Mortality measures--Mortality measures--

crude, specific, & crude, specific, & summary (the life table)summary (the life table)

Recall lesson about rates:Recall lesson about rates:crude, age-specific, summary stats crude, age-specific, summary stats

Crude death rate: Crude death rate:

conceals a lot because mortality varies conceals a lot because mortality varies greatly by age greatly by age

Age-specific death rates: Age-specific death rates: asdr = deaths ageasdr = deaths agexx/pop. at risk age/pop. at risk agex x

(for specific period and place)(for specific period and place)Summary statistic-->the life table: Summary statistic-->the life table:

life expectancy: mean years expected to live life expectancy: mean years expected to live from age x under current mortality from age x under current mortality conditionsconditions

The Life Table: demographic The Life Table: demographic fossil or “most useful tool”fossil or “most useful tool”

Laid out by John Graunt in 1662, when Laid out by John Graunt in 1662, when data and ability to calculate were scarce data and ability to calculate were scarce commoditiescommodities

The life table was designed to both “show The life table was designed to both “show the work”, how death rates by age are used the work”, how death rates by age are used to compute all stats of the table, and to to compute all stats of the table, and to “show the results”.“show the results”.

Notion can be extended beyond mortalityNotion can be extended beyond mortality

Brief history of the Life TableBrief history of the Life TableGraunt’s Observations (1662)Graunt’s Observations (1662)

Graunt “Observations on the London Bills of Mortality” Graunt Graunt “Observations on the London Bills of Mortality” Graunt speculated on the regularities of demographic events: more male speculated on the regularities of demographic events: more male births than female, higher male mortality than female, frequency births than female, higher male mortality than female, frequency of various causes of deaths, etc.of various causes of deaths, etc.

Graunt’s life table constructed without age or sex data Graunt’s life table constructed without age or sex data birthbirth 100 individuals100 individuals66 64 (based on deaths attributed to children64 (based on deaths attributed to children1616 40 (total conjecture: arithmetical formula)40 (total conjecture: arithmetical formula)2626 25253636 1616……7676 11

Brief history of the Life TableBrief history of the Life TableEdmond Halley (1656-1742)Edmond Halley (1656-1742)

Halley (1693), Halley (1693), ““An Estimate of the Degrees of the Mortality of Mankind, drawn An Estimate of the Degrees of the Mortality of Mankind, drawn from curious from curious Tables Tables of the of the Births Births and and Funerals Funerals at the City of at the City of Breslaw”Breslaw”

Critique of Graunt’s shortcomings: lacked the number of people, ages at death, London Critique of Graunt’s shortcomings: lacked the number of people, ages at death, London had too much migrationhad too much migration

Breslau (Poland) seemed to be a closed population with little migration and death data Breslau (Poland) seemed to be a closed population with little migration and death data were available by age (individual years)were available by age (individual years)

1 (birth) 1000 individuals1 (birth) 1000 individuals22 85585533 79879844 76076055 732732….….3333 507507……8484 2020

1675 – published his first paper on astronomy… in the Philosophical Transactions of the Royal Society… followed by many others

1680s principal editor of the Philosophical Transactions

1691 – denied professorship of astronomy at Oxford: charged with not accepting literal truth of the Bible

Identified the comet of 1682 as the same as 1531, 1607, … and 1305, 1380 and 1456

Predicted its return for Dec. 1758; he was proven correct when it appeared Dec. 25.

1704 – appointed professor of geometry at Oxford.

Brief history of the Life TableBrief history of the Life TableEdmond Halley (1656-1742)Edmond Halley (1656-1742)

Uses of life table, according to HalleyUses of life table, according to Halley

I.I. Proportion of men to bear armsProportion of men to bear arms

II.II. Show differing degrees of mortality by age; the odds that a person shall live from Show differing degrees of mortality by age; the odds that a person shall live from one age to anotherone age to another

III.III. Years that a person is likely to die (used the median)Years that a person is likely to die (used the median)

IV.IV. The price of insurance upon livesThe price of insurance upon lives

V.V. The valuation of annuitiesThe valuation of annuities

VI.VI. The valuation of joint annuities (husband+wife; wife+child, etc.)The valuation of joint annuities (husband+wife; wife+child, etc.)

