for each x, and the rest of the computations follow suit. From any given schedule of probabili- ties q0,q1,q2,..., the lx table is easily computed A life table is a tabular representation of central sequentially by the relation lx 1 lx(1 qx ) for x features of the distribution of a positive random 0, 1, 2,.... Much of the effort+ in= life-table− construc-= variable, say T , with an absolutely continuous tion therefore is concentrated on providing such a distribution. It may represent the lifetime of an schedule qx . individual, the failure time of a physical component, More conventional{ } methods of life-table construc- the remission time of an illness, or some other tion use grouped survival times. Suppose for sim- duration variable. In general, T is the time of plicity that the range of the lifetime T is subdivided occurrence of some event that ends individual into intervals of unit length and that the number of survival in a given status. Let its cumulative failures observed during interval x is Dx. Let the distribution function (cdf) be F (t) Pr(T t) corresponding total person-time recorded under risk = ≤ and let the corresponding be of failure in the same interval be Rx . Then, if µ(t) S(t) 1 F (t), where F(0) 0. If F( ) has the is constant over interval x (the assumption of piece- = − = · probability density function (pdf) f(), then the risk wise constancy), then the death rate µ D /R is · x x x of event occurrence is measured by the hazard the maximum likelihood estimator ofˆ this= constant. µ(t) f (t)/S(t), for t where S(t) > 0. Because Relation (1) can again be used to provide an estimator of its= sensitivity to changes over time and to risk qx 1 exp( µx ), (2) differentials between population subgroups, µ(t) is a ˆ = − −ˆ centerpiece of interest in empirical investigations. and the crucial first step in the life-table computation In applications to human mortality, which is where has been achieved. Instead of (2), µ 1 1 µ is life tables originated, the time variable normally is a x 2 x often used to estimate q . This solutionˆ is+ ofˆ older person’s attained age and is denoted x. The function x vintage and may be regarded as an%& approximation' µ(x) is then called the or death to (2). intensity (see Hazard Rate). The life-table function Two kinds of problems may arise: (i) the exact l 100 000 S(x) is called the decrement function x value of R may not be known, and (ii) the constancy and= is tabulated for integer x in complete life tables; x assumption for the hazard may be violated. in abridged life tables it is tabulated for sparser values When the exact risk time R is not known, some of x, most often for five-year intervals of age. The x approximation is often used. An Anglo-Saxon tra- radix l is selected to minimize the need for decimals 0 dition is to use the midyear population in the age in the l table; a value different from 100 000 is x interval. Alternatively, suppose that the number N sometimes chosen. Other life-table functions are the x of survivors to exact age x and the number W of expected number of d l l at age x x x x 1 withdrawals (losses to follow-up) in the age interval x (i.e. between age x and age =x 1),− the+ single- + are known. What has become known as the actu- year death probability qx Pr(T x 1 T > x) = ≤ + | = arial method then consists in approximating Rx by dx /lx, and the corresponding survival probability 1 Nx 2 (Dx Wx ). If there are no withdrawals and px 1 qx . Simple integration gives − + = − Nx is known, then Dx/Nx is the maximum likelihood x 1 + estimator of qx, and this provides a suitable starting qx 1 exp µ(s) ds .(1) point for the life-table computations. = − − ! "x # For the case where only grouped data are avail- Life-table construction consists in the estima- able and the piecewise-constancy assumption for tion and tabulation of functions of this nature from the intensity function is implausible, various meth- empirical data. If ungrouped individual-level data are ods have been developed to improve on (2). For available, then the Kaplan–Meier estimator can be an overview, see Keyfitz [12]. Even if single-year used to estimate lx for all relevant x and estimators age groups are used, mortality drops too fast in the of the other life-table functions can then be com- first year of life