|HDE Lecture Notes|
J. R. Carey
|Life Tables and Mortality|
The Life Table
Death rate is a primary vital statistic along with birth and marriage. It is important because it: (1)is a barameter of current health in a population; (2)identifies trends in health and mortality by indicating at what age most deaths are occuring and the causes of death; and (3)provides a baseline for prediction of trends for the future which can be used for life insurance, medicare, social security and business. The main analytical tool for examining mortality and longevity in populations is the life table. The life table is a product of actuarial science (life insurance). The first life table was developed in 1662 in England by John Graunt when he published "The Bills of Mortality". This was useful as a concept (ordering mortality data) but there was very unreliable mortality data available at that time. It was not until 1900 that reliable mortality data became available.
Knowledge of some background terms and concepts will be useful for understanding of life table techniques.
Some of the life table parameters express mortality or survival as a probability given as the number of individuals which experience an event (like death or marriage) divided by the number that are at risk of experiencing that event. For example, if 2 people died out of a pool of 100 people who were subjected to a toxic substance we would say that the probability of dying given this exposure is 0.02 or 2%. This is what is referred to as an a posteriori probability since the death rate is after the fact; that is, it is not possible to predict the death rate or probability without first conducting the "experiment". This is distinct from an a priori probability which expresses a probability before the fact (of the experiment). For example, we can easily estimate the probability of a coin landing heads (0.50 or 50%) without first flipping the coin 100 times. Mortality rates are a posteriori probabilities; that is, the number of deaths relative to the number of persons at risk must be determined from retrospective estimates.
A life table is a detailed description of the age-specific mortality, survival and expectation of life of a population. Life tables help to answer questions such as: What is the life expectancy of the average newborn? How many years remains for the average 75 year old woman in the U.S. Over what age groups do most deaths occur? What is the probability of dying in the next year given that you are age 18? What is this probability at age 80? What fraction of all 1990 newborn will live to age 85? What proportion will live from age 50 to age 90?
There are two general forms of the life table. The first is the cohort life table which provides a longitudinal perspective in that it includes the mortality experience of a particular cohort from the moment of birth through consecutive ages until none remains in the original cohort. The second basic form is the current life table which is cross-sectional. This table assumes a hypothetical cohort subject throughout its lifetime to the age-specific mortality rates prevailing for the actual population over a specified period and are used to construct a synthetic cohort. Both forms of the life table may be either single decrement or multiple decrement. The first of these lumps all forms of death into one and the second disaggregates death by cause.
Life Table Functions
The life table is organized in seven columns starting with an age x column from 0 through the age of the oldest person ever to live around 120 years.
Table 1. Hypothetical life table showing the main functions.
to age x
x to x+1
x to x+1
x to x+1
at age x
Column 1. The first column of a life table contains all age classes denoted x and ranges from 0 (newborn) through the oldest possible age, .
Column 2. This column gives the number of the original cohort alive at age x and is denoted Nx. The initial number is typically 100,000 and is known as the life table radix.
Column 3. This column contains cohort survival, lx, defined as the fraction of the original cohort surviving to age x. The general formula is given as
|= 1.0000||= 0.4000|
Column 4. The parameter defined in this column is known as period survival, px, defined as the fraction of individuals alive at age x that survive to age x+1. The general formula is given as
For example the period survival rates for ages x=0 and x=3 from Table 1 are
|= 0.90||= 0.25|
Thus the probability of surviving from age 0 to 1 is 0.90 and from age 3 to 4 is 0.25.