Halley’s observation: “Halley’s observation: “the Growth and Encrease of Mankind is not so much stinted the Growth and Encrease of Mankind is not so much stinted

by any thing in the Nature of the Species, as it is from the cautious difficulty most by any thing in the Nature of the Species, as it is from the cautious difficulty most People make to adventure on the state of Marriage, from the prospect of the Trouble People make to adventure on the state of Marriage, from the prospect of the Trouble and Charge of providing for a Family. Nor are the poorer sort of People herein to be and Charge of providing for a Family. Nor are the poorer sort of People herein to be blamed, since their difficulty of subsisting is occasioned by the unequal Distribution of blamed, since their difficulty of subsisting is occasioned by the unequal Distribution of Possessions, all being necessarily fed from the Earth, of which yet so few are Masters.Possessions, all being necessarily fed from the Earth, of which yet so few are Masters. ””

The Life Table (see spreadsheet): The Life Table (see spreadsheet): 3 fundamental mysteries revealed3 fundamental mysteries revealedHow to read a life table:How to read a life table:

from age specific death rates, from age specific death rates, derive life expectancy derive life expectancy How a table is constructed: How a table is constructed:

with deaths at age with deaths at age xx, derive the rest, derive the restWhat do life tables reveal about What do life tables reveal about

mortality transitions of the past mortality transitions of the past quarter millennia (quarter millennia (a few surprisesa few surprises))

Life Table: Sweden, 1751-55, females. Use death rates at age x to compute: life expectancy

mortality Quotient

Lives at age x

Deaths age x to x+n

central mortality rate years Lived

Total years lived

life Expectancy

Agex, x+n-1 nqx lx ndx nmx nLx Tx ex

data lx-n-dx, x+n qx*lx complex lx*n-dx*(n/2) Sum Lx Tx/lx

0 0.21116 100000 21116 88890 3843574 38.441 0.14165 78884 11174 287601 3754684 47.605 0.05859 67710 3967 328633 3467083 51.2010 0.03038 63743 1937 313849 3138450 49.2415 0.02922 61806 1806 304495 2824601 45.7020 0.03214 60000 1928 295155 2520106 42.0025 0.03966 58072 2303 284565 2224951 38.3130 0.05147 55769 2870 271607 1940386 34.7935 0.04942 52899 2614 257903 1668779 31.5540 0.06476 50284 3256 243190 1410876 28.0645 0.06360 47028 2991 227580 1167687 24.8350 0.07824 44037 3445 211453 940107 21.3555 0.09983 40591 4052 192649 728654 17.9560 0.14624 36539 5343 168985 536005 14.6765 0.19813 31196 6181 139959 367019 11.7770 0.30893 25015 7728 104568 227061 9.0875 0.41532 17287 7180 66888 122493 7.0980 0.52440 10107 5300 35660 55605 5.5085 0.66513 4807 3197 14613 19945 4.1590 1.00000 1610 1610 5332 5332 3.31

• key:key: x = age x; n = interval. x = age x; n = interval.

• qqxx = = nnddxx/l/lxx, the mortality quotient, the likelihood , the mortality quotient, the likelihood of dying at age x to age x+n. of dying at age x to age x+n. NOTE: every statistic in the entire life table is derived NOTE: every statistic in the entire life table is derived from from nnqqxx

• llxx = number of individuals alive at exact age x = number of individuals alive at exact age x

•nnddxx = number of deaths at exact age x to x+n = number of deaths at exact age x to x+n

•nnLLxx = years lived by individuals from exact age x = years lived by individuals from exact age x to age x+nto age x+n

• TTxx = total years lived from exact age x to = total years lived from exact age x to maximum age in life table: Tmaximum age in life table: Tx+nx+n + + nnLLxx

• eexx = life expectancy at exact age x: T = life expectancy at exact age x: Txx/l/lxx

The Life Table: The Life Table: 3 fundamental mysteries revealed3 fundamental mysteries revealedHow to read a life table:How to read a life table:

at age at age xx, 2 stats: , 2 stats: mortality rate, life expectancy mortality rate, life expectancy How a table is constructed: How a table is constructed:

with death rates, derive...with death rates, derive...What do life tables reveal about What do life tables reveal about

mortality transitions of the past mortality transitions of the past quarter millennia (quarter millennia (a few surprisesa few surprises))

