Life Table Analysis David Wesley METHODOLOGY

METHODOLOGY Life Table Analysis David Wesley

Abstract: Life table analysis is an effective way to present and evalu- Address: Cologne Life Reinsurance, ate survival data in a number of circumstances. The summary tables PO Box 300, 30 Oak Street, and survival curves that are commonly seen in the medical literature Stanford, CT 06904, 0300 are frequently derived through life table analysis. A basic understanding of life table construction is of benefit to the medical director Correspondence: David Wesley, MD, who wishes to abstract comparative mortality data from study Vice President & Chief reports. This article reviews the basic methodology of life table analy- Medical Direftor sis with a particular focus on handling right-censored study participants as withdrawals. Key Words: Life Table Analysis

Received: November 1, 1998

Accepted: November 20, 1998

Journal of Insurance Medicine 1998, 30:247-254

Previous explanations of life table analysis death and would yield only one outcome cat- have left readers of the Journal of Insurance egory. Medicine with a misunderstanding of withdrawals. We have been led to believe that To help explain these statements, I offer the there are at least three outcome categories for following chapter from the syllabus I have patients in a follow-up survival study: used in recent years for teaching mortality deceased, alive, and censored. Censored can methodology. This chapter precedes the subdivided into: lost to follow-up, switched chapter on mortality abstracts and provides treatment, ended treatment, and other dis- an understanding of life table analysis from qualifying outcomes as defined by the the perspective of the researcher. I use a nota- research protocol. But all censored patients tion for cumulative survival and numbers at are treated as withdrawals for purposes of risk that is different than typically seen in the life table analysis. In fact, for purposes of life Journal. This notation is from the clinical lit- table analysis, there are only two outcome erature and I find it easier to read because it categories: deaths and withdrawals. Those does not require the eye to distinguish upper who are alive at the end of a study are actual- and lower case p’s. It also avoids the confu- ly withdrawals. sion between NER and exposure. The entire syllabus is available upon request. When a study reports "total ascertainment" or a similar phrase, all this means is that all Survival Analysis the patients can be accounted for at the end of The researcher who is interested in survival the study period. This is not the same as total (or its complement, mortality) data for a par- ascertainment of survival. This would require ticular disease faces several problems. Usual- following each patient in the group untilly, only a small number of individuals can be 247 JOURNAL OF INSURANCE MEDICINE VOLUME 30 NUMBER 4 1998 studied. Enrollment in the study occurs ran- ods will provide a sound foundation for domly and withdrawal from the study can be doing comparative mortality abstracts, both unpredictable. Survival study methods must because many of the same concepts apply be chosen that maximize the amount of infor- and also because one gains an appreciation of mation that can be derived, given the above why the data is presented as it is in the med- constraints. An understanding of these meth- ical literature.

TABLE 1 Sarcoma X Registry (Registry of Patients with Sarcoma X)

P~¢i~t 1989 1990 1991 1992 1993 1994 1995 Outcome Yrs F/U A [-~ Death 3.7 B Alive 5.7 C [ -} Withdrawn 2.2 D [ Death 4.3 E [, Alive 5.3 [ -] Death 3.3 G -] Death 1.8 H Alive 4.0 Alive 3.5 J Withdrawn 1.7 Alive 2.8 r -} Withdrawn 1.0 Death 1.1 Alive 1.7 o Death 0.9 Alive 1.2 Q Alive 0.7 R Alive 0.5 To~l 45.4

Note Table 1. This displays the results of a enrolled over the next six years. The study survival study done on patients with a rare, ended 10/31/95 when the researcher decided lethal, and fortunately hypothetical cancer it was time to publish the results. Three that we will call sarcoma X. As is typical of patients (C, J, L) moved away and were lost to follow-up studies, enrollment occurs over a follow-up. Six patients (A, D, F, G, M, O) died considerable spread of time and the individu- during the study while the balance were still als in the cohort have very different terms of alive when the study was halted. exposure. Note that the study began on 11/1/89 with one patient and 17 others The sarcoma X researcher can choose one of 248 VOLUME 30 NUMBER 4 1998 JOURNAL OF INSURANCE MEDICINE