Column 5. This column contains the compliment of period survival and is known as period mortality, qx, defined as the fraction of individuals alive at age x that die prior to age x+1. The general formula is
qx = 1 - pxFor example, the period mortality rates for ages x=0 and x=3 from Table 1 are
|q0 = 1 - p0||q3 = 1 - p3|
|= 1.00 - 0.90||= 1.00 - 0.25|
|= 0.10||= 0.75|
Column 6. The parameter contained in this column is the fraction of the original cohort that dies in the interval x to x+1 and is denoted dx. It is the frequency distribution of deaths and is given by the formula
dx = lx - lx+1
For example, the value of dx for ages x=0 and x=3 from Table 1 are
|d0 = l0 - l1||d3 = l3 - l4|
|= 1.00 - 0.90||= 0.14 - 0.10|
|= .10||= .30|
Column 7. This column contains the parameter expectation of life, ex, which is defined as the average number of years (days, weeks) remaining to an individual age x. The general formula is given by
For example, the value of dx for ages x =0 and x=3 from Table 1 are
|= 2.40||= 0.75|
Current Life Table
A current life table is based on the concept of a synthetic cohort in which the probabilities of dying from one age class to the next are based on the death rates of each of the 115+ cohorts living at any one time. The idea is based on the notion that the probability of surviving to say, age 5, is the probability (or fraction) or newborn individuals living to age 1 times the probability of age 1 individuals living to age 2 times the probability of age 2 individuals living to age 3 and so forth. Thus the general form for reconstructing a survival curve and, in turn, all other life table parameters from period survival (or period mortality) data is
lx = p0 × p1 × p2 × ... × px-1
For example, suppose we wish to construct a survival curve from the death rates prevailing from 1990 to 1991 as given in Table 2. The probability of an individual age 0 in 1990 surviving to age 1 in 1991 is
Table 2. Number of persons in hypothetical census by age class and year.
Similarly for the probabilities for individuals surviving from age 1 to 2 and age 2 to 3
Thus a synthetic survival schedule can be constructed by multiplying progressively as
Thus out of 100,000 individuals age 0, 83,640 will survive to age 1. This is based on the 1990 to 1991 probability of survival from 0 to 1 years. Out of these 83,640 individuals alive at age 1, 0.8264 of these or 69,120 will survive to age 2. Out of these 69,120 individuals alive at age 2, 0.9182 or 63,470 will survive to age 3. In other words, the fraction that survive to age x equals the product of the fractions (or probabilities) that survive through each of the previous age classes.
Both the cohort life table and the current life table are based on assumptions that are violated in reality. The cohort life table is based on the mortality experience of the same group of individuals from birth through the age when the last individuals dies. Thus if we were to use only cohort life tables to characterize longevity in the U.S., then the only life tables available would be for cohorts in which all individuals have died, the earliest of which would be individuals born around 1880. First, there is the problem of obtaining accurate records of mortality rates over a century ago and second, there is the problem that mortality rates in 1880 were much higher at all age classes than are current mortality rates. Third, the centenarians of today when through much more difficult times than will future centenarians. Thus their mortality rates may be much higher than will be the future centenarians.
But the current life table also has problems (though many fewer). First, the current life table is cross sectional. Thus every cohort will have a different history. And it is well known that the experiences at younger ages has an impact on mortality rates at older ages. Take for example, two cohorts aged 10 and 50. Suppose the cohort age 50 was severely malnourished when they were age 10 years assumes that when the 10 year old cohort attains age 50 they will experience the same mortality rate as the 50-year old cohort does this year. Clearly they have different experiences that will have a major bearing on their mortality rates at older ages.
Table 1. Life table for 1970 birth cohort subjected to mortality rates of 1990 U.S. females.
from x to
from x to
dying x to
Importance of Mortality
The life table provides five different expressions or functions which describe the mortality and survival experience of a cohort. Because each of the functions can be independently derived from the original cohort data and all but expectation of life can be used to derive the other functions, it is often inferred that no single function has precedent. Although this is true algebraically, it is not accurate biologically, demographically, or actuarially. The age-specific mortality schedulethe series of probabilities that an individual alive at age x dies prior to age x+1serves as the actuarial foundation for all other functions.
Why mortality is fundamental:
The Gompertz Mortality "Law"
One of the most fundamental and important parameters in demography, gerontology and actuarial studies is the rate of change or slope of the age-specific mortality schedule with age. In 1825 the English actuary Benjamin Gompertz observed that "the number of living corresponding to ages increasing in arithmetical progression, decreased in geometrical progression." He proposed that a geometrical decrease in survival with age existed because of a geometric increase in the "force of mortality" with age. The Gompertz model has been the major mortality rate model in gerontology for more than 60 years. It is of the form:
where x is the mortality at age x, a is the initial mortality rate and b is the Gompertz parameter which denotes the exponential rate of change in mortality with age. The Gompertz model is important because:
The Gender Gap
While women generally outlive men by a margin of 4 to 10 years throughout the industrialized world a long standing question in biology is whether this female advantage in longevity is a general characteristic of most nonhuman species as well. The scientific literature contains conflicting views.
