Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

22

Survival and Longitudinal Analysis

CONTENTS

22.1Differences between Survival and Longitudinal Analysis

22.2Construction of Survival Summaries

22.2.1Sources of Numerator Losses

22.2.2Adjustment of Denominators

22.2.3Display of Results

22.2.4Choice of Methods

22.3Scientific Problems and Adjustments

22.3.1Informative Censoring

22.3.2Competing Risks

22.3.3Competing Outcomes

22.3.4“Left Censoring”

22.3.5Sensitivity Analysis for Lost-to-Follow-Up Problems

22.3.6Relative Survival Rates

22.4Quantitative Descriptions

22.4.1Median Survival Time

22.4.2Comparative Box Plots

22.4.3Quantile-Quantile Plots

22.4.4Linear Trend in Direct-Method Survival Rates

22.4.5Hazard Ratio

22.4.6Hazard Plots

22.4.7Customary Visual Displays

22.5Stochastic Evaluations

22.5.1Standard Errors

22.5.2Confidence Intervals

22.5.3Comparative Tests

22.5.4Sample-Size and Other Calculations

22.6Estimating Life Expectancy

22.6.1Customary Actuarial Technique

22.6.2Additional Procedures

22.7Dynamic (Age-Period) Cohort Analysis

22.7.1Methods of Construction and Display

22.7.2Uncertainties and Problems

22.8Longitudinal Analysis

22.8.1Confusion with Longitudinal Cross-Sections

22.8.2Applications of Longitudinal Analysis

22.8.3Statistical and Scientific Problems

References

Exercises

When free-living people are members of a cohort under long-term observation, a baseline “zerotime” status can be identified for each person, but special problems and challenges can arise thereafter. The final outcome may be unknown for members of the cohort who “drop out” or become “lost to

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follow-up.” The intermediate events may be unknown or difficult to interpret for persons who continue to participate in the study, but who make unauthorized changes in the assigned plans or fail to appear for scheduled examinations.

For events that are either unknown or occur under inappropriate circumstances or timing, the statistical analyses cannot be suitably managed with a regression equation that merely has outcome as one variable and time as the other. In other situations, where the follow-up period is aimed not at a single “survival” event, such as death, but at a series of events or changes, the analysis of repeated or multiple outcomes is also not amenable to any of the statistical structures discussed thus far in the text.

This chapter is devoted to methods of serial analysis that have been developed to cope with these challenges. The chapter begins and is mainly concerned with a method called survival analysis, which is also known as life-table or actuarial analysis. Used when each person is followed until the occurrence of a single “failure event,” such as death, which concludes the person’s period of observation “at risk,” the analysis produces the survival curves that commonly illustrate clinical trials and cohort studies. The main discussion of survival analysis is followed by brief accounts of two additional applications: measuring life expectancy and evaluating certain age-period cohort effects in a general population. The chapter concludes with an outline of another cohort procedure, called longitudinal analysis, that is devoted to repeatedly measured serial outcomes, which can be either recurrences of a binary event, such as episodes of streptococcal infection, asthma, or epilepsy, or changes in dimensional variables, such as blood pressure, serum cholesterol, or pulmonary function tests.

22.1 Differences between Survival and Longitudinal Analysis

Any form of cohort analysis can be regarded as “longitudinal,” but the particular methods called survival analysis and longitudinal analysis collect and analyze the information with different approaches. In a conventional survival analysis, each person’s outcome data contain a single pair of bivariate values: the duration of serial time until the person’s exit date and the concomitant binary “exit state,” which might be dead or alive. In a longitudinal analysis, each patient’s basic outcome data contain multiple pairs of bivariate values for the timing and concomitant value of each measurement.

Both the survival and the longitudinal procedures use bivariate temporal data for each cohort, but the bivariate relationship can become trivariate when an additional variable allows results to be compared for two or more cohort groups, such as recipients of Treatment A or B or patients in Stages I, II, and III of a particular disease. The analytic methods can become multivariate when data for additional conditions, such as age and baseline clinical severity, are used to “adjust” the bivariate or trivariate results.

