Life Table analysis

Dr Debjyoti MohapatraJunior resident

Dept. of Community MedicinePGIMS,Rohtak

Life expectancy at birth in India-65

Life expectancy at birth in Japan-83

Life expectancy at birth in United states-79

Life expectancy at birth in China-76

Life expectancy at birth in Sri lanka-75

Life expectancy at birth in Bangladesh -70

Life expectacy at birth in Iran-69

Life expectancy at birth in Nepal-68

Life expectancy at birth in Pakistan-67 Figures rounded of to nearest whole numbers.WHO report -2011

Latest report of life expectancy in India

MALES-67.3FEMALES-69.6

How long will I live ????????

Many babies born today will live past 100.

Joseph Brownstein ABC news medical unit(OCT 1,2009)

If you leave to be one hundred, you have got it made.Very few people die past that age.

George Burns American Comedian

(January 20,1896-March 9,1996)

How long do we live?

Can not answer for an individual , but can make a statement about a group.

Contents

Brief overview

Life table : A historical perspective

Types of Life table

Construction of a Life table

Applications of Life table

Conclusion

Brief overview

Edmond Halley was the first person to show us how to properly

calculate and construct the life table.

Halley was a british astronomer , geophysicist, mathematician,

meteorologist who is best known for computing the orbit of

Halley’s comet.

He calculated the first ever life table sometime 300 years back . Till

date the same methodology is followed with only slight variation.

Brief overview

Life table tells us how long people live on an average.

It converts a cross sectional information into a longitudinal cohort information.

Brief overview

Average life span : or how long do we live Suppose we have a population of 10 people and we

follow the till they all die. Here are their life spans.1,2,10,20,35,45,50,60,70,80

So the average life span is 1+2+10+20+35+45+50+60+70+80=37.3 10

0.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40 50 60 70 80 90

1*0.1+2*0.1+10*0.1+20*0.1+35*0.1+45*0.1+50*0.1+60*0.1+70*0.1+80*0.1=37.3

Survival curve or life table curve

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90

Per

cent

age

surv

ived

U5

AREA=average survival

Adult mortality=70-30/70=57%

Age

0 10 20 30 40 50 60 70 80 90

Age

Perc

en

tag

e S

urv

ived

Life table : A historical perspective

The first life table had been published in London in 1662 in a book

entitled Natural and Political Observations Made upon the Bills of

Mortality.

People still wonder nowadays if it was written by John Graunt, a

London merchant and author indicated on the book cover, or by his

friend William Petty, one of the founders of the Royal Society.

Their life table was subject to large errors.

At about the same time, Caspar Neumann, a theologian living in Breslau, was

collecting data about the number of birth, deaths and other vital statistics in his

city.

Breslau belonged to the Habsburg empire. (it is now in Poland and called

Wrocław)

Neumann sent to Henry Justel, the secretary of the Royal Society, his

demographic data from the city of Breslau for the years 1687–1691.

Justel died shortly after, and Halley got hold of the data, analyzed them and in

1693 published his conclusions in the Philosophical Transactions of the Royal

Society.

Edmond Halley

HALLEY’S LIFE TABLE OF BRESLAU CITY

We do not know what will happen to the 65 year olds in say 2050;

BUT we do know what happened to the 65 year old in past year, and the 66year old in past year etc.

Finally the definition

The life table methodology constructs a

life experience for a fictional cohort

subjected to current mortality rates as it

progresses through life, as if the current

rates do not change.

Types of Life Table

1. Current/Period vs Generation/Cohort.

2. Complete vs Abridged.

3. Multiple decremental tables.

4. Incremental –Decremental life tables

Current/Period vs Generation/Cohort

COHORT LIFE TABLE PERIOD LIFE TABLE

• The cohort life table presents the mortality experience of a particular birth cohort.

• All persons born in a year, from the moment of birth through consecutive ages in successive calendar years.

• The cohort life table reflects

the mortality experience of an actual cohort from birth until no lives remain in the group.