to merit an assumption of con- puted subsequently. Alternatively, a segment of the stancy over this interval. Demographers often use Nelson–Aalen estimator can be used to estimate µ0/[1 (1 a0)µ0] to estimate q0, where a0 is some x 1 ˆ + − ˆ + x µ(s) ds; (1) can then be used to estimate qx small figure, say between 0.1 and 0.15 [2]. If it is $ 2 Life Table possible to partition the first year of life into subin- then individuals who live at widely differing ages in tervals in each of which mortality can be taken as the period of observation cannot be expected to have constant, then it is statistically more efficient essen- the same risk structure, and the period table is said tially to build up a life table for this year. This leads to reflect the patterns of a synthetic (fictitious) cohort to an estimate like q0 1 exp( i µi ), where the exposed to the risk of the period at the various ages. sum is taken over theˆ = first-year− intervals.− ˆ See Dublin et al. [5, p. 24] for an example. ( The force of mortality is sometimes represented Multiple-decrement Tables by a function h(x; θ), where θ is a vector of When two or more mutually exclusive risks operate parameters. most often use the classical on the study population (see Competing Risks), one Gompertz–Makeham function h(x; a, b, c) a x = + may correspondingly compute a multiple-decrement bc for the force of mortality in their life tables table to reflect this. For instance, a period of sick- (see Parametric Models in ). ness can end in death or, alternatively, in recovery. When individual-level data are available, it would be Suppose that an integer K repre- statistically most efficient to estimate the parameters sents the cause of decrement and define Fk(t) by the maximum likelihood method, but most often = Pr(T t, K k), fk(t) dFk(t)/ dt, and µk(t) they are estimated by fitting h( ; θ) to a schedule ≤ = = = · fk(t)/S(t), assuming that all Fk( ) are absolutely of death rates µx , perhaps by least squares, · continuous. Then µk( ) is the cause-specific hazard minimum chi-square{ˆ } (see Ban Estimates), or some · (intensity) for risk cause k and µ(t) k µk(t) is method of moments. This approach is called analytic the total risk of decrement at time t. For= the multiple- graduation; for its statistical theory, see [11]. One of decrement table, we define the decrement( probability many alternatives to modeling the force of mortality (k) is to let [10] qx Pr(T x 1,K k T > x) = ≤ + = | 1 t qx (x B)C 2 x A + D exp[ E(ln x ln F) ] GH . exp µ(x s)ds µk(x t)dt. p = + − − + = − + + x "0 ! "0 # (3) So far we have tacitly assumed that the data (k) come from a group of independent individuals who For given risk intensities, qx can be computed by have all been observed in parallel and whose life- numerical integration in (3). The expected number times have the same cdf. Staggered (delayed) entries of decrements at age x as a result of cause k is (k) (k) into the study population and voluntary exits (with- dx lx qx . When estimates are available for the drawals) from it are permitted provided they contain cause-specific= risk intensities, one or two columns no information about the risk in question, be it death, can therefore be added to the life table for each cause (k) (k) recurrence of a disease, or something else. Never- to include estimates of dx and possibly qx . theless, the basic idea is that of a connected cohort Several further life-table functions can be defined of individuals that is followed from some significant by formal reduction or elimination of one or more of starting point (like birth or the onset of some disease) the intensity functions in formulas like those above. and which is diminished over time due to decrements In this manner, a single-decrement life table can be (attrition) caused by the risk’s operation. In demog- computed for each cause k, depicting what the normal raphy, this corresponds to following a birth cohort life table would look like if cause k were the only through life or a marriage cohort while their mar- one that operated in the study population and if it riages last, and the ensuing tables are called cohort did so with the risk function estimated from the data. life tables. The purpose is to see the effect of the risk cause Because such tables can only