Then and Now

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Then and Now

• What we did
• Tidy up direct/indirect standardization
• Types of mortality rates
• Introduction to life tables
• What we will do
• Make your own life table
• The guts of the calculations
• Where to find life tables
• Examples
• Homework

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Why the Life Table?

• Describe the patterns of mortality experience
  by age for an entire population
• Involves basic concepts and calculations used
  for other demographic measures
• Precursor to more advanced techniques of
  survival analysis

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Force of Mortality

• At any moment, there is a probability that death
  can occur. This is the force of mortality
• Changes with age
• For any mortal species, there is an age where the
  probability of death is 100.
• Useful way to describe and compare populations

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Example Life TableHandout US Life Table
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Available Inputs for a Life Table

• Deaths between age x and xn (nDx)
• This is a real number from a complete population
• Wont use big D in life table notation
• Mid-point Population size between age x and xn
  (nPx)
• M-type death rates
• nmx (nDx) /(nPx)
• nPx this notation will change when we discuss
  life tables

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Methuselah

• Methuselah was a descendant of Adam who lived to
  be 969 years old, according to the Bible.
• Methuselah is not a major figure in the Bible he
  is mentioned only in passing in Genesis, 1
  Chronicles 13, and Luke 337. According to the
  King James version of Genesis 525-27
• "And Methuselah lived an hundred eighty and seven
  years, and begat Lamech. And Methuselah lived
  after he begat Lamech seven hundred eighty and
  two years, and begat sons and daughters And all
  the days of Methuselah were nine hundred sixty
  and nine years and he died."
• Lamech was the father of Noah, making Metheselah
  Noah's grandfather.
• Biblical scholars say Methuselah died in the year
  of the Great Flood, though the Bible gives no
  indication whether the old man was a victim of
  the flood itself. Luke 337 traces the lineage of
  Jesus of Nazareth back through Noah and
  Methuselah to Adam.
• The name of Methuselah is now a synonym for any
  old or long-lasting person.

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Going From mx to qx (if necessary)

• If we had our choice we would use qx
• Often we end up calculating mx because this is
  calculable from vital records
• Use mx to get qx
• q (2m)/(2m) (unabridged) for single age groups
  (pg 15 in Hinde for proof)
• nqx (n . nmx)/(1 (n/2)(nmx)) for grouped age
  categories
• n is the width of the age interval

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Slope of Lines Plot of lx by age A vs B Same
number of deaths but force is greater in case B
because of smaller population at risk C vs B
Same number of deaths but force is greater in
case C because they occur faster Force of
mortality (qx) is a function of speed at which
deaths occur and number at risk of death
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Calculating q-type mortality rates

• assume a cohort of people born at the same time
  l0 (radix)
• lx is the number of people out of lo who are
  alive at exact age x

• where dx is the number of deaths between ages x
  and x1

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Calculating m-type mortality rates and Stationary
Populations

• If there is a constant number of people born
  into a population (radix of lo) and the
  age-specific death rates remain the same, then
  the Lx describe the number of people alive at age
  x in a stationary population (the age composition
  of a population).

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Calculating Person Years

• How many years are lived by people in a given
  population between ages x and x1?
• As a practical matter assume that people deaths
  are evenly distributed throughout the age
  interval

• Except at very young and old ages

• where ax is the average number of person years
  lived between ages x and x1 among those who die
  in that interval. Book often assumes ax is 0.1
  or 0.2.
• ax is usually .5 for most other age intervals

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Example

• Kids and parental mortality

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Total Person Years of Life Left to Live

• At a given age x, how many total person years of
  life do we have left to live?

• where x is a given age and ? is the limiting
  age (e.g. 122)

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What is Life Expectancy?

• A child born today can expect to live to age 77
• Life expectancy can be evaluated at any age
  (e.g., how many years you can expect to live as
  evaluated at age 50)
• To actually calculate this number, you can use
  the quantities we have developed

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Median Survival and eo

• eo is the average number of years lived by each
  person at birth
• Median survival is the age at which half the
  population has died
• In general, median survival will be higher than
  eo because eo is more susceptible to extreme
  values (death at young ages mostly) which
  suppresses its value
• In future populations (low infant and child
  mortality, increases in exceptional longevity),
  this pattern will be reversed.

