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Samuel Clark Department of Sociology, University of Washington Institute of Behavioral Science, University of Colorado at Boulder Agincourt Health and Population Unit, University of the Witwatersrand The Single Decrement Life Table

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1 Plan Demographic Probabilities Single Decrement Life Table

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2 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 <= P <= 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 T+n or A+n, unless they experience an event that removes them from the cohort

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3 Birth Cohort of 1995

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4 Symbols Used to Represent Event Counts B 0 (95)= 6 S D 0 (95)= 1 P D 0 (96)= 1 B 1 (96)= 4 P D 0 (95)= ? B 1 (95)= ?

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5 Notation - Definitions Probability of dying in the age interval x to x+n 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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6 Period & Cohort Rates & Probabilities

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7 n q x Examples

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8 Diagram of Synthetic n q x

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9 Synthetic Calendar Year Probabilities n q x is probability that person dies during calendar year of attaining age x PLUS probability that person survives the year when they attain their x th birthday TIMES probability that if person survives the year of their x th birthday, they die during the subsequent year

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10 Synthetic Calendar Year Probabilities Rearrange to only use deaths from one calendar year to create synthetic n q x :

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11 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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12 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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13 Life Table Columns: l x Number left alive at age x

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14 Life Table Columns: n d x Number dying between ages x and x+n

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15 Life Table Columns: n q x Probability of dying between ages x and x+n

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16 Life Table Columns: n p x Probability of surviving from ages x to x+n

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17 Life Table Columns: n L x Person-years lived between ages x and x+n

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18 Life Table Columns: T x Person-years lived at ages older than x

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19 Life Table Columns: e x 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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20 Life Table Columns: n m x Death rate in the cohort between ages x and x+n

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21 Life Table Columns: n a x Average number of years lived in the age interval by those dying in the age interval

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22 Example Life Table

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24 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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25 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, n M x 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 n q x n M x to n q x conversion Critical assumption is that n M x ~ n m x

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26 nmx nqxnmx nqx

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27 Strategies for Choosing n a x n m x n q x requires n a x … where do we get n a x ? From calculating it directly From smoothing (graduating) the death distribution within each age interval Borrowing values from another population Making one of two assumptions: – n a x is half the length of the age interval (n/2), or – n m x is constant in the interval which negates the necessity of using n a x because there is a direct formula to calculate n p x:

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28 n a x 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 n a x 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 n a x

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29 n a x for Very Young Ages

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30 The Open-ended Age Interval Because n is effectively infinite for the open (last) age interval, we cannot calculate n L x given the formulas we have

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31 Review Construct life tables from real data in Excel

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