## Timetoevent data

There are many examples in cancer where a time from one event to another is used. For example, the time a woman with ovarian cancer survives once the tumour has been removed by surgery, the time a man with testicular cancer treated with chemotherapy survives and remains free of disease. These times are triggered by an initial event: a surgical intervention, followed by a subsequent event, death, or recurrence of disease. The time between such events is known as the 'survival time.' The term survival is used because the first use of such techniques arose from the life insurance industry. The difference between 'survival' data and other types of numeric continuous data is that the time to the event occurring is not observed in all subjects. Thus in the above examples, in the lifetime of our study all the women with ovarian cancer may not die, or all the men with testicular cancer may not experience a recurrence of their disease. Such non-observed events are called 'censored.' Historically, much of the analysis of survival data has been developed and applied in relation to randomized cancer clinical trials in which the survival time is often measured from the date of randomization or commencement of therapy until death, and the early papers by Peto and colleagues [7,8] reflect this.

Estimation and summary statistics for time-to-event data

The Kaplan-Meier survival curve. The Kaplan-Meier estimate of the survival curve is often used as a means of summarizing 'survival' data. The survival rate estimate at time t, S(t), will start from 1 (100 per cent of patients alive) since S(0) = 1, and progressively decline towards 0 (all patients have died) with time. It is plotted as a step function, since the estimated survival curve remains constant between successive patient death times. It drops instantaneously at each time of death to a new level. The graph will only reach 0 if the patient with the longest observed survival time has died. Patients who are still alive are included in the formation of the curve but are 'censored' from the time they were last known to be alive. In general the overall probability of survival at time t, S (t), is given by:

where dt represents a death at time t and nt the number of patients still alive and being followed up (that is not censored) at time t. Thus the value of S(t), changes only on times (days) on which there is at least one death. As a consequence the curve does not change during the times (days) when there are no deaths. The succesive overall probabilities of survival S(1), S(2),..., S(t) are known as the Kaplan-Meier (or product-limit) estimates of survival. If all the patients have experienced the event before the data are analysed the estimate is exactly the same as the proportion of survivors plotted against time.

The overall survival curve is much more reliable than the individually observed survival probabilities at each of the timepoints of which it is composed. Spurious (large) falls or (long) flat sections may sometimes appear. These are most likely to occur if the proportion of censored observations is large or when the number of patients still alive and being followed up may be relatively small. A guide to the reliability of different portions of the survival curve can be obtained by recording the number of patients 'at risk' at various stages beneath the time axis of the survival curve. The 'number at risk' is defined as the number of patients who are known to be alive at that timepoint and therefore have not yet died nor have been censored before the timepoint. These individuals are therefore 'at risk' of the event in the subsequent time interval. At time zero, which is the time of entry of patients into the study, all patients are at risk and hence the number 'at risk' recorded beneath t = 0 is the number of patients entered into the study. The patient numbers obviously diminish over time, because of both deaths and censoring. The deaths which happen at various times can be seen on the plot by the fact that the survival curve drops down; to help indicate the censored observations the survival curve is sometimes annotated with vertical ticks at each timepoint at which an individual patient is censored.

The reliability of the Kaplan-Meier estimate of S(t) diminishes with increasing t. There is no precise moment when the right-hand side tail of the survival curve becomes unreliable. However, as a rule of thumb, the curve can be particularly unreliable when the number of patients remaining at risk is fewer than fifteen. The width of the confidence intervals (see below), calculated at these and other timepoints, will help to show the uncertainty in the estimate. Nevertheless, it is not uncommon to see the value of S(t), corresponding to the final plateau, being quoted as the 'cure' rate, especially if the plateau appears to be long. This can be seriously misleading as the rate will almost certainly be reduced with further patient follow-up. Clearly, if there are no censored observations preceding the end of a plateau, then the plateau will not disappear with more patient follow-up. Even in such cases the plateau should be interpreted with considerable caution.

### Median survival time

A commonly reported statistic in cancer studies is the median survival time. This statistic is particularly useful in studies of advanced disease, where prolongation of survival may be more relevant than cure. The median survival time is obtained by finding the time at which the proportion alive (and dead) is 0.5. This is done by reading across from the vertical axis at 0.5 and when the curve is hit, reading down to the timepoint at which this happens. As an example Fig. 9.1 shows the overall survival curves for the three groups (de Gramont, raltitrexed and Lokich) from the CR06 colorectal cancer trial discussed above. Reading across from 0.5, median survival for the de Gramont appears to be approximately nine months. More accurately, considering the data used to plot these curves, the estimated median survival is 294 days.

de Gramont

Lokich

Raltitrexed

Patients at risk de Gramont 303 Lokich 301

raltitrexed 301

197 197

12 Months

101 108 95

38 64 49

20 25 20

Fig. 9.1 Overall survival in the CR06 trial. Reprinted with permission from Elsevier Science (The Lancet, 2002, 359, 1555-63).