[Note to readers: This long chapter covers the extensive scope of the topic, but the main parts to learn are survival analysis (in Sections 22.2 through 22.5) and life expectancy (Section 22.6). If you are tired by the time you reach Section 22.7, you can go through the rest of the chapter quickly, and need not struggle with the details. — ARF]

22.2 Construction of Survival Summaries

Survival analysis is concerned with an event that concludes the person’s period of time “at risk.” In the following discussion, the “event” will be a “failure,” i.e., death, but the analytic methods are equally pertinent and applicable for other types of concluding events, such as development of a myocardial infarction, stroke, unwanted pregnancy, or even a desired pregnancy.

If time to death were known for each person, the cohort’s results could easily be summarized as a mean survival time. When most studies are ended, however, some members of the cohort may still be alive. Their unknown duration of survival will preclude calculation of an accurate mean for the group. (If the unknown durations are omitted for still-alive persons, the mean survival time for the group will be too low.)

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This problem can often be avoided, as noted earlier in Section 3.6.4, by summarizing results with a median survival time. It has the double advantage of being easily determined in most groups despite unknown durations for the still-alive persons, while also avoiding distortions of the mean by outlier members with particularly short or long survivals. In fact, as noted later, the median survival time is probably the best simple descriptive index for the results.

Nevertheless, neither the mean nor the median would show dynamic features of the survival curve, and neither index would permit effective predictions. If the median survival is 2.3 years, we can predict that 50% of the cohort will be alive at 2.3 years, but we would not know what the survival rates might be at serial times such as 6 months, 1 year, 3 years, or 5 years.

If survival time were known for everyone except those still alive at the end of the study, a summary survival curve could easily be constructed, as shown earlier in Figure 18.5 for the data in Figure 18.4, to depict the proportion of persons alive at each successive time interval. The choice of a suitable “parametric” algebraic expression for these curves has enriched statistical literature with the names of mathematical models called gamma, lognormal, Weibull, Rayleigh, and Pareto distributions.

A simpler algebraic model is possible, however, if the survival-curve pattern resembles the type of exponential decay that occurs for radioactivity. In this frequent situation, the data can often be fitted with a descending exponential expression, Y = e−ct, in which t = time, Y = the corresponding survival proportion (which is 1 at t = 0), and c is a constant appropriate for each curve. This expression can promptly be logarithmically transformed into a straight-line model, ln Y = −ct.

The summary expression for Y at each time point, however, is a proportion constructed as (n − d)/n, where n is the number of pertinent people at risk and d is the number of persons who have died at or before the cited time. Because the original denominator, n, is always known, this proportion could easily be determined if the numerator status were also known as dead or alive at each time point for each person. Unfortunately, in the realities of human follow-up studies, this status may not always be known; and even if known, the person’s condition may not always be easily classified in the simple dichotomy of dead/alive. This difficulty creates the problem of numerator losses — the prime challenge in survival analysis for medical phenomena.

22.2.1Sources of Numerator Losses

Everyone is known to be alive when the cohort is assembled at each person’s zero-state baseline. The cohort is then followed thereafter until a selected “closing” duration, T, which might be five years of serial follow-up for each patient. If a relevant death occurs before time T, the status of the patient is always known for each time point thereafter. If a relevant death has not occurred before timeT, however, problems are created by patients who have not been followed for as long as T. Such patients are called censored; and the censoring can arise from three (or four) mechanisms.

22.2.1.1 Insufficient Duration — In most cohort studies and clinical trials, the group is assembled by accrual during a period of calendar time, rather than being collected and entered into the study all at once on the same day. The research will therefore take much longer in calendar time than the shorter duration, T, that is usually chosen for the maximum length of each patient’s serial observation.