be terminated at the in question without interference from other causes. end of a cohort’s life, it is more common to compute Some demographers call this abstraction the risk’s age-specific attrition rates µx from data collected for pure effect. No assumption is made that in practice the members of a populationˆ during a limited period the total attrition risk can actually be reduced to the and to use the mechanics of life-table construction to level of the one which is in focus or that this cause produce a period life table for the population from operates independently of other causes. For instance, such rates. If mortality patterns are tied to cohorts, a single-decrement life table of recovery from an Life Table 3 illness reflects the pure timing effect of the duration If L 1 l dt, we get L 1 (l l ) by the x 0 x t x ∼ 2 x x 1 structure of the intensity of recovery even though the trapezoidal= rule+ of numerical integration,= + and+ elimination of mortality is unattainable. $

A single-decrement life table is at an extreme end ∞ ∞ lx t 1 of a class of tables produced by deleting one (or e˚x Lx t + ,(5) = + ∼ l − 2 t 0 = t 0 x more) of the cause intensities in formulas like those )= )= above. To obtain a cause-deleted life table, where which is normally used to compute values for e˚x . only cause k has been eliminated, one may introduce Equivalent names for the life expectancies are µ k(t) µ(t) µk(t), ˚ − = − mean survival time for e0 and mean residual survival 1 t time at age x for e˚ . The median length of life is the ( k) x qx− exp µ k(x s)ds µ k(x t)dt median in the distribution of T ; it used to be called = 0 − 0 − + − + " ! " # the probable length of life (see Median Survival x 1 + Time). Correspondingly, the median residual length 1 exp µ k(s) ds ,(4) = − − of life at age x used to be called the probable residual !"x # length of life. If we denote the latter by ξx , then it is and so on, and a “normal” life table may be 1 defined by the relation lx ξx lx . computed with µ(t) replaced by µ k(t) everywhere. + = 2 A corresponding cause-deleted multiple-decrement− The above functions can be computed for cohort life table may be based on reduced cause-specific life tables and for period life tables. Figure 1 shows ˚ decrement probabilities like plots of the function ex according to the mortal- ity experience for Swedish women in 1891–1900 1 t and 1990–1994. The at birth has exp µ k(x s)ds µj (x t)dt, 0 − 0 − + + increased from 53.6 years in the older table to 80.8 " ! " # some one hundred years later. Note that in the for j k. #= older table e˚x increases with x up to age 2 and Such a table would show what a normal table remains above e˚0 up through age 11. When mor- would look like if it were possible to eliminate tality is high at very young ages, surviving the cause k without changing the risk of any other first part of life increases your expected remaining cause. Again no assumption needs to be made about lifetime. As a consequence of mortality improve- the feasibility of such elimination in real life nor ments for very young children, these features have about cause independence. The computations are based on a pure abstraction. The interpretation for real-life applications must be based on substantive 90 considerations and is a different matter. 80

70 Life Expectancy 1990−1994 60 1891−1900 An individual’s life expectancy (at birth) is the 50 expected value

Years 40 ∞ ∞ lx e˚0 E(T ) [1 F (x)]dx dx = = − = l 30 "0 "0 0 of his or her lifetime T , computed for the probability 20 distribution F( ) operating at the time of birth. When the individual· has survived to (exact) age x, his or 10 her remaining lifetime, U T x, is positive and = − 0 has the survival function Sx(u) S(x u)/S(x) 0 10 20 30 40 50 60 70 80 90 100 = + = Age lx u/lx, and the residual life expectancy is + ∞ ∞ lx u Figure 1 Residual life expectancy for Swedish women, e˚x E(T x T > x) Sx(u) du + du. = − | = = l 1891–1900 and 1990–1994 "0 "0 x 4 Life Table disappeared in the younger table. Note that the References expected total lifetime, x e˚x, always increases + with x throughout the human lifespan. (One can [1] Bernoulli, D. (1766). Essai d’une nouvelle analyse de show that the derivative of this function is always la mortalite´ causee´ par la petite verole,´ et des avantages positive.) The longer you have lived already, the de l’inoculation pour la prevenir.