The Single Decrement Life Table

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The Single Decrement Life Table
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Plan

• Demographic Probabilities
• Single Decrement Life Table

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Demographic Probabilities

• Probability Number of events occurring in a
  given number of trials
• Number of successes cannot exceed number of
  trials
• P between 0 and 1 0.0 lt P lt 1.0
• Events of interest must be related to the trials
  undertaken
• Demographic probabilities must be related to a
  cohort
• Surviving members of a cohort at time T or age A
  are the trials or the at risk population to
  which events may occur over the next period of
  time
• All members of a cohort are exposed to the risk
  of an event for the same duration of time between
  Tn or An, unless they experience an event that
  removes them from the cohort

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Birth Cohort of 1995
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Symbols Used to Represent Event Counts

• B0(95) 6
• SD0(95) 1
• PD0(96) 1
• B1(96) 4
• PD0(95) ?
• B1(95) ?

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Notation - Definitions

• Probability of dying in the age interval x to xn
  for those who survive to age x

• Total deaths at age x (last birthday) in year Y

• Separation factor for age x in year Y deaths at
  age x to individuals attaining age x in year Y
  over all deaths age x in year Y

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Period Cohort Rates Probabilities
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nqx Examples
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Diagram of Synthetic nqx
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Synthetic Calendar Year Probabilities

• nqx is
• probability that person dies during calendar
  year of attaining age x
• PLUS
• probability that person survives the year when
  they attain their xth birthday
• TIMES
• probability that if person survives the year of
  their xth birthday, they die during the
  subsequent year

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Synthetic Calendar Year Probabilities

• Rearrange to only use deaths from one calendar
  year to create synthetic nqx

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Infant Mortality Rate

• Take deaths from year Y cohort and year Y-1
  cohort divided by births in year Y
• Not a true probability because events in the
  numerator are not all associated with trials in
  the denominator
• However, is a reasonable approximation, is easy
  to calculate with available data, and is very
  standard

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The Life Table

• One of the most important demographic techniques
• Describes the dying out of a cohort
• Age or more generally duration is the most
  important dimension along which a life table is
  organized
• Contains a number of columns
• Age (age groups),
• Numbers of deaths in each age group
• Probability of dying in each age group
• Number of survivors to the beginning of each age
  group
• Number of person years lived in each age group
• Average additional years to live for those who
  survive to beginning of each age group, etc.

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Life Table Columns lx

• Number left alive at age x

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Life Table Columns ndx

• Number dying between ages x and xn

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Life Table Columns nqx

• Probability of dying between ages x and xn

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Life Table Columns npx

• Probability of surviving from ages x to xn

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Life Table Columns nLx

• Person-years lived between ages x and xn

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Life Table Columns Tx

• Person-years lived at ages older than x

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Life Table Columns ex

• Expectation of life at age x average additional
  years of life that someone who survives to age x
  can expect to live

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Life Table Columns nmx

• Death rate in the cohort between ages x and xn

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Life Table Columns nax

• Average number of years lived in the age interval
  by those dying in the age interval

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Example Life Table
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(No Transcript)
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Period Life Table

• Cohort data not very common
• Need ability to use period data that describe
  age-specific mortality in a given year or period
• A period life table is exactly the same as a
  cohort life table except it describes the dying
  out of a synthetic cohort that experiences at
  each age the age-specific mortality associated
  with a given period
• A hypothetical group of people survives through
  the age-specific risk of dying associated with a
  period

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Creating a Period Life Table

• The key to this is the fact that the hypothetical
  cohort experiences the age-specific probabilities
  of dying associated with the period
• The data available are usually observed
  age-specific mortality rates, nMx
• The trick then is to convert these observed
  age-specific mortality rates into one of the
  columns of a life table
• The most convenient choice is to convert to nqx
• nMx to nqx conversion ?
• Critical assumption is that nMx nmx

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nmx ? nqx
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Strategies for Choosing nax

• nmx ? nqx requires nax where do we get nax ?
• From calculating it directly
• From smoothing (graduating) the death
  distribution within each age interval
• Borrowing values from another population
• Making one of two assumptions
• nax is half the length of the age interval (n/2),
  or
• nmx is constant in the interval which negates the
  necessity of using nax because there is a direct
  formula to calculate npx

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nax in Practice

• Usually use n/2 for all age groups except the
  first
• Mortality rate between ages 0 and 5 changes very
  rapidly, falling very quickly at first and then
  flattening out
• Consequently most deaths early in life occur
  closer to 0 than to 5 and hence nax is
  significantly less than n/2 in the first two age
  groups (0, 1-4)
• In general in other age groups where mortality is
  changing less rapidly, the overall life table is
  very insensitive to the exact choice of nax

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nax for Very Young Ages
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The Open-ended Age Interval

• Because n is effectively infinite for the open
  (last) age interval, we cannot calculate nLx
  given the formulas we have

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Review

• Construct life tables from real data in Excel ?