For example, suppose a particular study began on January 1, 1993 and ended on December 31, 1999. If T was set at 5 years, everyone who entered the study before January 1, 1995 had a chance to be followed for five years when the study ended; but someone who entered on July 1, 1997 was followed for only 2 1/2 years. If still alive on December 31, 1999, the latter person would have been censored at a follow-up duration of 30 months. In this type of “terminal censoring,” the ultimate status of the patient, who is still alive and under observation when the study closes, might easily be determined if the calendar time of the study were extended.

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22.2.1.2Intermediate Lost-to-Follow-Up — A more substantial problem is created by intermediate rather than terminal censoring. Intermediate censoring occurs when patients are last seen or known to be alive at a serial time before T, but are then “lost” or “dropped out” with nothing known of their status thereafter. Such intermediate lost-to-follow-up patients are censored (and called “withdrawn alive”) as of the last date on which they were known to be alive.

22.2.1.3Withdrawn by “Competing Risk” — If any kind of death is the failure event, the problem of a “relevant” death need not be considered. In many cohort studies, however, the failure event is death due to a specific disease. For example, in the follow-up of patients treated for cancer of the breast, the relevant deaths under analysis are usually those ascribed to cancer of the breast. If someone dies of an apparently unrelated myocardial infarction or automobile accident, the death is regarded as part of the “competing risk” of an incidental, non-relevant event. Such patients are censored at the date of death and are also regarded as “withdrawn alive.”

22.2.1.4Altered Therapy — A formidable problem in many randomized clinical trials is what to do about patients who abandon the originally assigned treatment, who may continue it in a poorly maintained schedule, or who may even transfer to the competing opposite therapy. For example, in a trial of medical vs. surgical treatment, some of the patients originally assigned to the medical therapy may later decide to have surgery. In a trial of an active vs. placebo pharmaceutical agent, some of the patients assigned to the active drug may comply so ineffectively that they essentially become an untreated counterpart of the placebo group. A further problem in any trial is the unauthorized use of additional treatment that may affect or obscure the actions of the main assigned agent(s). For example, if antibiotics A and B are being compared for their prophylactic ability to prevent a particular infection, the results may be distorted by patients who also take antibiotic C incidentally for some other reason.

The solution to the altered-therapy problem is controversial. In one popular statistical approach, called “intent-to-treat” (ITT) analysis, any therapeutic changes or supplements are ignored, and everyone is counted as though he or she had received the initially assigned treatment, in the exact regimen or schedule in which it was assigned. The main argument for this approach is that it is statistically “unbiased”: the original therapeutic classification is not affected by anything that happened after the initial “sanctifying” randomization. Consequently, the ITT approach makes no adjustments for altered therapy.

The counter-argument is that the ITT approach, although perhaps statistically unbiased, is scientifically improper. With scientific common sense, someone would not be counted as having received surgical treatment if the surgery was not done, nor would patients be counted as having received only medical therapy, if they later had the operation.

Because the ITT controversy has not yet been resolved, a possibly acceptable compromise is to withdraw the patient as censored when the major therapeutic alteration began. This type of withdrawal, if used, becomes a fourth mechanism for censoring.

22.2.2Adjustment of Denominators

Although patients who have died can be “followed” thereafter for any selected duration of time, the original size of the cohort is progressively reduced when patients are withdrawn as censored. Therefore, the calculated summary expressions at each time point should reflect the smaller numbers of people for whom suitable data are available. This problem — the need to change denominators sequentially after censored losses in the numerators — cannot be readily managed by parametric algebraic models. Accordingly, survival summaries are usually constructed with “non-parametric” methods that do not have an algebraic format and that fit the data in an ad hoc manner. For the postcensoring adjustments, the denominators of the remaining cohort can be decremented by two main mechanisms: a direct or an interval method; and the intervals can have a fixed or varying duration.

