DETERMINING MEDIAN SURVIVAL TIME

Figure 19.2 The Kaplan-Meier survival curve drawn from the data in Table 19.2 (the dotted line indicates median proportion surviving – see text below)

Exercise 19.1 The data in Table 19.3 shows the survival times (in days) of eight patients with acute myocardial infarction, treated with a new reperfusion drug Explase, as part of a fibrinolytic regimen. Patients were followed up for 14 days. Calculate survival probabilities and plot Kaplan-Meier survival curves. Comment on your results.

Table 19.3 The survival times (in days) of eight patients with acute myocardial infarction. Patients were followed up for 14 days

Determining median survival time

One of the consequences of not knowing the actual survival times of all of those subjects who survive beyond the end of the study period is that we cannot calculate the mean survival time of the whole group. However, if you interpret the probabilities on the vertical axis of

a Kaplan-Meier chart as proportions or percentages, you can often easily determine median survival times. It is that value which corresponds to a probability of 0.5 (i.e. 50 per cent). In Figure 19.2, the median survival time is 8 months. At this time, half of the patients still survived. Obviously the survival time of any proportion of the sample can be determined in this same way, including the interquartile range values, provided that the Kaplan-Meier curve goes down far enough (unfortunately it often doesn’t).

Figure 19.3 Kaplan-Meier curves for overall survival for three groups of patients in a comparison of three chemotherapy regimes in the treatment of colorectal cancer. Reprinted from The Lancet, 2002, 359, 1559 with permission from Elsevier

Exercise 19.2 Figure 19.3 shows Kaplan-Meier curves for progression-free survival, for three groups of patients in a comparison of three chemotherapy regimes used for the treatment of colorectal cancer (Maughan et al. 2002). The three regimes were: the de Gramont regimen; the Lokich regimen; and Raltitrexed. What were the approximate median survival times for progression-free survival with each of the three regimes?

Comparing survival with two groups

Although the survival curve for a single group may sometimes be of interest, much more usual is the desire to compare the survival experience of two or more groups. For example, Figure 19.4 is taken from a study of chemotherapy for the treatment of bladder cancer (Medical Research Council Advanced Bladder Working Group 1999). One group of patients (n = 485) was randomly assigned to receive conventional radical surgery (cystectomy) or radiotherapy, while a second group (n = 491) received the conventional treatment plus chemotherapy. The

Figure 19.4 Survival curves for two groups of patients with bladder cancer, one group given conventional surgery or radiotherapy, the other group given the conventional treatment plus chemotherapy. Reprinted coutresy of Elsevier (The Lancet, 1999, Vol No. 354, p. 533–9)

question asked, ‘Was the survival experience of the chemotherapy group any better over the five year follow up?’

The two Kaplan-Meier curves seem to show that the proportions surviving in the chemotherapy group was larger than those in the conventional group throughout the duration of the study, since the survival curve for the former was higher than that of the latter. In fact, the authors of this study report median values for disease-free survival of 20 months for the chemotherapy group and 16.5 months for the no-chemotherapy group. The 95 per cent confidence interval for the difference in medians was (0.5 to 7.0) months, so the difference in medians was statistically significant.

Notice that the authors have provided a table showing the numbers at risk at each time interval. This is to remind us that the smaller numbers of survivors towards the end of a trial produce less reliable results. As a direct consequence of this effect, you should not assume that just because the gap between two survival curves gets progressively larger (as it is often seen to do), that this is necessarily due to an actual divergence in the survival experiences in the two groups. It might well be caused simply by the low numbers of subjects still at risk. This can make the ends of the curves unreliable.

The log-rank test

If you want to compare the overall survival experience of two (or more) groups of patients (rather than say comparing just the median survival times as we did above), then one possible approach is to use the non-parametric log-rank test. Essentially, the null hypothesis to be tested is that the two samples (the two groups) are from the same population as far as their survival experience is concerned. In other words there is no difference in the survival experiences.

The log-rank test of this hypothesis uses a comparison of observed with expected events (deaths, say), given that the null hypothesis is true.2 If the p value is less than 0.05 you can reject the null hypothesis and conclude that there is a statistically significant difference between the survival experience of the groups. You can then use the Kaplan-Meier curves to decide which group had the significantly better survival. A limitation of the log-rank test is that it cannot be used to explore the influence on survival of more than one variable, i.e. the possibility of confounders – for this you need Cox’s proportional regression, which we’ll come to shortly.

