The document discusses life tables and how they are used to calculate survival rates and life expectancy. It explains that the first four columns of a life table show age, mortality rate, population alive, and deaths. Column 5 shows years lived, column 6 shows remaining years of life, and column 7 shows life expectancy. The document provides an example of how to calculate these values and discusses how to handle censored data when not all outcomes are observed.

1. SURVIVAL AND LIFE TABLES
Nigel Paneth

3. THE FIRST FOUR COLUMNS
OF THE LIFE TABLE ARE:
1. AGE (x)
2. AGE-SPECIFIC MORTALITY RATE (qx)
3. NUMBER ALIVE AT BEGINNING OF
YEAR (lx)
4. NUMBER DYING IN THE YEAR (dx)

4. PROCEDURE:
We use column 2 multiplied by column 3 to
obtain column 4.
Then column 4 is subtracted from column 3 to
obtain the next row’s entry in column 3.

5. EXAMPLE:
100,000 births ( row 1, column 3) have an infant
mortality rate of 46.99/thousand (row 2, column 2),
so there are 4,699 infant deaths (row 3, column 4).
This leaves 95,301 left (100,000 – 4,699) to begin
the second year of life (row 2 column 3).

6. If we stopped with the first four columns, we
could still find out the probability of
surviving to any given age.
e.g. in this table, we see that 90.27% of non-
white males survived to age 30.

7. THE NEXT THREE COLUMNS OF
THE LIFE TABLE ARE:
Column:
5. THE NUMBER OF YEARS LIVED BY
THE POPULATION IN YEAR X (Lx)
6. THE NUMBER OF YEARS LIVED BY
THE POPULATION IN YEAR X AND IN
ALL SUBSEQUENT YEARS (Tx)
7. THE LIFE EXPECTANCY FROM THE
BEGINNING OF YEAR X (ex)

8. WE CALCULATE COLUMN 5
FROM COLUMNS 3 AND 4 IN THE
FOLLOWING WAY:
The total number of years lived in each year
is listed in column 5, Lx. It is based on two
sources. One source is persons who
survived the year, who are listed in column
3 of the row below. They each contributed
one year. Each person who died during the
year (column 4 of the same row) contributed
a part of year, depending on when they
died. For most purposes, we simply
assume they contributed ½ a year.

9. The entry for column 5, Lx in this table for age 8-9 is
94,321. Where does this number come from?
1. 94,291 children survived to age 9 (column 3 of
age 9-10), contributing 94,921 years.
2. 60 children died (column 4 of age 8-9) , so
they contributed ½ year each, or 30 years.
3. 94,921 + 30 = 94,321.

10. EXCEPTION TO THE ½
YEAR ESTIMATION RULE
Because deaths in year 1 are not
evenly distributed during the year
(they are closer to birth), infants
deaths contribute less than ½ a year.
Can you figure out what fraction of a
year are contributed by infant deaths
(0-1) in this table?

11. 1. Lx = 96,254
2. 95,301 contributed one year
3. 96,254 - 95,301 = 953 years, which must come
from infants who died 0-1
4. 4,699 infants died 0-1
5. 953/4,699 = .202 or 1/5 of a year, or about 2.4 months

12. HOW DO WE GET COLUMN 6, Tx
The top line of Column 6, or Tx=0 , is
obtained by summing up all of
the rows in column 5. It is the total
number of years of life lived by all
members of the cohort.
This number is the key calculation in
life expectancy, because, if we divide it
by the number of people in the cohort,
we get the average life expectancy at
birth, ex=0, which is column 7.

13. COLUMN 7, LIFE
EXPECTANCY, or ex=0
For any year, column 6, Tx, provides
the number of years yet to be lived by
the entire cohort, and column 7, the
number of years lived on average by
any individual in the cohort. (Tx/lx)
Thus column 7 is the final product of
the life table, life expectancy at birth,
or life expectancy at any other
specified age.

14. WHAT IS LIFE
EXPECTANCY?
Life expectancy at birth in the US
now is 77.3 years. This means that a
baby born now will live 77.3 years
if…………..
that baby experiences the same age-
specific mortality rates as are
currently operating in the US.