Life Table: Sweden, 1751-55, females. Use death rates at age x to compute: life expectancy

mortality Quotient

Lives at age x

Deaths age x to x+n

central mortality rate years Lived

Total years lived

life Expectancy

Agex, x+n-1 nqx lx ndx nmx nLx Tx ex

data lx-n-dx, x+n qx*lx complex lx*n-dx*(n/2) Sum Lx Tx/lx

0 0.21116 100000 21116 88890 3843574 38.441 0.14165 78884 11174 287601 3754684 47.605 0.05859 67710 3967 328633 3467083 51.2010 0.03038 63743 1937 313849 3138450 49.2415 0.02922 61806 1806 304495 2824601 45.7020 0.03214 60000 1928 295155 2520106 42.0025 0.03966 58072 2303 284565 2224951 38.3130 0.05147 55769 2870 271607 1940386 34.7935 0.04942 52899 2614 257903 1668779 31.5540 0.06476 50284 3256 243190 1410876 28.0645 0.06360 47028 2991 227580 1167687 24.8350 0.07824 44037 3445 211453 940107 21.3555 0.09983 40591 4052 192649 728654 17.9560 0.14624 36539 5343 168985 536005 14.6765 0.19813 31196 6181 139959 367019 11.7770 0.30893 25015 7728 104568 227061 9.0875 0.41532 17287 7180 66888 122493 7.0980 0.52440 10107 5300 35660 55605 5.5085 0.66513 4807 3197 14613 19945 4.1590 1.00000 1610 1610 5332 5332 3.31

The Life Table: The Life Table: 3 fundamental mysteries revealed3 fundamental mysteries revealedHow to read a life table:How to read a life table:

at age at age xx, 2 stats: , 2 stats: mortality rate, life expectancy mortality rate, life expectancy How a table is constructed: How a table is constructed:

with deaths at age with deaths at age xx, derive the rest, derive the restWhat do life tables reveal about What do life tables reveal about

mortality transitions of the past mortality transitions of the past quarter millennia (quarter millennia (a few surprisesa few surprises))

Life table Life table mortality probabilities and mortality probabilities and

life expectancies life expectancies in historical perspective:in historical perspective:

The case of Sweden, 1751-1999The case of Sweden, 1751-1999

Changing demographic patterns Changing demographic patterns of dying: Sweden, 1751-1996of dying: Sweden, 1751-1996

Indicator: Indicator: nnqqxx, or mortality quotient, or mortality quotient

graphics of graphics of nnqqxx by: by:

age (0, 1, 5...85+),age (0, 1, 5...85+),

sex andsex and

time (1751, 1851, 1951, 1996-9) time (1751, 1851, 1951, 1996-9)

nnqqxx Females: 1751 Females: 1751n

qx

Sweden 1751 nQx (mortality quotient): femaleage

0 10 20 30 40 50 60 70 80

.0005

.001

.01

.1

1

female

11qq0 = 0 = 0.2110.211

55qq15 = 15 = 0.0290.029

55qq65 = 65 = 0.1980.198

nnqqxx Females and Males: 1751 Females and Males: 1751

Sweden 1751 nQx (mortality quotient): female vs. maleage

q1751f q1751m

0 10 20 30 40 50 60 70 80

.0005

.001

.01

.1

1

femalemale

11qq0 = 0 = 0.2110.211

55qq15 = 15 = 0.0290.029

55qq65 = 65 = 0.1980.198

Sweden females nQx: 1751, 1851age

q1751f q1851f

0 10 20 30 40 50 60 70 80

.0005

.001

.01

.1

1

1751

1751

1851

1851

nnqqxx 1751- 1851: Sweden Females 1751- 1851: Sweden Females

11qq0 = 0 = 0.2110.211

55qq15 = 15 = 0.0290.029

55qq65 = 65 = 0.1980.198

1751-1851: 1751-1851: gains for youth; gains for youth; losses for elderslosses for elders

