SURVIVAL AND LIFE TABLES

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SURVIVAL AND LIFE TABLES

• Nigel Paneth

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(No Transcript)
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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)

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• 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 rows entry in column 3.

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• 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).

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• 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.

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THE NEXT THREE COLUMNS OF THE LIFE TABLE ARE

• Column
• 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)

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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.

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• 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.
• 94,921 30 94,321.

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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?

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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

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HOW DO WE GET COLUMN 6, Tx

• The top line of Column 6, or Tx0 , 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, ex0, which is
  column 7.

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COLUMN 7, LIFE EXPECTANCY, or ex0

• 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.

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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.

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

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SOME OTHER MEASURES OF SURVIVALAND THE PROBLEM
OF CENSORED DATA
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• 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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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• 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

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• 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 slides
  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

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• 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

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• 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

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• 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

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

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• 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.

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

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• 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.