22.2.2.1 Arrangement of Data — In the ordinary arrangement of data, each person will have a calendar date for the time of randomization or some other event (such as date of diagnosis or admission) that acts as the zero time for calculating serial duration until the calendar date of exit. The exit state

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will be listed as the failure event (such as death) or one of the diverse sources of censoring. After the serial durations are determined, the persons can be listed in ranked order according to the durations. An excerpt of such a list might show the following:

Such a list is used in the two “interval” methods of calculation. For the “direct” type of calculation, however, another entry is needed to denote the maximum duration for which dead patients might have been followed during the calendar time of the study. For example, the patient ranked 29 in the foregoing list died at 18 months, but potentially might have been followed for 48 months, whereas patient 30, who died at 19 months, may have entered the study only two years before its end, so that the maximum survival duration would have been only 24 months.

22.2.2.2 Direct Method — In the direct method of adjustment, the denominator should contain only those persons who could have been followed for the appropriate length of time. Thus, one-year survival rates are calculated only for those persons who could have been followed for at least a year; two-year rates are calculated only for persons who could have been followed for at least two years; and so on.

Persons who died are counted as dead in the interval when they died, and in all subsequent eligible intervals thereafter. For example, someone who entered the cohort 18 months ago and who died at 5.4 months of serial time would be eligible for being counted as dead in the 6-month and 1-year but not in the 2-year survival rates. Persons who were censored are counted in the denominators of only the pertinent eligible durations. Thus, someone who was censored at 6.3 months would not be counted in the denominator for the 1-year or 2-year survival rates, but could appear in a 6-month survival denominator. Persons who entered the study 12 months before it ends can be counted in all 6-month or 1-year survival rates, but are not eligible for inclusion in a rate for any subsequent duration, even if they died soon after admission.

If pertinent eligibility for being counted is ignored, substantial biases can be introduced. For example, the direct 2-year survival rates might include the dead but not the still-living members of a group eligible for only 1 year of follow-up.

Table 22.1 shows the way the data of a direct analysis might be summarized for 150 persons who entered a study that lasted for three years, with 50 persons accrued at the beginning of each annual interval. The first row shows the annual occurrence of deaths and losses in the 50 persons who could have been followed for three years. The next two rows show analogous results for persons who could have been followed for only two years or for only one. The third line shows 43 patients remaining and terminally censored in the one-year group after 3 deaths and 4 intermediate losses. The second line shows 39 patients remaining in the second-year group after a total of 5 deaths and 6 intermediate losses during the two-year period.

Table 22.2 shows the direct arrangement of survival for the 150 persons cited in Table 22.1. All 150 persons were potentially eligible for 1-year survival calculations, but the 6 persons lost to follow-up during that year are removed from the denominator. The mortality rate at one year is thus calculated as 9/144 = .0625. Only 100 persons are potentially eligible for inclusion in 2-year survival rates, but 13 of them have become lost to follow-up (2 in the first year and 11 in the second). With 12 deaths (= 3 + 3 + 4 + 2) having occurred in the two-year period, the 2-year mortality is 12/87 = .1379. Finally, among the 50 persons potentially eligible for 3-year survival calculations, 10 (= 1 + 6 + 3) have been lost and 9 (= 3 + 4 + 2) have died. The 3-year total mortality rate would be 9/40 = .2250. The corresponding total

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TABLE 22.1

Summary of Data that Precede a Direct Survival Analysis for 150 Persons

TABLE 22.2

Direct Arrangement of Survival Data for 150 Persons in Table 22.1

survival rates at 1, 2, and 3 years respectively are thus .9375, .8621, and .7750. These results are reasonably close to those noted later with the interval methods.