The period life table presents what would happen to a hypothetical (or synthetic) cohort if it experienced throughout its entire life the mortality conditions of a particular time period.

Complete vs Abridged

COMPLETE ABRIDGED

A complete life table contains

data for every year of age.

An abridged life table

typically contains data by 5-

or 10-year age intervals .

A separate group is made for

age group 0-1 years .

In India a 5 year interval is

selected.

Single and Multiple Decremental Life Tables

SINGLE DECREMENTAL MULTIPLE DECREMENTAL

Multistate or Incremental-decremental Life table

Construction of Life table

To construct a life table, two things are required:

1.Population living at all individual ages in a selected year.

2.Number of deaths that occurred in these ages during the selected year.

Age interval(x tox+n)

Probability dying(n q x)

Number surviving((lx)

Number Dying(nd x)

Person-years lived between exact agesx and x+n

Person –years lived above age x (Tx)

Life expectancy at age x(ex)

This coloumn shows the age interval between two exact ages indicatedProbability of dying in the age group(x-

x+n)given survival upto age xx

This column shows the number of persons, starting with a cohort of 100,000 live births, who survive to the exact age marking the beginning of each age interval

This column shows the number dying in each successive age interval out of 100,000 live births.

Person-years lived between exact ages x and x+nThis takes into consideration the years that would have been lived by the people that are dying during this time interval

. It is number of years lived by group from age x until all of them die

It is done by calculating the sum of the previous coloumn data staring from particular age group till all cohort dies.

Average remaining lifetime for a person who survives to age x

Calculated by the formula Tx/lx

5+536

2.5

Calculation of probability of dying in a particular age group

Application of life tables

A Age expectancy at age 60=355234+310042+250443+180901+113743+95820/74611=

1306183/74611=17.50

Applications of life table

Preston

Newton

Boston

BostonCatholic

Application of life table

RA

Q) From the abridged SRS based life table, India,1976-77 for males and females, find:

a. What proportion of men entering services at 20 will be eligible for pension at age 55.

b. Calculate life expectation at age 55 for men.

c. If 20% of deaths occuring in men between the ages of 60 and 70 inclusive are due to cancer ,what proportion of men aged 60 yrs are likely to die of cancer before reaching their 70th birthday.

l20=78368L55=64886Proportion eligible for pension=64,886/78,368*100=82.79

l20=77433L55=58985Proportion58985/77433*100=76.17%

Life expectancy at 55=

Sum of lx coloumn from lx 55 onwards/number who started at lx 55 onwards + 2.5=

(58,985+51,345+41,150+30,952)*5/64886+2.5=16.55

Now calculating by the methoddiscussed by us

276812+232232+180976+282667/58985=16.49

Calculating life expectancy at age 40 by adopting authors method(71,977+68856+64886+58985+51345+41150+30092)*5/73801+(2.5)=28.73

Now adopting the method discussed by us

352655+335021+310579+276812+232232+180976+282667/71977= 27.38

No of deaths in age group (60-70 yrs)= 51345-30952=20393

Cancer deaths=20*20393= 4078

Therefore proportion of men aged 60 likely to die of cancer by 70 yrs=4078/51345 *100= 7.942

100

This method is modification of usual life table for calculating the survival rates after specific treatment or operation or at any point of time after that, It can be explained by an example of tuberculosis.

A total of 23 patients of tuberculosis started treatment in a T.B. clinic. Their number becomes less due to defaulters.

Out of 23 who started treatment in January, only 22 reported in February,79 in March & so on till there were only 9 left in the month of November.

Modified life table

Follow-up results of tuberculosis patients

Month

Pt. Feb

Mar Apr May Jun Jul Aug Sep Oct

Nov

Jan 23 22 13 12 12 10 9 9 9 9 9

Feb 9 7 4 4 4 4 3 2 2 2

Mar 20 12 12 9 9 8 8 8 7

Apr 17 15 10 10 10 9 7 6

May 23 20 17 12 11 11 9

Jun 17 12 12 11 9 8

Jul 20 12 11 9 9

Aug 22 20 14 13

Sep 18 16 15

Oct 16 12

Month

Pt. 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10t

h

Jan 23 22 13 12 12 10 9 9 9 9 9

Feb 9 7 4 4 4 4 3 2 2 2

Mar 20 12 12 9 9 8 8 8 7

Apr 17 15 10 10 10 9 7 6

May 23 20 17 12 11 11 9

Jun 17 12 12 11 9 8

Jul 20 12 11 9 9

Aug 22 20 14 13

Sep 18 16 15

Oct 16 12Total no.