several approaches to summarizing the sur- that it takes into account all the survival vival results of this study: simple survival, n-durations. It also yields an annualized mor- year survival, and life table analysis. tality or survival rate that is easier to compare with other studies of varying length but for Simple Survival which an annualized mortality can be calcu- Mean survival, median survival and the lated. The limitation of the person-year overall survival rate are the least accurate approach is that it does not allow for situa- methods for summarizing survival data. tions where the mortality risk varies over They fail to adequately account for outliers, time. For example, in most cancers the mor- follow-up duration and the survival experi- tality is extremely high in the first few years ence of patients withdrawn or still alive at the after diagnosis but then drops off rapidly to end of the study. The weakest survival statis- standard or near standard mortality rates. On tic is mean survival. The reader is probably the other hand, early stage prostate cancer familiar with a form of mean survival known has a long latent period, so the mortality is as "life expectancy." not so high shortly after diagnosis but climbs later. For the sarcoma X data, the mean survival, median survival and the overall survival rate Life Table Analysis are 2.5 years, 2.6 years and 67% respectively. Life table analysis utilizes a stratified person- years approach and can give mortality or sur- n - Year Survival Rate vival rates for any interval or overall. Actuar- In this case, the duration is explicit. However, ial life table analysis is especially apt for life the denominator remains a problem. Consid- insurance mortality studies because the er a 5-year survival rate. If we include in the analysis and the results are based upon year- denominator all the patients who did not die ly intervals. Using different end-points such during the study, the result is an overly pes- as the state of disability or the state of reha- simistic 2/18 = 11% (only 2 patients survived bilitation, one can use actuarial life table >_ 5 years). If we exclude from the denomina- analysis methods to determine rates that can tor the withdrawals and those who were assist insurers in underwriting disability alive but followed for less than 5 years, then products fairly. we have 1/7 = 14% which is still too low since it ignores the survival experience of the 11 Life table analysis requires well-defined start- excluded patients. points (time-zero), end-points and exposure. For mortalitN we will consider the outcome Person-Years Survival Rate (end-point) to be death and exposure to be Here the person-years mortality rate is calcu- defined as exposure to the excess mortality lated and then subtracted from unity to give associated with an impairment. In disability its complement, the person-years survival studies, these points and exposures are more rate. difficult to define. In all cases, consistency throughout the study is important. The denominator for the person-years mortality rate is expressed in units of person- Life table analysis employs four assump- years. In the example study, the total number tions: of observed person-years is 45.4. The person- 1. The mortality risk is independent of the years mortality then would be 6/45.4 -- 0.132 calendar, i.e. no seasonal variation. There or 0.132 deaths per person-year and the per- are a small number of circumstances son-years survival would be 86.8%. where this assumption is not valid. 2. Withdrawals (and study cessation) are The strength of the person-years approach is independent of mortality risk. This

249 JOURNAL OF INSURANCE MEDICINE VOLUME 30 NUMBER 4 1998

assumption is frequently invalidated. For eliminate front-end intervals of very high example, if a study is meant to determine mortality, i.e. the first 30 days after CABG. the mortality of patients post percuta- Since life table analysis is stratified by neous transluminal coronary angioplasty interval, there is no need for constant mor- (PTCA), but those who later require emer- tality between intervals. gency coronary artery bypass grafting 4. Start and end points are well defined. (CABG) are censored (treated as with- Death is usually easily defined but out- drawals), then the survival for PTCA comes other than death can be studied. patients will be positively biased. The beginning of exposure to the risk mu~t also be clearly identifiable. 3. The mortality risk remains constant within the study intervals. While yearly intervals Life table construction is made easier if we are most convenient for our purposes, an first re-arrange the study table to start every- impairment with rapidly changing mortal- one out at the same time-zero. This requires ity may require smaller intervals to satisfy the assumption that calendar year is not a this assumption. A common need is to factor.