Sweden females nQx: 1751, 1851, 1951age

q1751f q1851f q1951f

0 10 20 30 40 50 60 70 80

.0005

.001

.01

.1

1

1751

1751

1851

1851

1951

1951

nnqqxx 1751- 1951: Sweden Females 1751- 1951: Sweden Females

11qq0 = 0 = 0.2110.211

55qq15 = 15 = 0.0290.029

55qq65 = 65 = 0.1980.198

1851-1951: 1851-1951: major gains to 65 major gains to 65 small gains for 65+small gains for 65+

Sweden females nQx: 1751, 1851, 1951, 1996age

q1751f q1851f q1951f q1996f

0 10 20 30 40 50 60 70 80

.0005

.001

.01

.1

1

1751

1751

1851

1851

1951

1951

1996

1996

nnqqxx 1751- 1996: Sweden Females 1751- 1996: Sweden Females

11qq0 = 0 = 0.2110.211

55qq15 = 15 = 0.0290.029

55qq65 = 65 = 0.1980.198

1951-1999: 1951-1999: major gains all agesmajor gains all ages

including 65+ including 65+

Sweden 1951 nEx: 1751, 1851, 1951, 1996 (females)age

e1751f e1851f e1951f e1996f

0 10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

90

100

1751

1751

1851

1851

1951

1951

1996

1996

eexx 1751, 1851, 1951 and 1996: 1751, 1851, 1951 and 1996:

Sweden FemalesSweden Femalesage 0age 0

81817373

44443838

age 10age 10

71716565

49494949

age 65age 652020151510101212

Life expectancy and the Life expectancy and the mortality transitionmortality transition

First improvements: infant mortality; 1751-First improvements: infant mortality; 1751-1851: e1851: e00 increases 6 years; e increases 6 years; e6565 decreasesdecreases 2 2

yearsyears

Second, 1851-1951: infant and child Second, 1851-1951: infant and child mortality, emortality, e0 0 increases 29 and eincreases 29 and e1010 16 years; e 16 years; e6565

increases 5 yearsincreases 5 years

Last, 1951-: eLast, 1951-: e00 increases 8 years; e increases 8 years; e1010

increases 6; eincreases 6; e6565 increases 5 years. increases 5 years.

Life expectancy and the Life expectancy and the mortality transitionmortality transition

First improvements: infant mortality; 1751-First improvements: infant mortality; 1751-1851: e1851: e00 increases 6 years; e increases 6 years; e6565 decreases 2 decreases 2

yearsyears

Second, 1851-1951: infant and child Second, 1851-1951: infant and child mortality, emortality, e0 0 increases 29 and eincreases 29 and e1515 16 years; e 16 years; e6565

increases 5 yearsincreases 5 years

Last, 1951-: eLast, 1951-: e00 increases 8 years; e increases 8 years; e1515

increases 6; eincreases 6; e6565 increases 5 years. increases 5 years.

Increase in additional years of life: Sweden, Females

Period e0 e10 e65

1751-1851 +6 0 -2

1851-1951 +29 +16 +5

1951-1999 +8 +6 +5

Life expectancy and the Life expectancy and the mortality transitionmortality transition

First improvements: infant mortality; 1751-First improvements: infant mortality; 1751-1851: e1851: e00 increases 6 years; e increases 6 years; e6565 decreases 2 decreases 2

yearsyears

Second, 1851-1951: infant and child Second, 1851-1951: infant and child mortality, emortality, e0 0 increases 29 and eincreases 29 and e1515 16 years; e 16 years; e6565

increases 5 yearsincreases 5 years

Last, 1951-: eLast, 1951-: e00 increases 8 years; e increases 8 years; e1515

increases 6; eincreases 6; e6565 increases 5 years. increases 5 years.

Net increases in additional years of life: Sweden, Females

Period e0 e10 e65

1751-1851 +6 0 -2 +6 +2 -2

1851-1951 +29 +16 +5+13 +11 +5

1951-1999 +8 +6 +5 +2 +1 +5

e0 in the Americas, 1900-2005unequal in 1900; converging since 1960

1900

192019401960198071 76

7975

782005

ee00: 2005: 2005source: www.prb.orgsource: www.prb.org

EndEnd