An important point to bear in mind is that if the direct method is used without suitable attention to the “eligible death” group, the mortality rates will become excessively high after the first interval. For example, the group eligible for only 1 year in Table 22.1 had 3 deaths. If those 3 deaths are carried into the 2nd year calculation in Table 22.2, the mortality numerator becomes 12 + 3 = 15 and the denominator would be 87 + 3 = 90, so that the total mortality rate at 2 years would be raised to 15/90 = .1666. The group eligible for only 2 years in Table 22.1 had 5 deaths in those two years. If those 5 deaths plus the other 3 deaths from the only 1-year eligible group were included for the 3rd-year calculation in Table 22.2, the total mortality numerator would be 9 + 8 = 17 and the denominator would be 40 + 8 = 48. The total mortality rate would rise to 17/48 = .3542 and the corresponding survival rate, .6458, would be substantially lower than the correct value of .7750.

When properly calculated, the direct method is simple and obvious, and it has the scientific advantage of being straightforward and promptly understood. The survival rate at each time point indicates the exact number of persons who were eligible to be counted and classified at that point. The direct method, however, does not acknowledge the survival contributions of censored people to the interval in which they were censored. For example, persons who are censored at 23 months make no contribution to the 2-year survival rate, although they survived for 23/24ths of that period.

The direct results thus offer easy comprehension, but an underestimation of the dynamic survival rates. The latter problem could be substantially reduced by calculating the rates at frequent intervals, such as monthly rather than yearly, but a slight underestimate would still remain. Another disadvantage of the direct method is a paradox that can occur when the survival rate is relatively high in the few persons who have (or could have) been followed for the longest durations of time. In such situations, the direct 5-year survival rate may be higher than the corresponding rates at earlier time points.

Perhaps the main disadvantage that makes the direct method seldom used, however, is mathematical. Being calculated directly at each time point, the total survival rates lack the mathematical appeal of a series of interval survival rates, discussed in the next few sections, that are multiplied to form each total survival rate as a cumulative product.

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22.2.2.3 Actuarial (Fixed-Interval) Method — Long before the outcome of clinical cohorts became a challenge in biostatistics, the actuaries of life insurance companies had developed a method, called life tables, to manage survival analyses. The actuaries particularly wanted to know about life expectancy, so that their companies could set appropriately profitable “premiums” for the cost of policies sold to the public. The derivation of life expectancy from a life-table survival analysis is discussed later in Section 22.6. The analyses themselves, however, rely on constructing a cumulative product of fixedinterval survival rates.

In the latter process, the total observation period for the group is divided into a series of follow-up intervals, such as 0–1 year, 1–2 year, 2–3 year, etc. At the end of each interval, the numbers are catalogued according to persons who were “at risk” when the interval began. They then died or were censored during the interval, or they were alive and under follow-up when it ended. An interval mortality rate is calculated as the number who died divided by an appropriate denominator of those at risk during the interval. The interval mortality rate, which is called the hazard for the interval, is then converted to an interval survival rate. The final result, called the cumulative survival rate, is the product of all the component interval survival rates.

For example, suppose pt represents the survival proportion for each interval. When the observations begin, everyone is alive and p0 = 1. If p1 is the interval survival at the end of the first interval, the cumulative survival is S1 = 1 × p 1 = pl. At the end of the second interval, with p2 as its survival proportion, the cumulative survival is S2 = S1 × p 2. At the end of the fifth interval, S5 = S4 × p 5 = p1 × p 2 × p 3 × p4 × p 5. This construction should be intuitively evident, but can be illustrated as follows:

In a cohort of 100 persons for whom the outcome is always known to be either alive or dead, suppose 10 die in the first interval, 13 in the second, and 24 in the third. The interval survival rates will be (100 − 10)/100 = .900 for the first interval, (90 − 13)/90 = .856 for the second interval, and (77 − 24)/77 =

.688 for the third. The cumulative survival rate at the end of the third interval is the product of these three interval rates: .900 × .856 × .688 = .530. [The direct method of calculation yields the same result: 47 persons have died at the end of the third interval, and so total survival rate at that time is (100 − 47)/100 = .530.]