Table 2: Life Table Preparation, Sarcoma X

Time.> 0 1 2 3 4 5 A Death 3.7 B Alive 5.7 C -} Withdrawn 2.2 D -] Death 4.3 E Alive 5.3 F Death 3.3 G Death 1.8 H Alive 4.0 I Alive 3.5 Withdrawn 1.7 K Alive 2.8 L Withdrawn 1.0 M Death 1.1 N Alive 1.7 O Death 0.9 P Alive 1.2 ^ Q Alive 0.7 R ^ Alive 0.5

250 VOLUME 30 NUMBER 4 1998 JOURNAL OF INSURANCE MEDICINE We can now create the following actuarial life table.

Table 3: Actuarial Life Table interval I interval number interval interval interval interval entrants withdrawals at risk deaths mortality survival cumulative x to x+l Ix wx rx dx qx Px Sx 0- 1 18 2 17 1 0.059 0.941 0.941 1 - 2 15 4 13 2 0.154 0.846 0.796 2 - 3 9 2 8 0 0.000 1.000 0.796 3 - 4 7 2 6 2 0.333 0.678 0.540 4 - 5 3 0 3 1 0.333 0.678 0.3~6 5 - 6 2 2 1 0 0.000 1.000 0.366

The interval goes from time x to time x + 1 Sx = Px ° Sx-1 since we are using 1 year intervals, lx is the number of those alive to enter the interval. Note the convention of assigning a withdrawal occurring at an exact multiple of the wx is the number of withdrawals, also called yearly interval (e.g. 1.0, 4.0) to the nearest censored participants. Note that those alive at even numbered interval (up to second, down the end of the study period are treated as to fourth). This avoids bias from always withdrawals. rounding in one direction.

The number of persons at risk is labeled rx . In considering mortality risks we most often Its calculation is based upon the assumption are concerned with annual mortality rates. A that withdrawals occur evenly throughout relatively ineffective approach would be the the interval. Thus, on average, each with- arithmetic average in which we take the 6- drawal is at risk for ½ of the withdrawal year cumulative mortality (1- $6) and divide interval and by 6. How can we test the accuracy of the answer, 0.106? We use the complement of rx = Ix- Wx/2 0.106 as a interval survival rate for 6 intervals and see if the resulting cumulative survival is Interval mortality is calculated from the dx, 0.366: the number of deaths, and rx. S = (1 - 0.106) ¯ (1 - 0.106) ¯ (1 - 0.106) ¯ (1 - qx = dx 0.106) (1 - 0.106) ¯ (1 - 0.106) ¯ (1 - 0.106) rx S = (0.894)6 = 0.511 Interval survival, Px, is the complement of qx for each interval. One can calculate Sx the Not very accurate, but it does point in the cumulative survival by multiplying the inter-right direction. If we solve the following val survival rates: equation for ]~ we have what is called the geometric mean survival rate and its complement Sx = (pl)(P2)(P3)...(px) would be the geometric mean mortality rate:

However, another way to calculate Sx that SN = (/~)N = (1- ~)N can be quicker is: /~ = 1- q = (s)7N = XY$-

251 JOURNAL OF INSURANCE MEDICINE VOLUME 30 NUMBER 4 1998 where N is the number of intervals.

In this case, S = 0.366 and N = 6 years:

~ = 1- (S)7N = 1- ~- = 0.154 For the Sarcoma X study each interval was one year in length, so the exposure for each The resulting q^ is an annualized figure. The interval was equivalent to the rx . Table 4 geometric mean is to be used whenever only a shows the aggregate mean mortality for Sar- cumulative survival is available, which iscoma X over the entire 6 year interval. often the case. It can be used for cumulative intervals other than the entire study, e.g. the Table 4: Aggregate Mean Mortality first three years of this study. However, use of Aggregate Aggregate Aggregate Aggregate the geometric mean assumes that mortality Interval Exposure Deaths Mean Mortali- remains constant from interval to interval. ty x to x+t ~E ~d Where we have stratified interval data, we can x x -~ use much better estimates of the mean: the 0 - 6 48 6 0.125 aggregate mean mortality, and the aggregate mean survival. In contrast to the case of the geomet- Again, the average mortality (or its comple- ric mean figures, the aggregate mean mortali- ment, survival) is an annualized figure since ty must be calculated first and the aggregate exposure is in person-years. mean survival found as its complement. Before we can calculate q 2, we must deter- Kaplan-Meier (Product-Limit) Table mine E, the exposure, which is measured in But before moving on to mortality abstracts, person-years. Ex is the product of rx, the num- we should have a brief review of the Kaplan- ber at risk during an interval, and the length of Meier life table methods. The construction of the interval in years: a Kaplan-Meier life table begins with orga- nizing the experience data as in Table 2, just Ex = rx ¯ tx as we did for the actuarial life table method. The major difference is that instead of look- and the aggregate mean ing at evenly spaced intervals and considering deaths and withdrawals that occurred during the timed intervals, in the Kaplan- Meier method one looks at intervals defined by the times of deaths. In the Sarcoma X Table 5: Kaplan-Meier Life Table The following Kaplan-Meier life table can be constructed from the data in the sarcoma X study, number mortality interval cumulative time ’ at risk deaths rate survival survival t rt dt qt = dtlrt Pt = 1 - qt St = (Pl)(P2)’’’