185 148 108

80 64 50 36 25 18 11 9

Next mont

h

185 136 93 67 55 42 27 19 11 9

Follow-up results of tuberculosis patients

No. of patients in 0th (xeroth) month/starting month = 185

Reported for treatment in next/1st month = 148

So, probability of coming in next/1st monthpx = 148/185 = 0.80

Out of 148, history of 12 patients was not available, so no. of patients left = 136

Reported for follow up in 2nd month =108So, probability of coming in next month p = 108/136 = 0.79

Probability of defaulting qx = 1 - dx

Contd…

How to calculate dx?dx = lx × qxe.g. dx at 1 month follow up =

1000×0.20 = 200So, lx at 1 month follow up

1000 – 200 = 800 Similarly dx at 2 month follow up =

800×0.21 =168So, lx at 2 month follow up = 800-168 =

632, so on…How to fill up Lx (no. of months attended by

starters) column? Lx=lx + 1/2dx e.g. at the end of 1st month 800+100=900 at the end of 2nd month 632 + 84 =716, so on…

Contd…

Month of Rx

x

Probability of

Reporting px

Probability of

Defaulting qx

No. available in every month

lx

No. of defaulte

rs in every

month dx

No. of months

attended by

starters Lx

Total months

attended at all ages

Tx

Expected to attend

at any month

ex

0 0.8 0.20 1000 200 900 5,162 5.16

1 0.79 0.21 800 168 716 4,262 5.33

2 0.86 0.14 632 68 588 3,546 5.61

3 0.96 0.04 544 22 533 2,958 5.43

4 0.91 0.09 522 47 499 2,425 4.64

5 0.86 0.14 475 67 442 1,926 4.05

6 0.93 0.07 408 29 394 1,484 3.63

7 0.95 0.05 379 19 370 1,090 2.87

8 1.00 0.00 360 0 360 720 2.00

9 1.00 0.00 360 0 360 360 1.00

10 - - 360 - -

Table 4 Follow-up results in life table form

How to calculate ex (expected no. of months for which a person is likely to attend at any month)?

ex = Tx/lx e.g.

at xeroth month ex = 5162/1000 = 5.16 months at the end of 6th month ex = 1484/408 = 3.63

Contd…

Uses & Applications

To find the number of survivors out of 1,000 or

10,000 or over birth or at any age thereafter say,

At the age of 5, to find number of children likely to enter primary school.

At the age of 15, to find number of women entering fertile period.

At age of 18, to find number of persons become eligible for voting.

Contd…

• To estimate the number likely to die after joining service till retirement, helping in budgeting for payment towards risk or pension.

• To find expectation of life or longevity of life at birth or any other age.

• Increase in longevity of life means reduction in mortality, thus life table is another method applied to compare mortality of two places, periods, professions or groups.

Contd…

• To find survival rate after treatment in chronic disease like tuberculosis, cancer or after cardiac surgery by modified life table.

• Helps to project population estimates by age & sex.

• Calculate failure rate of contraceptive.

ConclusionThe Life table methodology was first adopted some 300 years back. Credit must be given to Edmond Halley who used his innovative mind to create this statistical tool.

In an era of sophisticated and advanced statistical applications, Life tables have survived the test of time.

Life table is an old method and probably in this era may not be gold but definitely it has not lost its shine and is still being used to calculate some vital parameters(life expectancy, contraceptive failure).

Conclusion

Who knows one day with certain modifications this old

methodology may ascertain its value as the gold

methodology.

HAPPY HOLI