### Confidence intervals

Confidence intervals (CI) calculated at relevant points along the Kaplan-Meier survival curve will give an indication of the reliability of the curve at those points. There are several ways of calculating confidence intervals at given points. Popular approaches include those proposed by Greenwood, Peto and Rothman (see pp. 36-40 in Ref. [9]). We first present our preferred method - the transformation method. One of the reasons this a preferred method is because it constrains the confidence interval within zero and one, something not all methods will do.

We first transform S( t) onto a scale which more closely follows a Normal distribution. It can be shown that logg {— logg [S (t)]} has an approximately Normal distribution, with SE given by r i1/2

where dj is the number of deaths on day j and nj is the number of patients alive on follow-up at the beginning of day j.

To calculate the 95 per cent CI on this transformed scale we calculate the following two values:

logg{- loge[S(t)]} - 1.96SEtr[S(t)] to loge{- loge[S(t)]} + 1.96SE^[S(i)].

We now return to the original scale by using S(t)exp (+L96setr) which is the upper 95 per cent confidence limit and S(t)exp(-L96setr), the lower 95 per cent confidence limit.

A simpler but less reliable way of obtaining a confidence interval is a method suggested by Peto. In this approach we first calculate a standard error as follows:

where Rt is the number of patients recruited to the trial minus the number censored by the time under consideration. Thus Rt = N - ct, where N is the number of patients in the trial and ct is the number of patients censored before time t. Rt is sometimes termed the effective sample size. A 95 per cent confidence interval is then given by S(t) - 1.96SEtr[S(t)] to S(t) + 1.96SEtr[S(t)].

### Comparison of two survival curves

Above, we described how the Kaplan-Meier estimate of a single survival curve is obtained. Within most clinical trials we shall want to compare two or more survival curves. For example, we may wish to compare the survival experiences of patients who receive different treatments. Before we describe the methods available to compare two survival curves, we note that it is inappropriate and usually very misleading to compare survival curves only at a particular point of the curve, for example, making comparisons of 1-year survival rates. The reason for this is that each individual point on the curve is not in itself a good estimate of the true underlying survival. Comparing two such estimates (one from each of the two curves) leads to an unreliable comparison of the survival experience of the two groups. Further, it makes very poor use of all the data available by concentrating on particular points of each curve and ignoring the remainder of the survival experience. The comparison of two (or more) survival curves is usually done by the logrank test, or some variant of it.

The Logrank test The Logrank test, like all hypothesis tests, is based on comparing the observed against the expected data under the null hypothesis of no difference between the experimental and control arms. To compare two survival curves, one for the experimental treatment and one for the control, we use the following test statistic:

Ee Ec where Oe is the observed number of deaths on the experimental arm, e, Ee, the expected number of deaths on the experimental arm, e, Oc, the observed number of deaths on the control arm, c, Ec, the expected number of deaths on the control arm, c.

This is a format which is similar to that for the standard chi-square test for a 2 x 2 table. The only difference is that the expected numbers of deaths in groups e and c are actually a summation of the expected numbers of deaths in each group calculated on each occasion that a death happens in either group. The expected number of deaths on the experimental and control arms is obtained by forming a 2 x 2 table at each timepoint at which there is an death in the trial. Under the null hypothesis of no difference between the groups the probabilty of an event in either group at a given time is proportional to the number of event free in each group at that time. Thus at each timepoint at which there is an event we can calculate the probability that the event should have been in either the experimental or control groups. Multiplication of this probability by the actual number of events (usually one) at that timepoint gives the expected number of deaths in the two groups at that timepoint. Summation of these expected values over all timepoints at which there is an event, gives the Ee and Ec in equation 9.18. The observed number of deaths Oe and Oc are simply the number of deaths seen in the trial on the experimental and control arms.

When there are two treatment groups the /¡2ogrank statistic is compared against a chi-square distribution with one degree of freedom. In general, for G treatment groups the XL2ogrank statistic is compared with the chi-square distribution with G — 1 degrees of freedom. The logrank test is a hypothesis test, testing the (null) hypothesis that there is no difference in survival between the groups. We also need an estimate of the difference between the curves, and this is given by the hazard ratio.

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