Моделирование смертности Л.А. Гаврилов Center on Aging NORC and the University of Chicago Chicago, Illinois, USA
Questions of Actuarial Significance • How far could mortality decline go? (absolute zero seems implausible) • Are there any ‘biological’ limits to human mortality decline, determined by ‘reliability’ of human body? (lower limits of mortality dependent on age, sex, and population genetics) • Were there any indications for ‘biological’ mortality limits in the past? • Are there any indications for mortality limits now?
How can we improve the actuarial forecasts of mortality and longevity ? By taking into account the mortality laws summarizing prior experience in mortality changes over age and time: Gompertz-Makeham law of mortality Compensation law of mortality Late-life mortality deceleration
The Gompertz-Makeham Law Death rate is a sum of age-independent component (Makeham term) and age-dependent component (Gompertz function), which increases exponentially with age. μ(x) = A + R e αx A – Makeham term or background mortality R e αx – age-dependent mortality; x - age risk of death
Gompertz Law of Mortality in Fruit Flies Based on the life table for 2400 females of Drosophila melanogaster published by Hall (1969). Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Gompertz-Makeham Law of Mortality in Flour Beetles Based on the life table for 400 female flour beetles (Tribolium confusum Duval). published by Pearl and Miner (1941). Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Gompertz-Makeham Law of Mortality in Italian Women Based on the official Italian period life table for 1964-1967. Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
How can the Gompertz-Makeham law be used? By studying the historical dynamics of the mortality components in this law: μ(x) = A + R e αx Makeham component Gompertz component
Historical Stability of the Gompertz Mortality ComponentHistorical Changes in Mortality for 40-year-old Swedish Males • Total mortality, μ40 • Background mortality (A) • Age-dependent mortality (Reα40) • Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Predicting Mortality CrossoverHistorical Changes in Mortality for 40-year-old Women in Norway and Denmark • Norway, total mortality • Denmark, total mortality • Norway, age-dependent mortality • Denmark, age-dependent mortality Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Predicting Mortality DivergenceHistorical Changes in Mortality for 40-year-old Italian Women and Men • Women, total mortality • Men, total mortality • Women, age-dependent mortality • Men, age-dependent mortality Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Historical Changes in MortalitySwedish Females Data source: Human Mortality Database
Extension of the Gompertz-Makeham Model Through the Factor Analysis of Mortality Trends Mortality force (age, time) = = a0(age) + a1(age) x F1(time) + a2(age) x F2(time)
Factor Analysis of Mortality Swedish Females Data source: Human Mortality Database
Implications • Mortality trends before the 1950s are useless or even misleading for current forecasts because all the “rules of the game” has been changed
Preliminary Conclusions • There was some evidence for ‘ biological’ mortality limits in the past, but these ‘limits’ proved to be responsive to the recent technological and medical progress. • Thus, there is no convincing evidence for absolute ‘biological’ mortality limits now. • Analogy for illustration and clarification:There was a limit to the speed of airplane flight in the past (‘sound’ barrier), but it was overcome by further technological progress. Similar observations seems to be applicable to current human mortality decline.
Compensation Law of Mortality(late-life mortality convergence) Relative differences in death rates are decreasing with age, because the lower initial death rates are compensated by higher slope (actuarial aging rate)
Compensation Law of MortalityConvergence of Mortality Rates with Age 1 – India, 1941-1950, males 2 – Turkey, 1950-1951, males 3 – Kenya, 1969, males 4 - Northern Ireland, 1950-1952, males 5 - England and Wales, 1930-1932, females 6 - Austria, 1959-1961, females 7 - Norway, 1956-1960, females Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Compensation Law of Mortality (Parental Longevity Effects) Mortality Kinetics for Progeny Born to Long-Lived (80+) vs Short-Lived Parents Sons Daughters
Compensation Law of Mortality in Laboratory Drosophila 1 – drosophila of the Old Falmouth, New Falmouth, Sepia and Eagle Point strains (1,000 virgin females) 2 – drosophila of the Canton-S strain (1,200 males) 3 – drosophila of the Canton-S strain (1,200 females) 4 - drosophila of the Canton-S strain (2,400 virgin females) Mortality force was calculated for 6-day age intervals. Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991
Implications • Be prepared to a paradox that higher actuarial aging rates may be associated with higher life expectancy in compared populations (e.g., males vs females) • Be prepared to violation of the proportionality assumption used in hazard models (Cox proportional hazard models) • Relative effects of risk factors are age-dependent and tend to decrease with age
The Late-Life Mortality Deceleration(Mortality Leveling-off, Mortality Plateaus) The late-life mortality deceleration law states that death rates stop to increase exponentially at advanced ages and level-off to the late-life mortality plateau.