The tricky statistical decision in this approach is the choice of an appropriate denominator for those at risk during each interval. Everyone who died in an interval or who lived through it was at risk throughout the interval, but those who were intermediately censored were not at risk for the entire time. According to an arbitrary but customary convention, half the censored persons are assumed to have been at risk for the interval. (Alternatively phrased, if the interval is one-year long, each intermediately censored person is assumed to have been at risk for half a year. Thus, if 4 persons are censored respectively at 2, 5, 7, and 10 months after the one-year interval began, their total duration of observation in that interval is 24 months, which yields an average of 6 months, or 1/2 year, for each of the 4 persons).

With the actuarial approach for fixed-interval survival rates, the data in Tables 22.1 and 22.2 can be rearranged as shown in Table 22.3. The first line in Table 22.3 begins with all 150 patients, of whom 6 (= 4 + 1 + 1) were intermediately censored during the interval and 9 died. The denominator is adjusted only for the 6 lost patients, and it becomes 150 – (6/2) = 147. The interval mortality rate is 9/147 = .0612 and the corresponding survival rate is .9388. Because 43 patients of the “1-yr.-only” group were withdrawn at the end of the first interval, the second-year interval is begun by 92 persons (= 150 − 6 − 9 − 43), of whom 11 are lost, and 6 die. The adjusted denominator is 92 – (11/2) = 86.5; the interval mortality and survival rates are respectively 6/86.5 = .0694 and .9306. The cumulative survival rate at the end of the second year is .9388 × .9306 = .8737.

After the 39 remaining persons in the 2-year-only group are censored at the end of that interval, the third-year interval begins with 36 persons at risk, of whom 3 are intermediately censored and 2 die. The interval mortality rate is 2/[36 − (3/2)] = .0580; and the interval survival rate of .9420 leads to a threeyear cumulative survival rate of .8737 × .9420 = .8230. Note that the cumulative survival rates with the actuarial method are only slightly higher than with the direct method for the same data. The differences in the two methods can sometimes be more substantial, however, as shown in Exercise 22.1.

In the actuarial method, each interval usually has the same duration, but sometimes the lengths can vary so that a few intervals might be one year, and others, 6 months long. Regardless of equality in durations, however, the temporal location of the intervals is usually fixed in advance, before the analysis

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TABLE 22.3

Fixed-Interval Actuarial Arrangement of Survival Data for 150 Patients in Table 22.2

begins. In medical activities, the fixed-interval approach is sometimes called the Berkson-Gage method,1 commemorating the Mayo Clinic statisticians who introduced the actuarial or “life table” method into medical research. The Berkson-Gage eponym also serves to differentiate the fixed-interval method from the Kaplan-Meier method, discussed in Section 22.2.2.5, which uses intervals of varying length for the actuarial life-table procedure.

22.2.2.4 Hazard Functions — Actuaries and demographers who specialize in analyzing the “forces of mortality” in a cohort use the term hazard for the interval mortality rates. For example, if a particular population has an annual mortality rate of .007, the hazard of death is .007 for that year.

For interval analyses, the hazard is the proportionate change in cumulative survival rate between any two intervals. Thus, if the cumulative survival rate is S4 at interval 4 and S3 at interval 3, the change is

(S3 − S4)/S3. Because S4 = p4S3, the hazard is (S3 − p4S3 )/S3 = S3(1 – p4)/S3 = 1 − p4, which is the interval mortality rate. Consequently, if n3 people are followed for the third interval, and if d3 persons die during

that interval, the hazard is d3/n3.

In radioactive decay, the rate of decay at any moment is constantly proportional to the amount of radioactivity, Y, at that moment. The expression for the rate of radioactive decay over time is dy/Y = –c, which (if you remember the calculus) becomes converted to ln Y = −ct, and Y = e−ct. The value of c in this expression is the constant hazard, or the logarithmic rate of decay, over time.

With your consent to some more mathematics, we can note that if successive temporal values of the cumulative survival curve are denoted as St, the general formula for the type of change just discussed is (St − St − 1)/St − 1. This expression (again recalling the calculus) is the time derivative of the natural logarithm of St. Thus, if St has an exponential format, ln St = −ct. To make the latter expression positive, we can examine −ln St = ct. If we then take the time derivative, we get d( −ln St) = c, which is the value of the hazard.

