1. Chapter 4 SURVIVAL AND LIFE TABLES
Dr. A. PHILIP AROKIADOSS
Assistant Professor
Department of Statistics
St. Joseph’s College (Autonomous)
Tiruchirappalli

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:
THE NUMBER OF YEARS LIVED BY THE POPULATION IN YEAR X (Lx)
THE NUMBER OF YEARS LIVED BY THE POPULATION IN YEAR X AND IN ALL SUBSEQUENT YEARS (Tx)
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
The entry for column 5, Lx in this table for age 8-9 is 94,321. Where does this number come from?
94,291 children survived to age 9 (column 3 of age 9-10), contributing 94,921 years.
60 children died (column 4 of age 8-9) , so they contributed ½ year each, or 30 years.
94, = 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. Lx = 96,254
95,301 contributed one year
96, ,301 = 953 years, which must come from infants who died 0-1
4,699 infants died 0-1
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
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:
( = 197)/375 = 52%
Number Number alive in
Treated
Total 375

26. What is survival in year two, if the patient survived year one?
( = 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
Total 375

27. By the same logic, survival in the third year (for those who survived two years) is:
( = 36)/( = 55) = 65%
 Number Number alive in
Treated
Total 375

28. In year 4, survival is(10 + 6)/(36-13) = 70%
Number Number alive in
Treated
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 = .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
Changes over time can be looked at within the data. For example, note survival to one year, by year of enrollment: % % % % %
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.