Mortality deceleration at advanced ages. • After age 95, the observed risk of death [red line] deviates from the value predicted by an early model, the Gompertz law [black line]. • Mortality of Swedish women for the period of 1990-2000 from the Kannisto-Thatcher Database on Old Age Mortality • Source: Gavrilov, Gavrilova, “Why we fall apart. Engineering’s reliability theory explains human aging”. IEEE Spectrum. 2004.
Mortality Leveling-Off in House FlyMusca domestica Based on life table of 4,650 male house flies published by Rockstein & Lieberman, 1959
Non-Aging Mortality Kinetics in Later Life Source: A. Economos. A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 1979, 2: 74-76.
Invertebrates: Nematodes, shrimps, bdelloid rotifers, degenerate medusae (Economos, 1979) Drosophila melanogaster (Economos, 1979; Curtsinger et al., 1992) Housefly, blowfly (Gavrilov, 1980) Medfly (Carey et al., 1992) Bruchid beetle (Tatar et al., 1993) Fruit flies, parasitoid wasp (Vaupel et al., 1998) Mammals: Mice (Lindop, 1961; Sacher, 1966; Economos, 1979) Rats (Sacher, 1966) Horse, Sheep, Guinea pig (Economos, 1979; 1980) However no mortality deceleration is reported for Rodents (Austad, 2001) Baboons (Bronikowski et al., 2002) Mortality Deceleration in Animal Species
Existing Explanations of Mortality Deceleration • Population Heterogeneity (Beard, 1959; Sacher, 1966). “… sub-populations with the higher injury levels die out more rapidly, resulting in progressive selection for vigour in the surviving populations” (Sacher, 1966) • Exhaustion of organism’s redundancy (reserves) at extremely old ages so that every random hit results in death (Gavrilov, Gavrilova, 1991; 2001) • Lower risks of death for older people due to less risky behavior (Greenwood, Irwin, 1939) • Evolutionary explanations (Mueller, Rose, 1996; Charlesworth, 2001)
Testing the “Limit-to-Lifespan” Hypothesis Source:Gavrilov L.A., Gavrilova N.S. 1991. The Biology of Life Span
Implications • There is no fixed upper limit to human longevity - there is no special fixed number, which separates possible and impossible values of lifespan. • This conclusion is important, because it challenges the common belief in existence of a fixed maximal human life span.
Latest Developments Was the mortality deceleration law overblown? A Study of the Real Extinct Birth Cohorts in the United States
Challenges in Hazard Rate Estimation At Extremely Old Ages • Mortality deceleration may be an artifact of mixing different birth cohorts with different mortality (heterogeneity effect) • Standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high • Ages of very old people may be highly exaggerated
Challenges in Death Rate Estimation at Extremely Old Ages • Mortality deceleration may be an artifact of mixing different birth cohorts with different mortality (heterogeneity effect) • Standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high • Ages of very old people may be highly exaggerated
U.S. Social Security Administration Death Master File Helps to Relax the First Two Problems • Allows to study mortality in large, more homogeneous single-year or even single-month birth cohorts • Allows to study mortality in one-month age intervals narrowing the interval of hazard rates estimation
What Is SSA DMF ? • SSA DMF is a publicly available data resource (available at Rootsweb.com) • Covers 93-96 percent deaths of persons 65+ occurred in the United States in the period 1937-2003 • Some birth cohorts covered by DMF could be studied by method of extinct generations • Considered superior in data quality compared to vital statistics records by some researchers
Quality Control Study of mortality in states with better age reporting: Records for persons applied to SSN in the Southern states, Hawaii and Puerto Rico were eliminated
Crude Indicator of Mortality Plateau (2) Coefficient of variation for life expectancy is close to, or higher than 100% CV = σ/μ where σ is a standard deviation and μ is mean
What are the explanations of mortality laws? Mortality and aging theories
Additional Empirical Observation:Many age changes can be explained by cumulative effects of cell loss over time • Atherosclerotic inflammation - exhaustion of progenitor cells responsible for arterial repair (Goldschmidt-Clermont, 2003; Libby, 2003; Rauscher et al., 2003). • Decline in cardiac function - failure of cardiac stem cells to replace dying myocytes (Capogrossi, 2004). • Incontinence - loss of striated muscle cells in rhabdosphincter (Strasser et al., 2000).
Like humans, nematode C. elegans experience muscle loss Herndon et al. 2002. Stochastic and genetic factors influence tissue-specific decline in ageing C. elegans. Nature 419, 808 - 814. “…many additional cell types (such as hypodermis and intestine) … exhibit age-related deterioration.” Body wall muscle sarcomeres Left - age 4 days. Right - age 18 days
What Should the Aging Theory Explain • Why do most biological species including humans deteriorate with age? • The Gompertz law of mortality • Mortality deceleration and leveling-off at advanced ages • Compensation law of mortality