In the realities of medical events, the survival curve seldom has a perfect exponential shape and the hazard is seldom fixed at a constant value. In customary nomenclature, however, St is called the cumulative survival function, −ln St is called the cumulative hazard function, and d(−ln St) is called the hazard function. The values of the hazard function at different time points are usually approximated by the interval mortality rate, qt = 1 − pt.

These ideas about hazards are not particularly important or necessary for the simple life tables under discussion in this chapter, but the concept becomes important in more advanced statistics, when a multivariable procedure, called proportional hazards analysis (or “Cox regression”), is used to evaluate the effect of different baseline covariate factors — such as age, stage, anemia — on survival curves. The concept of a hazard is also used in Section 22.4.5 to designate an index called the hazard ratio.

22.2.2.5 Variable-Interval Method — The variable-interval life-table method was created as a result of some “matchmaking” by John Tukey.2 Knowing about a common interest in time-to-failure events, he brought together workers in two different fields. The workers were Edward L. Kaplan, then at Bell Laboratories, who was studying the “survival” time of vacuum tubes, and Paul Meier, then a biostatistician at Johns Hopkins Medical School, who was interested in analyzing post-therapeutic human survival.

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Kaplan and Meier were troubled by the arbitrary fixed-duration intervals and adjustments of the Berkson-Gage method and were especially worried about the effects in small samples. The alternative Kaplan-Meier strategy3 was to let the intervals be defined by the observed events: a new interval would be demarcated every time someone died. Anyone who was censored during one of those death-defined intervals would be simply eliminated from the denominator of that interval, as in the direct method of calculation discussed in Section 22.2.2.2. (Because no adjustments are made for a contribution to the interval, the contribution may not be missed if the deaths occur at small enough intervals. Problems can arise, however, when patients are censored during the long intervals that often appear near the right end of many survival curves.)

Because the events must be exactly timed in the Kaplan-Meier method, they cannot be grouped for fixed intervals as in Tables 22.2 or 22.3. Instead, the serial durations and corresponding exit states must be individually examined for each person. To illustrate these entities for the cohort of 150 people under scrutiny in Table 22.3, suppose the nine deaths in the first year occurred at 0.2, 0.3, 0.4, 0.4, 0.5, 0.5, 0.6, 0.7, and 0.9 years, and the six intermediate censorings at 0.3, 0.4, 0.6, 0.6, 0.7, and 0.9 years. To avoid analytic ambiguity, deaths and censorings cannot take place at exactly the same time. Accordingly, when a death and censoring seem to occur identically (as in many time points in this example), the death is assumed to have happened just before the censoring. The death thus terminates the interval, and the censoring is ascribed to the next interval. For example, for the death and censoring that both occurred at 0.3 yr, the death is counted in that interval; the censored person is removed from the denominator of the next interval.

For the remaining persons in Table 22.3, the deaths occurred at 1.1, 1.4, 1.5, 1.5, 1.6, 1.9, 2.3, and 2.8 years; and the intermediate censorings at 1.1, 1.1, 1.4, 1.4, 1.5, 1.5, 1.5, 1.9, 1.9, 1.9, 1.9, 2.3, 2.8, and 2.8 years. In addition, 43 persons of the 1-year-only group were terminally censored at 1.0 year, and 39 persons of the 2-year-only group were terminally censored at 2.0 years.