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Working With Life Tables

• 5q60 prob of dying between 60 and 65
• 5p10 prob of surviving between 10 and 15
• l50 no. of people in the life table pop at 50
• 5d10 no. of deaths to life table pop between 10
  15
• 5L30 no. of person-years lived by life table
  pop between 30 35
• T40 no. of person-years lived above exact age
  40
• e20 life expectancy at 20

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Common Calculations
Prob of survival from x to y ly/lx probability
of surviving between 40 55 or This can also
be calculated by multiplying npx between the
exact ages(5p40 x 5p45 x 5p50) Probability of
dying between exact ages by calculating nqx if we
already know ndx and lx nqx ndx / lx
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Some Practical Matters

• If you know mx (based on the raw data), then you
  can get qx
• If you know qx, then you can map out lx
• If you know lx, you can generate Lx and then Tx
• If you know Tx and lx, you can get ex

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Quantities Used in the Life Table
Survival probability
Abridged
Unabridged
In common
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Quantities Used in the Abridged Life Table

• Problem of dealing with extreme age groups
  (under age 5 and over age 90) when using grouped
  data.
• Closing out the abridged life table
• To get the L estimate for the last category,
  assume an e figure to generate L
• e.g. nL80 l80e80
• Use the fact that for the last category, nqx1
  which means that

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Mapping the lx is the same as the survival curve
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1990
Period Life Tables
1900
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1990 Female
1990 Male
1900 Male
1900 Female
Gender Comparison
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Whites
Blacks
1900
Whites
1990
Blacks
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Projected life expectancy at birthSelected
sub-Saharan countries
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WHO/Utah Life Tables

• http//www3.who.int/whosis/life_tables/life_tables
  .cfm?pathevidence,life_tableslanguageenglish
• http//governor.utah.gov/dea/demographics/liftab/l
  ifetable.html

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WHO LIFE TABLE FOR 1999 RUSSIAN
FEDERATION Males x nMx nqx lx ex 0
0.0201 0.0198 100,000 62.67 1 0.0011 0.0042
98,024 62.93 5 0.0005 0.0027 97,612 59.19 10
0.0006 0.0028 97,347 54.34 15 0.0018 0.0089
97,077 49.48 20 0.0036 0.0178 96,211 44.91 25
0.0042 0.0205 94,500 40.68 30 0.0053 0.0261
92,559 36.48 35 0.0068 0.0336 90,145 32.39 40
0.0093 0.0454 87,120 28.42 45 0.0128 0.0620
83,163 24.66 50 0.0176 0.0841 78,011 21.12 55
0.0253 0.1190 71,453 17.83 60 0.0345 0.1588
62,953 14.90 65 0.0495 0.2204 52,953 12.24 70
0.0652 0.2805 41,283 9.99 75 0.0930 0.3774
29,704 7.92 80 0.1345 0.5033 18,494 6.20 85
0.2021 1 9,186 4.95
WHO LIFE TABLE FOR 1999 UNITED STATES OF
AMERICA Males x nMx nqx lx ex 0
0.0069 0.0068 100,000 73.80 1 0.0004 0.0014
99,318 73.31 5 0.0002 0.0009 99,175 69.41 10
0.0002 0.0012 99,081 64.48 15 0.0010 0.0048
98,958 59.55 20 0.0013 0.0066 98,482 54.83 25
0.0014 0.0069 97,829 50.18 30 0.0018 0.0089
97,158 45.51 35 0.0023 0.0115 96,295 40.89 40
0.0031 0.0155 95,191 36.34 45 0.0044 0.0216
93,712 31.87 50 0.0066 0.0322 91,688 27.52 55
0.0102 0.0496 88,732 23.35 60 0.0167 0.0800
84,332 19.44 65 0.0251 0.1182 77,589 15.92 70
0.0394 0.1795 68,419 12.71 75 0.0595 0.2590
56,139 9.95 80 0.0963 0.3881 41,597 7.55 85
0.1738 1 25,455 5.75
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Russian lx
Age
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Russian qx
Age
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Russian ex
Age
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Russian and US qx
Russia
US
Age
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Ratio of Russian to US qx
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Russian and US lx
Russia
US
Age
40
Example Life Table
In the first interval, assume average yrs lived
among the dead is a0.1 In the last interval,
assume e80 is 8 years