Supportive Views: Hazzard (1990) states "The greater longevity of females than males appears to have a fundamental biological basis. Studies of comparative zoology suggest that greater female longevity is virtually universal". Hamilton and Mestler (1969) begin their paper with the statement, "Males tend to die at an earlier age than females in most species of animals for which data are available". Brody and Brock (1985) state, "...there are basic and fundamental questions posed by the fact that female survival [advantage] seems to be one of the most pervasive findings within the animal kingdom."
Conflicting Views: The paper by Lints and co-workers (Lints et al. 1983) made 218 comparisons between female and male life span in Drosophila melanogaster and concluded that mean life span of females exceeds that of males in only about half the cases. Smith (1989) notes, "...while there is some evidence that adult populations of many animal species contain more females than males, most of these studies do not consider survival to an age approaching the potential limit for the species as implied by the word longevity." Gavrilov and Gavrilova (1991) state "The hopes connected with the search for general biological mechanisms underlying these [sex] differences seem to be in vain, since, despite the wide-spread opinion to the contrary, the greater life span of females is not in itself a general biological regularity".
In 1900 the life expectancy at birth of U.S. women was around 49 years and for men around 47 years--a 2 gap referred to as the gender gap. Mortality rates have decreased over the past 90 years although the rate of decrease for females has been greater than that for males so that currently the gender gap is 7 years--females have a life expectancy of 79 years while for males life expectancy is 72 years.
General determinants of sex mortality differentials:
Donner Party Mortality|
Out of 90 individuals in the Donner party 32-of-55 males or 58% died whereas 10-of-32 females or 29% of females died. Source: S. McCurdy, Western J. Medicine 160, 338-342 (1994)
Mortality crossover--an attribute of the relative change and level of age-specific mortality rates in two population groups: one group is advantaged (lower relative mortality) and the other disadvantaged (higher relative mortality). The disadvantaged population must manifest age-specific mortality rates markedly higher than the advantaged population through middle age at which time the rates change.
Two examples of mortality crossovers
Elimination of Mortality
The table below shows how much will life expectancy change if eliminated mortality to various ages from 0 (no elimination) to age 80 years. The point here is that life expectancy at birth will change very little if we eliminate all mortality at early ages or even late ages. Individuals who all live to age 80 are then subject to the mortality experience of all 80 year olds and, on average, would only live 6.2 more years. There are very few 'person-years' to be gained from eliminating more death from younger ages.
Table 1. Effect on expectation of life if mortality up to different ages was completely eliminated.
Cause of Death
Deaths are typically grouped into manner of death, types of causes and classification of main cause. The manner of death refers to natural, accident, suicide or homicide. Four types of causes are commonly recognized:
For example, a person who has diabetes his entire life may be diagnosed with inoperable lung cancer 14 months before his death, with pneumonia 10 days before death and septicemia 2 days before death. Thus septicemia would be the immediate cause, pneumonia an intervening cause, lung cancer the underlying cause and diabetics as a contributory cause.
The underlying cause is the cause used for statistical reporting and is defined as "i)the disease or injury which initiated the train of morbid events leading directly to death; or ii)the circumstances of the accident or violence which produced the fatal injury".
Cause of Death
Persons involved in a death: i)Attending Physician--usually pronounces dead; ii)medical examiner--M.D. who is trained clinician; iii)coroner--usually political appointee; used when suspicious circumstances or homicide, suicide etc. iv)forensic pathologist--M.D. who specializes in pathologies associated with morbidity and death.
Example: i)26% or about 1 out of 4 deaths there was not sufficient cause of death; ii)the autopsy for 88 year old male with Alzheimer's disease provides no clear answers to cause of death.
* denotes the underlying cause of death
Table 2. Autopsy findings in an 88 year old male with Alzheimer's disease (from Martin 1988).
Chiang, C. 1984. The Life Table and its Application. Robert E. Krieger Publishing Co., Malabar, Florida.
McCurdy, S. A. 1994. Epidemiology of disaster. the Donner party (1846-1847). Western Journal of Medicine 160:338-342.
Moriyama, I. M. 1956. Development of the present concept of cause of death. Amer. J. Public Health 46:436-441.
Return to the Course Syllabus