Demography: Life Tables and Mortality

• Study notes on demography, actuarial science, and mortality, focusing on life table plots, comparisons between countries, and the force of mortality

Introduction to Life Tables

• Key aspects of life table functions include survival curves and sex-based differences.
• Mortality probabilities are compared between males and females.
• There are mortality trends across age groups, typically showing higher mortality among males.

Life Tables: Mexico 2019

• Life tables are provided for males and females in Mexico for 2019.
• Separation factors are derived using the Coale-Demeny relationship, assuming uniform mortality.

Comparison of Life Tables by Sex (MEX2019)

• Females generally have a greater life expectancy than males at any given age.
• The difference in life expectancy at birth between sexes is approximately 6 years.
• Probabilities of death are consistently higher for males than females.
• The manner in which the synthetic cohort decreases with age is affected.

Survival Curves

• Mexico survival curves are shown for 2019, with males in blue and females in orange.
• Until age 20, survival curves are similar for both sexes.
• After age 20, males tend to die more rapidly.
• The rate at which survival curves decrease becomes faster as ages approach 60.
• Survival curves are taking longer to fall rapidly as time passes.
• This indicates an increase in rectangularization, where the survival curve becomes more rectangular for both sexes.
• Rectangularization: A trend toward a more rectangular survival curve due to increased survival and concentration of deaths around the mean age at death.

Probabilities of Death

• Each group shows greater probabilities of death for men versus women.
• Probabilities of death are at their minimum for both sexes close to age 10.
• At the point near age 10, probabilities of death are practically the same for both sexes.
• Behavior of survival curves is consistent.
• There is an impact gap in mortality between the ages of 10 and 50 in males and females.
• From approximately age 50 onward, probabilities of death begin to increase faster.

Males' Overmortality

• The figure illustrates the ratio between the probabilities of death for males and females by age.
• There is a local minimum at approximately age 10, where probabilities of death are roughly the same for both sexes.
• A maximum difference is reached between ages 20 and 40, where probabilities of death for males are three times higher than for women.
• The probability of death becomes the same for both sexes after age 90.
• The probability of dying becomes 1 for both beyond age 85, making the ratio equal to 1.

Male-Female Differences in Life Expectancy at Birth

• From 1980 to 1996, differences in life expectancy at birth decreased.
• From 6.6 to 6.2, on average, being the difference in life expectancy at birth.
• The dispersion of data decreased from 1.156 to 0.987 in terms of standard deviation.
• The range confirms the same tendency.
• The observed reduction in the difference in life expectancy between men and women reflects improvements in male health through better prevention, care, and awareness of risks.
• Alignments occur with the decrease of the sex ratio as age progresses.
• Overall development progress is not necessarily indicated, considering social and economic factors.

Sex Mortality Ratios by Age for Selected Countries

• Patterns are similar across countries.
• Poland had larger differences in mortality between males and females at the time (data from the 1990s).
• In France, Germany, and Denmark, Poland was not as highly developed, and these differences may have since diminished.
• The 2019 pattern in Mexico is very similar.

Continuous Life Table Functions

• Age-specific death rates (ASDRs) are used to construct life tables that allow calculation of remaining functions and life expectancy at various attained ages.
• Deriving these functions involves approximations of a continuous process.
• Continuous survival function frames all this.

Intensity of Mortality

• For measuring "intense" mortality at age can use the derivative of the survival function l(u) at x ; i.e., l ′ (x)
• The effect of that rate of change l ′ (x) will be different depending on the number of survivors at two points Xo and x₁ .

Force of Mortality

• The force of mortality is positive or zero, and expressed as µ(u) = − {ln [l (u)]}′.
• This means it is the derivative of the natural logarithm of the survival function l(u).
• Intensity of mortality at u relates to the number of survivors ! (u)!
• Integrating the force of mortality means integrating over a generic interval.
• Eq.(2) is the probability of surviving, over a specific interval.
• One can express the probability of dying in [x, x + h] in terms of the force of mortality .
• One must solve integral to get values.
• Person-years function continuous expresses other life table functions and allows calculation easier.
• Life expectancy can be expressed through this too!

Expressions for µ(u)

• Finding an analytic expression for which to calculate for involves finding an expression for µ (u).
• Gompertz and Makeham have proposed two expressions.

Gompertz Proposal

• At any age u, force of mortality follows exponential curve given by µG (u) = BC^u, where B and C are fixed and depend on age.
• The values of B are generally between 10−6 and 10−3.
• C is often between 1.06 and 1.12 for human populations.
• Gompertz is useful in determining life table survivors for ages beyond available data.

Makeham's Modification

• Gompertz' law shows progression of mortality for adults but not at younger ages.
• In 1860 Makeham modifies Gompertz adding a constant term.
• A is usually within 0.001 to 0.003.
• This increases the force of mortality younger ages, where BCᵘ is relatively insignificant.
• Eventually, BCᵘ swamps (A), making the Makeham and Gompertz curves practically the same.