The authors in the bladder cancer study reported a log-rank test p value of 0.019 for the difference in survival times at three years, but unfortunately don’t give the results of the test over the whole five year duration of the study.

Exercise 19.3 What do you conclude about the statistical significance of the difference in three year survival times of the chemotherapy and non-chemotherapy groups from the results given in the previous paragraph?

Figure 19.5 Kaplan-Meier curves from a study to assess the clinical efficacy of azithromycin for prophylaxis of Pneumocystitis carinii pneumonia in HIV-1 infected patients. Reprinted courtesy of Elsevier (The Lancet, 1999, Vol No. 354, p. 1891–5)

An example of the log-rank test in practice

Figure 19.5 shows the Kaplan-Meier curves from a study to assess the clinical efficacy of azithromycin for prophylaxis of Pneumocystitis carinii pneumonia in HIV-1 infected patients (Dunne et al. 1999). Patients were randomly assigned to one of three treatment groups: the first group given azithromycin, the second rifabutin and the third a combination of both drugs. The figure shows the event-free (no Pneumocystitis carinii pneumonia) survival experiences over an 800 day period for the three treatment groups.

2You may have spotted the similarity with the chi-squared test considered earlier in the book. In fact the calculations are exactly the same.

The log-rank test was used to test the hypothesis that there is no difference in the percentage event-free between the azithromycin and rifabutin groups (p value = 0.033), and between the azithromycin and the combination groups (p-value = 0.026). The authors concluded that azithromycin as prophylaxis for Pneumocystitis carinii pneumonia, provides additional protection over and above standard Pneumocystitis carinii pneumonia prophylaxis. However, these results should be treated with caution because of the very small size of the survivor group towards the end of the study.

Figure 19.6 Kaplan-Meier curves of percentage number of subsequent ischaemic events from a randomised controlled trial into the relative effectiveness of aspirin and oral anticoagulants (coumadin), used for antiplatelet treatment, following myocardial infarction. Reprinted courtesy of Elsevier (The Lancet 2002, 360, 109–14, Fig. 3, p. 111)

Exercise 19.4 Figure 19.6 shows the Kaplan-Meier curves for the percentage number of ischaemic events from a randomised controlled trial into the relative effectiveness of aspirin and oral anticoagulants (coumadin) for antiplatelet treatment following myocardial infarction (van Es et al. 2002). The object was to investigate which of these two drugs is more effective for the long-term reduction of subsequent ischaemic events, and whether the combination of the two drugs offers greater benefit than either drug alone. Is there a statistically significant difference in mortality between the three possible treatments? Which treatment seems to offer the best survival?

The hazard ratio

The log-rank test is limited by the fact that it is just that – a test. It will tell you if there is a significant difference between the survival experience of two (or more) groups, but does not quantify that difference. For this we need what is called the hazard ratio (based on the ratio

overall survival (only about half as well, HR = 0.49), and cancer-related survival (just over a quarter as well, HR = 0.29). Both of these results are statistically significant. Note that type of surgery; laparoscopy-assisted versus open colectomy, is not statistically significant in terms of overall survival as the confidence interval of (0.23 to 1.01) includes 1.

Table 19.5 Hazard ratios due to a number of risk factors in a univariate (unadjusted), and multivariate (adjusted) cohort analysis of the risk to HIV+ women of vulvovaginal and perianal condylomata acuminata and intraepithelial neoplasia. Reproduced courtesy of Elsevier (The Lancet, 2002, Vol No. 359, page 108–14

In univariate analysis, vulvovaginal lesion was the outcome variable.

†355 HIV-1-positive and 325 HIV-1-negative women were included in the multivariate analysis, with vulvovaginal or perianal lesion as the outcome variable and HIV-1 infection, CD4 T lymphocyte count, human papillomavirus infection, less than a highschool education, cigarette smoking, and history of injection of two or more drugs three or more times per week as covariates.

‡ HIV-1 negative women were presumed to have a CD4 count >500 cell/, μL. § Does not include a history of genital warts.

Exercise 19.6 Table 19.5 shows the hazard ratios (unadjusted and adjusted) due to a number of risk factors in a cohort analysis of the risk to HIV+ women of vulvovaginal and perianal condylomata acuminata and intraepithelial neoplasia (Conley et al. 2002). Interpret the multivariate results. How do these differ from the univariate results?

Checking the proportional hazards assumption

The proportional hazards assumption can be checked graphically using what is known as the log-log plot. Unfortunately, this procedure is beyond the scope of this book.