustrated in Table 1 and a plot of this function is given in Figure 1. The estimates of the mean and median survival times are 35.6 and 30 months respectively.

Of course, the best use of such a plot is to show the difference in survival rates for two al-ternative treatments. The survival rates plotted in Figure 2 have been calculated from a previ-ously published dataset1 and are the survival times of prostatic cancer patients. The marks on the plotted lines identify the censored times. The plot in Figure 2 suggests that the probability of survival is consistently better on treatment. In addition to the survival times, the data on these patients include a number of potentially useful explanatory variables. Among these, the two that are most correlated with survival time are the size of the tumour and an index of its stage and grade.

Testing for a treatment difference and further modelling of the survival data

Having seen, in Figure 2, an apparent differ-ence between the estimated survival rates, the natural thing to consider doing next is to test if treatment really is better than placebo. A simple signifi cance test is the so-called log-rank test described by Collett1. However, for the dataset

that produced Figure 2, we prefer to take a mod-elling approach that accounts for the inclusion of potential explanatory variables. This requires an understanding of the hazard function, de-scribed below.

The hazard function

Statistical modelling of survival data is usually done via the hazard function. The hazard for a patient at time t is the probability that the patient dies at time t, given that he or she was alive the instant before time t. Conditioning

on survival up to the instant just before time t is an important part of the defi nition. To see this consider the probability that someone will survive to age 100. In the general population, this probability will be quite small because most people die before reaching that age. However, if someone has survived to the instant before age 100, then their probability of survival to age 100 will be much higher than the overall probability. The function h(t), which gives the value of the hazard at any time t, is referred to as the hazard function. For the exponential p.d.f. given earlier, h(t) = λ, a constant that does not depend on t. This is the memoryless property we mentioned earlier. Often, however, the hazard will vary over time and be dependent on other patient and treatment characteristics, such as age, gender and dose of drug. Then statistical modelling can be used to estimate the hazard regression equa-tion that best explains the data. If the functions that measure the hazard under two different treatments are proportional, i.e. h1(t) = ch2(t), then a popular method of modelling the survival data from these two treatments is known as Cox’s proportional hazards regression1. This method allows explanatory variables, such as age and gender, to be included in the model.

For our example dataset, the important vari-ables are the size and index of the tumour. When these are included in the regression equation, we fi nd that the estimated ratio of the hazards (treatment/placebo) is about a third and the dif-ference between treatment and placebo is not signifi cant (p-value = 0.355). A 95% confi dence interval for the ratio is (0.03, 3.47). Collett1 should be consulted for further details of the da-taset (in Table 1.4) and of the other explanatory variables that were also considered for inclusion in the model.

Conclusion

Time to death is just one example of a variable that measures the time to an event. More gen-erally, similar techniques can be used to model other event times such as, for example, the time a patient has to wait to be admitted to hospital for surgery, the time to recover from an illness and the time to give birth. Methods such as those described in this article have widespread application in a range of disciplines.

Reference1. Collett, D. (1994) Modelling Survival Data in

Medical Research. London: Chapman and Hall.

Byron Jones is a Senior Director in the Statistical Re-search and Consulting Centre at Pfi zer Ltd, Sandwich, Kent, UK. He also holds honorary professorships at University College London and at the London School of Hygiene and Tropical Medicine.

Survival Time in Months

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Figure 2. Kaplan–Meier plot of data given in Collett1 (Table 1.4)

The hazard function tells us that a 99-year-old is more likely than a new-born child to survive to the

age of 100