15. Life expectancy is a shorthand
way of describing the current
age-specific mortality rates.

16. SOME OTHER MEASURES OF
SURVIVAL
AND
THE PROBLEM OF CENSORED
DATA

17. • 5-year survival. Number of people still
alive five years after diagnosis.
• Median survival. Duration of time until
50% of the population dies.
• Relative survival. 5-year survival in the
group of interest/5-year survival in all
people of the same age.
• Observed Survival. A life table approach
to dealing with censored data from
successive cohorts of people. Censoring
means that information on some aspect of
time or duration of events of interest is
missing.

18. THREE KINDS OF CENSORING
COMMONLY ENCOUNTERED
• Right censoring
• Left censoring
• Interval censoring
Censoring means that some
important information required to
make a calculation is not available to
us. i.e. censored.

19. RIGHT CENSORING
Right censoring is the most common
concern. It means that we are not
certain what happened to people after
some point in time. This happens when
some people cannot be followed the
entire time because they died or were
lost to follow-up.

20. LEFT CENSORING
Left censoring is when we are not
certain what happened to people
before some point in time. Commonest
example is when people already have
the disease of interest when the study
starts.

21. INTERVAL CENSORING
Interval censoring is when we know
that something happened in an interval
(i.e. not before time x and not after
time y), but do not know exactly when
in the interval it happened. For
example, we know that the patient was
well at time x and was diagnosed with
disease at time y, so when did the
disease actually begin? All we know is
the interval.

22. DEALING WITH RIGHT-
CENSORED DATA
Since right censoring is the
commonest problem, lets try to find
out what 5-year survival is now for
people receiving a certain treatment
for a disease.

23. OBSERVED SURVIVAL IN
375 TREATED PATIENTS
Number Number alive in
Treated 1999 00 01 02 03
1999 84 44 21 13 10 8
2000 62 31 14 10 6
2001 93 50 20 13
2002 60 29 16
2003 76 43
Total 375

24. WHAT IS THE PROBLEM
IN THESE DATA?
We have 5 years of survival data only
from the first cohort, those treated in
1999.
For each successive year, our data is
more right-censored. By 2003, we have
only one year of follow-up available.

25. What is survival in the first year after treatment?
It is:
(44 + 31 + 50 + 29 + 43 = 197)/375 = 52%
Number Number alive in
Treated 99 00 01 02 03
1999 84 44 21 13 10 8
2000 62 31 14 10 6
2001 93 50 20 13
2002 60 29 16
2003 76 43
Total 375

26. What is survival in year two, if the patient
survived year one?
(21 + 14 + 20 + 16 = 71)/154 = 46%
Note that 154 is also 197 (last slide’s numerator) –
43, the number for whom we have only one year of
data
Number Number alive in
Treated 96 97 98 99 00
1995 84 44 21 13 10 8
1996 62 31 14 10 6
1997 93 50 20 13
1998 60 29 16
1999 76 43
Total 375

27. By the same logic, survival in the third year
(for those who survived two years) is:
(13 + 10 + 13 = 36)/(71 - 16 = 55) = 65%
Number Number alive in
Treated 99 00 01 02 03
1999 84 44 21 13 10 8
2000 62 31 14 10 6
2001 93 50 20 13
2002 60 29 16
2003 76 43
Total 375

28. In year 4, survival is(10 + 6)/(36-13) = 70%
In year 5, survival is 8/16-6 = 80%
Number Number alive in
Treated 99 00 01 02 03
1999 84 44 21 13 10 8
2000 62 31 14 10 6
2001 93 50 20 13
2002 60 29 16
2003 76 43
Total 375

29. The total OBSERVED SURVIVAL over
the five years of the study is the
product of survival at each year:
.54 x .46 x .65 x .70 x .80 = .08 or 8.8%

30. Subsets of survival can also be
calculated, as for example:
2 year survival = .54 x .46
= .239 or 23.9%

31. Five-year survival is averaged over
the life of the study, and improved
treatment may produce differences in
survival during the life of the project.
The observed survival is an average
over the entire period.

32. Changes over time can be looked at within
the data. For example, note survival to
one year, by year of enrollment:
1999 - 52.3%
2000 - 50.0%
2001 - 53.7%
2002 - 48.3%
2003 - 56.6%
Little difference is apparent.

33. These data also do not include any
losses to follow-up, which would
make our observed survival
estimates less precise. The
calculation is only valid if those
lost to follow-up are similar in
survival rate to those observed.