Ординатура / Офтальмология / Английские материалы / Using and Understanding Medical Statistics_Matthews, Farewell_2007

6.3. Computing the Kaplan-Meier Estimate

In this penultimate section of chapter 6 we intend to discuss, in some detail, a simple method for calculating the Kaplan-Meier estimate by hand. The technique is not complicated; nevertheless, some readers may prefer to omit this section.

To calculate a Kaplan-Meier estimated survival curve, it is first necessary to list all observations, censored and uncensored, in increasing order. The convention is adopted that if both observed and censored survival times of the same duration have been recorded, then the uncensored observations precede the corresponding censored survival times in the ordered list. For the simple example presented in figure 6.2a, the ordered list of observations, in days, is

1*, 3, 4*, 5, 5, 6*, 7, 7, 7*, 8*

where * indicates a censored survival time.

The second step in the calculation is to draw up a table with six columns labelled as follows:

tobserved distinct survival time, in days

nthe number of individuals still under observation at t days

rthe number of deaths recorded at t days

For convenience, the initial row of the table may be completed by entering t = 0, n = number of individuals in the study, r = 0, pc = 1, Pr(T 1 t) = 1 and s.e. = blank. This simply indicates that the estimated survival curve begins with Pr(T 1 0) = 1, i.e., the estimated probability of survival beyond Day 0 is one. The first observed survival time in the ordered list of observations is then identified, and the number of observations which exceed this observed survival time is recorded, along with the number of deaths occurring at this specific time. Notice that the number of individuals still under observation at t days is equal to the number of individuals in the study minus the total number of observation times, censored or uncensored, which were less than t days. Thus, both deaths and censored observations reduce the number of individuals still under observation, and therefore at risk of failing, between observed distinct survival times.

In our example, the first observed survival time is t = 3; therefore, the initial two rows of the table would now be:

of t, draw a horizontal section at a height equal to the corresponding value of Pr(T 1 t). Each horizontal section will extend from the current value of t to the next value of t, where the next decrease in the graph occurs.

From the columns labelled t and Pr(T 1 t) in the table for our example, we see that the graph of the K-M estimated survival curve has a horizontal section from t = 0 to t = 3 at probability 1.0, a horizontal section from t = 3 to t = 5 at probability 0.89, a horizontal section from t = 5 to t = 7 at probability 0.64, and a horizontal section from t = 7 at probability 0.32. Since the final entry in the table for Pr(T 1 t) is not zero, the last horizontal section of the graph is terminated at the largest censored survival time in the ordered list of observations, namely 8. Therefore, the final horizontal section of the graph at probability 0.32 extends from t = 7 to t = 8; the estimated probability of survival is not defined for t 1 8 in this particular case.

If the final censored observation, 8, had been an observed survival time instead, then the completed table would have been:

In this case, the graph of the K-M estimated survival curve would drop to zero at t = 8 and be equal to zero thereafter.

6.4. A Novel Use of the Kaplan-Meier Estimator

Despite the fact that estimation of the statistical distribution of the time to an endpoint like death is routinely referred to as the survival curve or survivor function, not all studies in which K-M estimators are used require subjects to die or survive. James and Matthews [11] describe a novel use of the K-M estimator in which the endpoint of interest is an event called a donation attempt. Their paradigm, which they call the donation cycle, represents the use of familiar tools like K-M estimator in an entirely new setting, enabling transfusion researchers to study the return behaviour of blood donors quantitatively – something that no one had previously recognized was possible.

According to the paradigm that they introduce, each donation cycle is defined by a sequence of four consecutive events: an initiating donation attempt,

Despite the size of each sample, this cycle-specific data is exactly what modern statistical software requires to generate K-M estimates of the so-called survival curve for each sample. The resulting graphs are displayed in figure 6.3, and illustrate how, for the type O, whole blood donors that contributed through the GCRBC, the intervals between donation attempts depend on a history of previous donations and the elapsed time since the initial or index donation attempt in a cycle. In each case, the estimated probability of making a subsequent donation attempt remains at or near 1.0 for the first eight weeks of the observation period – precisely the length of the mandatory deferral interval – and only thereafter begins to decrease. The graphs also show that the estimated survival function decreases with each additional donation attempt. Thus, for any fixed time exceeding eight weeks, the proportion of donors who had already attempted a subsequent donation is an increasing function of the number of previously completed donation cycles. In contrast to the usual staircase look of most K-M estimated survival curves, the smooth appearance of these estimated functions reflects the effect of using very large sample sizes to estimate, at any fixed value for t, the proportion of donors who had not yet attempted a subsequent donation. One year after the index donation, some 68% of first-time donors had yet to return for a subsequent attempt; at five years, approximately one-third of first-time donors had not attempted a second whole blood donation. The corresponding estimated percentages for donors in their fifth cycle were 36 and 10%, respectively.

The marked localized changes in the graph at 52 and 104 weeks are a particularly distinctive feature of these estimated curves; a third drop appears to occur at 156 weeks, although this decrease is less evident visually. James and Matthews speculate that these parallel changes ‘were the result of two GCRBC activities: blood clinic scheduling and a donor incentive programme which encouraged such clinic scheduling.’ The GCRBC offered an insurance-like scheme that provided benefits for donors and their next-of-kin, provided a donation had been made during the preceding 12 months. Similar localized changes have been seen in studies of other forms of insurance. For example, return-to- work rates usually increase markedly at or near the end of a strike or the expiration of unemployment insurance coverage (see Follmann et al. [13]).

In general, the K-M estimated survival curve is used mainly to provide visual insight into the observed experience concerning the time until an event of interest such as death, or perhaps a subsequent attempt to donate a unit of blood, occurs. It is common, however, to want to compare this observed experience for two or more groups of patients. Although standard errors can be used for this purpose if only a single fixed point in time is of scientific interest, better techniques are available. These include the log-rank test, which we describe in chapter 7.

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The Log-Rank or Mantel-Haenszel Test for the Comparison of Survival Curves

7.1. Introduction

In medical research, we frequently wish to compare the survival (or relapse, etc.) experience of two groups of individuals. The groups will differ with respect to a certain factor (treatment, age, sex, stage of disease, etc.), and it is the effect of this factor on survival which is of interest. For example, figure 7.1 presents the Kaplan-Meier estimated survival curves for the 31 lymphoma patients mentioned in chapter 6, and a second group of 33 lymphoma patients who were diagnosed without clinical symptoms. According to standard practice, the 31 patients with clinical symptoms are said to have ‘B’ symptoms, while the other 33 have ‘A’ symptoms. Figure 7.1 shows an apparent survival advantage for patients with A symptoms.

In the discussion that follows, we present a test of the null hypothesis that the survival functions for patients with A and B symptoms are the same, even though their respective Kaplan-Meier estimates, which are based on random samples from the two populations, will inevitably differ. This test, which is called the log-rank or Mantel-Haenszel test, is only one of many tests which could be devised; nevertheless, it is frequently used to compare the survival functions of two or more populations.

The log-rank test is designed particularly to detect a difference between survival curves which results when the mortality rate in one group is consistently higher than the corresponding rate in a second group and the ratio of these two rates is constant over time. This is equivalent to saying that, provided an individual has survived for t units, the chance of dying in a brief interval following t is k times greater in one group than in the other, and the same statement is true for all values of t. The null hypothesis that there is no differ-