🪵Intro to Demographic Methods Unit 7 – Life Tables & Survival Analysis in Demography

Key Concepts

• Life tables summarize mortality experiences of a population over a specified period
• Survival analysis examines the time until an event occurs (death, marriage, divorce)
• Cohort life tables track the mortality experience of a specific group of individuals born during the same time period
• Period life tables represent the mortality experience of a population during a specific time period, often a calendar year
• Abridged life tables contain data for age intervals of 5 or 10 years, while complete life tables have single-year age intervals
• Life expectancy at birth ($e_0$) represents the average number of years a newborn is expected to live given the mortality rates of a specific period
• Survival curves graphically represent the proportion of individuals surviving to each age, with the area under the curve equal to life expectancy

Historical Context

• John Graunt's 1662 book "Natural and Political Observations Made Upon the Bills of Mortality" laid the foundation for modern life tables and demographic analysis
• Early life tables were constructed using data from specific cities or regions (London, Breslau)
• Edmond Halley developed the first modern life table in 1693 using data from Breslau, Germany
• 19th-century improvements in data collection and mathematical techniques led to the development of national life tables
• Life tables played a crucial role in the development of actuarial science and the insurance industry
• Advances in public health and medical technology during the 20th century led to significant increases in life expectancy and changes in the shape of survival curves

Types of Life Tables

• Period life tables represent the mortality experience of a population during a specific time period, often a calendar year
  • Assume that age-specific mortality rates remain constant throughout the lifetime of a hypothetical cohort
  • Useful for comparing mortality patterns across populations or over time
• Cohort life tables track the mortality experience of a specific group of individuals born during the same time period
  • Require data on the actual mortality experience of the cohort over their entire lifetime
  • Provide a more accurate representation of the cohort's mortality but are less timely than period life tables
• Multiple decrement life tables analyze the impact of competing risks (causes of death) on mortality
• Cause-deleted life tables estimate the impact of eliminating a specific cause of death on life expectancy
• Multistate life tables model transitions between different states (marital status, health status) over the life course

Data Sources and Collection

• Vital registration systems provide data on births, deaths, and causes of death
  • Coverage and quality of vital registration data vary across countries and time periods
  • Incomplete or inaccurate reporting of deaths can lead to biases in life table estimates
• Census data provide information on population size and structure by age and sex
  • Used to estimate the population at risk of dying in each age group
  • Undercount or age misreporting can affect the accuracy of life table estimates
• Sample surveys (demographic and health surveys) collect data on mortality and health in populations lacking reliable vital registration systems
• Indirect estimation techniques (Brass methods, model life tables) are used when direct data on mortality are unavailable or incomplete

Constructing Life Tables

• Begin with age-specific mortality rates (${}_{n}M_{x}$) calculated from death counts and population estimates for each age group
• Convert mortality rates to probabilities of dying (${}_{n}q_{x}$) using assumptions about the distribution of deaths within age intervals
• Calculate the number of survivors to each age ($l_{x}$) starting with a hypothetical cohort of 100,000 live births ($l_{0}$)
  • $l_{x+n} = l_{x} * (1 - {}_{n}q_{x})$
• Compute the number of person-years lived in each age interval (${}_{n}L_{x}$) using assumptions about the distribution of deaths
  • ${}_{n}L_{x} = n * (l_{x} + l_{x+n}) / 2$ for most age groups
  • Special formulas used for the first and last age intervals
• Calculate the total number of person-years lived above each age ($T_{x}$) by summing ${}_{n}L_{x}$ values from the oldest to the youngest age
• Compute life expectancy at each age ($e_{x}$) by dividing $T_{x}$ by $l_{x}$

Survival Analysis Techniques

• Kaplan-Meier estimator is a non-parametric method for estimating survival curves from censored data
  • Calculates the probability of surviving to each event time, conditional on surviving to the previous event time
  • Useful when the underlying distribution of survival times is unknown or does not follow a specific parametric model
• Cox proportional hazards model is a semi-parametric regression method for analyzing the effect of covariates on survival times
  • Assumes that the hazard ratios between groups are constant over time
  • Estimates the baseline hazard function non-parametrically and the effects of covariates parametrically
• Parametric survival models (exponential, Weibull, log-normal) assume that survival times follow a specific probability distribution
  • Allows for the estimation of survival curves and hazard functions based on the chosen distribution
  • More efficient than non-parametric methods when the distributional assumption is valid

Interpreting Results

• Life expectancy at birth ($e_0$) is a summary measure of mortality representing the average number of years a newborn is expected to live
  • Higher values indicate lower overall mortality
  • Comparisons across populations or over time should consider differences in the age structure and mortality patterns
• Age-specific mortality rates (${}_{n}M_{x}$) and probabilities of dying (${}_{n}q_{x}$) provide insights into the age pattern of mortality
  • Typically high in infancy, low in childhood and early adulthood, and increasing exponentially at older ages
  • Deviations from this pattern can indicate unusual mortality risks or data quality issues
• Survival curves depict the proportion of individuals surviving to each age
  • Steeper curves indicate higher mortality and shorter life expectancy
  • Shifts in the curve over time or differences between populations reflect changes in mortality patterns
• Hazard ratios from Cox models quantify the relative risk of an event between groups with different covariate values
  • Ratios greater than 1 indicate increased risk, while ratios less than 1 suggest decreased risk
  • Statistical significance and confidence intervals should be considered when interpreting hazard ratios

Real-World Applications

• Life insurance and annuity pricing rely on accurate life tables to determine premiums and benefits
• Pension and social security systems use life tables to project future costs and adjust eligibility ages based on changes in life expectancy
• Public health researchers use life tables to monitor population health, identify disparities, and evaluate the impact of interventions
• Epidemiologists employ survival analysis to study the incidence and progression of diseases and the efficacy of treatments
• Demographers use life tables and survival analysis to project population size and structure, estimate healthy life expectancy, and analyze the determinants of longevity
• Actuaries apply life table and survival analysis techniques to assess and manage risks in various industries (insurance, finance, healthcare)
• Government agencies and policymakers rely on life table indicators to inform resource allocation, set public health priorities, and develop social policies