´ Histoire de l’Acad´emie longer you can expect the total length of your life Royale des Sciences, M´emoires, Ann´ee 1760, pp. 1–45. [2] Chiang, C.L. (1984). The Life Table and its Applications. to be. Krieger, Malabar. In a multiple-decrement situation, formula (5) can [3] Cournot, A. (1843). Exposition de la th´eorie des chances ( k) be used to compute a residual life expectancy e˚x− et des probabilit´es. Hachette, Paris. from the decrement series of the cause-deleted life [4] Deparcieux, A. (1746). Essai sur les probabilit´es de la ( k) dur´ee de la vie humaine. Guerin´ Freres,` Paris. table for risk k. The difference e˚x− e˚x is the gain one would get in residual life expectancy− at [5] Dublin, L.I., Lotka, A.J. & Spiegelman, M. (1947). age x if it were possible to eliminate risk cause Length of Life. Ronald Press, New York. [6] Dupaquier,ˆ J. (1996). L’invention de la table de mor- k without changing the risk intensity of any other talit´e. Presses Universitaires de France, Paris. cause of decrement. Dublin et al. [5, p. 96] note [7] Duvillard, E. (1806). Analyse des tableaux de l’influence that according to the cause-specific mortality of the de la petite verole,´ et de celle qu’un preservatif´ tel que US in 1939–1941 the gain would be 9.01 years la vaccine peut avoir sur la population et la longevit´ e.´ for white men and 8.80 years for white women at Paris. age 0 if one could eliminate the risk of death due to [8] Elandt-Johnson, R.C. & Johnson, N.L. (1980). Survival cardiovascular–renal diseases at all ages (and change Models and Data Analysis. Wiley, New York. [9] Graunt, J. (1662). Natural and Political Observations no other cause-specific mortality risks). The gains Made Upon the Bills of Mortality. London. from eliminating the risk of death in cancer alone [10] Heligman, L. & Pollard, J. (1980). The age pattern were much less (1.39 years for men and 2.05 years of mortality, Journal of the Institute of Actuaries 107, for women). 49–75. [11] Hoem, J.M. (1972). Analytic graduation, in Proceed- ings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 1970, Vol. 1, L.M. Le Cam, History and Literature J. Neyman & E.L. Scott, eds. University of California Press, Berkeley, pp. 569–600. The first step toward the development of the life [12] Keyfitz, N. (1982). Keyfitz method of life-table con- table was taken when Graunt [9] published his struction, in Encyclopedia of Statistical Sciences, Vol. famous Bills of Mortality. There were subsequent 4, S. Kotz & N.L. Johnson, eds. Wiley, New York, pp. contributions by Halley, Huygens, Leibniz, Euler, 371–372. [13] Manton, K.G. & Stallard, E. (1984). Recent Trends in and others. Deparcieux [4] clarified the definition Mortality Analysis. Academic Press, New York. of the life expectancy and identified the need for [14] Price, R. (1783). Observations of Reversionary Pay- separate tables for men and women. Wargentin [17] ments: On Schemes for Providing Annuities for Widows, was the first to publish real age-specific death rates, and for Persons in Old Age; and on the National Debt. and the first to do so for a whole country. Price [14] Cadell & Davies, London. included most of the columns now associated with the [15] Seal, H. (1977). Studies in the history of probability and life table, and the tables by Duvillard [7] contained statistics, XXV: multiple decrements or competing risks, Biometrika 64, 429–439. them all. The basic notions of cause-eliminated life [16] Smith, D. & Keyfitz, N. (1977). Mathematical Demog- tables go back to Bernoulli [1]. Cournot [3] developed raphy: Selected Papers. Springer-Verlag, Heidelberg. the essentials of their mathematics. See Dupaquierˆ [17] Wargentin, P. (1766). Mortaliteten i Sverige, i anledning [6] and Seal [15] for historical overviews. Smith af Tabell-Verket. Kongl. Vetenskaps-Academiens Hand- & Keyfitz [16] have collected extracts from many lingar, Stockholm. original texts. Life-table techniques are described in most intro- (See also ; Vital Statistics, Overview) ductory textbooks on the methods of actuarial statis- tics, biostatistics, demography, or . See JAN M. HOEM for example, Chiang [2], Elandt-Johnson & Johnson [8], or Manton & Stallard [13].