0.9 16 1 0.062 0.938 0.938 1.1 14 1 0.071 0.928 0.870 1.8 10 1 0.100 0.900 0.783 3.3 7 1 0.143 0.857 0.671 3.7 5 1 0.200 0.800 0.537 4.3 3 1 0.333 0.667 -0.358

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study, the first death occurred at 0.9 years. approaches lead to fairly similar results, so go This marks the end of the first interval. The with whichever one is on your computer." second interval ends at 1.1 years, the time of the second death, etc. Conclusion The medical director who wants to abstract Withdrawals are censored from the interval mortality information from the medical liter- in which they are withdrawn. Thus the num- ature is most often faced with a graphical ber at risk consists of those alive and at risk summary of the survival data as shown in fig- (not withdrawn) at the end of the interval ures 1 and 2. plus the death(s). By convention, when a death and a withdrawal occur at exactly the same time point, the death is assumed to 1 have occurred first, during the preceding interval and the withdrawal is assumed to 0.8 have occurred during the following interval. 0.6

t is the time since time-zero, when the indi- 0.4 vidual enrolled in the study. 0.2 rt is the number at risk, those alive at time t 0 plus the interval death(s). 0 1 2 3 4 5 6 dt is the number of deaths occurring at time t. Figure 1: Survival Curve, Actuarial Life Table Since no deaths occurred simultaneously in this study, the dt is 1 for each interval. qt, Pt, and St are straightforward calculations. The Kaplan-Meier method is optimal from the perspective of the deaths since they define the interval and the assumption of constant mortality during the interval is not necessary. However, the exposure of withdrawals during their last interval is ignored which may cause an overly pessimistic esti- Figure 2: Kaplan-Meier Survival Curve mate of the mortality. If a curve is all that is available, one must set- Kaplan-Meier survival curves are frequently tle for measuring the cumulative interval sur- published in the medical literature. They vivals off the y-axis for survival curve points have a characteristic staircase pattern with that correspond to yearly (or other regular vertical drops in the curve corresponding to interval) points on the x-axis. The geometric the times when deaths occurred. Which is mean survival (and corresponding mortality) better, actuarial life table analysis or Kaplan can then be calculated. This works adequate- Meier survival analysis? Theoretically, the K- ly for Kaplan-Meier curves as well, despite M approach works best where the number of their steps at irregular intervals. study participants is less than 50. Where the N is > 50, the life table strengths make it a bet- Fortunately, it is increasingly common for ter methodology. However, to quote Norman study reports to include numbers of entrants and Streiner, "in most cases the two to intervals under the corresponding points

253 JOURNAL OF INSURANCE MEDICINE VOLUME 30 NUMBER 4 1998 on ’the x-axis. Authors use different terms for References the numbers (exposed, number at-risk, etc.) 1. Cutler SJ, Ederer F. Maximum utilization of the life table method in analyzing survival. J Chron Dis 1958;8:699-712. but usually they are in fac~ interval entrants 2. Kramer MS. Clinical Epidemiology and Biostatistics. A primer (Ix’s). The lx’s and interval survivals can be for clinical investigators and decision-makers. New York: used to solve for interval withdrawals. Springer-Verlag, 1988. 3. Lancaster HO. An introduction to medical statistics. Sydney: Armed with an understanding of life table John Wiley 8c Sons, 1974. construction, the medical director can then 4. Norman GR, Streiner DL. Biostatistics: the bare essentials. re-construct the full life table summarized by Toronto: Mosby, 1994. 5. Spiegel MR. Schaum’s outline of theory and problems of sta- the graph. In particular, one needs to remem- tistics. New York: McGraw-Hill Book Company, 1961. ber that patients alive at the end of the study are treated as withdrawals.

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