Table 22.4 shows an appropriate Kaplan-Meier life-table arrangement for the foregoing data. The first interval in Table 22.4 begins with a cumulative survival rate of 1.000 and ends with the death at 0.2 year. The death drops the interval survival rate to 149/150 = .9933; and so the second interval begins with a cumulative survival rate of 1.000 × 0.9933 = 0.9933. After the death that ends the second interval at 0.3 yr, the interval survival rate is 148/149 = .9933. The third interval thus begins with a cumulative survival rate of .9933 × .9933 = .9866, and the denominator is reduced to 147 when decremented for the 1 censoring at 0.3 yr. With 2 deaths at 0.4 yr, the interval survival rate for the third interval is 145/147 = .9864, which makes the cumulative survival .9866 × .9864 = .9732.

The procedure then continues step by step, so that the seventh interval begins with a cumulative survival of .9461 and 137 persons at risk. The interval ends with a death at 0.9 yr. The one person censored at 0.9 yr and the 43 people followed for only 1 year are removed from the denominator of the eighth interval, which ends with a death at 1.1 yr. As the process continues, the thirteenth interval begins with a cumulative survival rate of .8750 and with a “risk group” (number alive) that has been reduced to 36, after removal of all the previous deaths, intermediate censorings, and terminal censorings. The fourteenth interval begins with a cumulative survival of .8507 and with 34 people who are alive and being followed. The interval survival rate of .9706 would make the 15th interval begin with a cumulative survival rate of .8257 and with 33 persons at risk, but no further deaths have occurred, so the table ends here. Note that the sum of deaths (17) plus censorings (102) plus final number at risk (31) in Table 22.4 equals the 150 persons who began. Note also that the Kaplan-Meier cumulative survival rates of .9392,

.8750, and .8257 correspond to the respective Berkson-Gage results of .9388, .8737, and .8230 in Table 22.3.

The Kaplan-Meier approach is sometimes called a product-limit method, because the cumulative survival is a product of all the component interval survival rates. Because the multiplicative tactic is also used in the Berkson-Gage method, variable-interval is a better generic title to distinguish the KaplanMeier procedure. The term actuarial or life-table analysis can be applied to either method, but is usually reserved for the fixed-interval technique.

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TABLE 22.4

Variable-Interval (Kaplan-Meier) Arrangement of Data in Table 22.3*

* Additional data regarding timing are provided in Section 22.2.2.5 of text

22.2.3Display of Results

Graphs for the fixed-interval and variable-interval survival summary curves of Tables 22.3 and 22.4 are shown in Figure 22.1. The fixed interval points are usually connected directly with straight lines; if the intervals are close enough, the lines may resemble a relatively smooth curve. The variable-interval results are usually shown as the flat lines of a “step function.” With a drop occurring for the death(s) at each step in the irregular intervals, the curve looks like a staircase with uneven steps. In a strict mathematical portrait, however, the Kaplan-Meier curve would be the horizontal lines of a “step function,” without connecting vertical lines. The vertical lines are usually added for visual esthetics, to avoid a ghostly staircase.

The fixed-interval and variable-interval methods usually produce relatively similar results, as in Figure 22.1, but disparities can sometimes occur.

22.2.4Choice of Methods

As in many other issues in statistical communication, the choice of a survival-arrangement method often depends on the background and viewpoint of the data analyst and reader. Because the denominators are reasonably well decremented for censored patients in all three methods, any one of the three offers a reasonably satisfactory approach. Contemporary biostatisticians (perhaps in deference to Meier) usually recommend the variable-interval method.

To provide prompt and easy-to-understand scientific communication, the direct method is most desir - able because it shows the exact constituents of each survival rate, rather than a multiplicative product whose components are not displayed. The survival rates in the direct method are slightly underestimated because the censored patients do not contribute to the interval in which they are censored, but this same disadvantage occurs with the variable-interval method. The variable-interval method gets its advantage only if deaths occur at frequent intervals so that a small duration is available for each censored loss. This same advantage can be gained for either the direct or the fixed-interval method if the rates are determined at arbitrarily frequent intervals. Because the frequency of intervals in the Kaplan-Meier method is data-dependent and cannot be altered, this method can have substantial disadvantages for long intervals that have no deaths but many censorings.

© 2002 by Chapman & Hall/CRC