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Life Table Analysis David Wesley METHODOLOGY JOURNAL OF INSURANCE MEDICINE Copyright © 1998 By Journal of Insurance Medicine 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 under- standing 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 partici- pants 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 with- drawals. 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 mor- tality 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.
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