Review of Cohort Studies Life Table Analysis

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Review of Cohort Studies Life Table Analysis

• Advanced Epidemiology II
• Spring 2002

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Measures of Association

• How much of a disease can be attributed to a
  causative factor?
• What is the potential benefit from intervening to
  modify the factor?

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Measures of Association

• Relative perspective
• what proportion of the risk is attributable to
  the factor
• what proportion of the cases of the disease might
  be avoided if the factor were absent?

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Measures of Association

• RR measures compare the risk (or rate) in an
  exposed group to that in an unexposed group
• in a manner that assesses the strength of
  association between the exposure and outcome
• for the purpose of evaluating whether the
  association is a causal one

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Measures of Association

• Absolute
• how much of the risk is attributable to the
  factor?
• how many cases might be avoided if the factor
  were absent (risk difference)
• Attributable fractions answer
• the so what question

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Measures of Association

• Now that I am 35 years old, my CHD risk from
  taking OC is 2 as great as when I was 25. But
  how much more risk do I have due to taking the
  pill?
• How many cases of asthma are due to ambient
  sulfur dioxide?
• Attributable fractions measure the impact or
  attributable risk

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Measures of Association

• If we observe incidence I1 in an exposed
  population and a lower incidence I0 in a
  comparable unexposed population,
• and we make the assumption that the exposure is
  causing the higher incidence in the exposed
  population,
• then it is logical to suppose that the difference
  I1-I0 is the amount of incidence that is due to
  the exposure

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Measures of Association

• caused by contrasted with attributed to
• among smokers exposed to X, what proportion of DX
  is caused by X
• among smokers exposed to X, what proportion of
  disease is attributable to X

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Measures of Association

• If the action of one causal factor can preempt
  the opportunity for another factor to cause the
  disease, then there is no way to know from
  epidemiologic data which factor caused the
  disease in a person or population exposed to both
  causal factors

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Measures of Association

• Interpreting attributable fractions
• basic concepts intuitively meaningful -
• subtleties and nuances
• confusing
• if you find yourself being confused by something
  you are reading in this area, always consider the
  possibility that what you are reading may be
  confused as well

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Analysis of cohort study data

• Cumulative Incidence Ratio
• CI1 / CI0
• Incidence Density Ratio
• ID1 / ID0 incidence density ratio
• Cumulative incidence difference
• CID C1 - C0
• Incidence density difference
• IDD ID1 - ID0

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Person time

• How do we calculate incidence density or
  incidence when there is variability in follow-up
  time.
• What if the risk of disease is not constant over
  time is a simple RR appropriate or is the
  effect modified by time?
• And how do we adjust for time as a
• confounder / interaction term / effect modifier

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Incidence Measures

• Choice is based on pragmatic considerations
• Cumulative incidence
• study of acute disease with restricted risk
  periods
• fixed populations
• Incidence density
• suited for studies of chronic diseases with
  extended risk periods
• dynamic populations (adjust follow-up time on a
  person-by-person basis

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Cohort life-table analysis

• Life table analysis actuarial method
• provides for one way of handling longitudinal
  data
• the probability of dying during each interval is
  the number dying in the interval divided by the
  number alive at the beginning of the interval
• (the probability of dying during an interval
  among those alive at the beginning of the
  interval)

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Life table analysis
the probability of surviving an interval but in
the probability of surviving 6 months or 12
months or some more specific period of time
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Life table analysis
Probability of dying 1-(30/300) .90 To
account for differences in follow-up 1-.0989
.90 (not much difference no lost to follow-up
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Life table analysis
Probability of surviving 5/300
0.02 Probability of dying 0.98 After adjusting
for lost to follow-up and deaths probability of
dying .90
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Life table analysis
Rather than withdrawals coming at the beginning
have withdrawals evenly spaced through the
interval.
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Comparison of survival curves

• Mantel-Haenszel procedure
• Compare two survival curves over entire period
  being analyzed
• each time interval is treated as a strata
• summary OR is calculated over all strata
• with the MH chi square calculated as usual

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Comparison of survival curves

• Logrank test - depends on the number number at
  risk at the beginning of each interval
• Logrank is easy to compute and an approximate
  summary chi square can be derived but this
  method can be misleading if you don not avoid
  intervals of high risk.

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High risk strata are not avoided First strata
193/297 and 165/297 Thus log-rank inappropriate
method
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Now to make things confusing

• The log-rank test in Stata is the MH test
• most appropriate when the hazard functions are
  thought to be proportional across the groups
• the hypothesis is that the survivor functions are
  the same
• if p gt0.05 we cannot reject the null
• with the life table command (ltable) the only
  comparison statistic is the log-rank test (the MH
  stratified test for equality)

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ltable timevar deadvar, by (groupvar) test
ltable studytim died, by(drug) test Likelihood-ra
tio test statistic of homogeneity
(groupdrug) chi2( 2 ) 20.02446, P
.00004485 Logrank test of homogeneity
(groupdrug)     Log-rank test for equality of
survivor functions   Events
Events drug observed expected ---------
---------------------- 1 19
7.25 2 6 8.20 3
6 15.56 ---------------------------
---- Total 31 31.00  
chi2(2) 30.19 Prgtchi2
0.0000
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Kaplan-Meier or Product-limit method

• Life table groups data into convenient intervals
• Special case of life table Kaplan-Meier
• intervals contain only persons with exactly the
  same survival time.
• no need for assumption of time of death or
  withdrawal or loss

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sts graph
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sts grapy, by(drug)
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Comparison of curves

• st test drug
• gives the MH comparison (appropriate with hazards
  are proportional across groups)
• st test drug, wilcoxon
• generalized Wilcoxon test of Breslow and Gehan.
  Appropriate when hazard functions are thought to
  vary in ways other than proportionally. Gives
  heavier weight to early strata

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• sts test drug, tware
• weights equal to the square root of the number of
  subjects at risk - heavier weight to early strata
• sts test drug, peto
• alternative to the wilcoxon test - weight
  estimate of the overall survivor function.
  appropriate when hazards are thought to vary in
  ways other than proportionally - not affected by
  differences in censoring patterns across groups.

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• sts test drug, trend
• can be preformed with any of the test functions
• when groups have a natural order (age, drug
  dosage)
• test null of no difference to alternative of
  ascending or descending differnences

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General Life Table Assumptions

• Is the surviving (or risk of dying) independent
  of time?
• Should subject within each group have similar
  distribution of survival time

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Major assumptions of actuarial vs product-limit
methods

• Product of conditional probabilities leads to the
  survival estimate
• Kaplan-Meier events and losses to follow-up do
  not occur simultaneously
• Actuarial losses occur at mid-interval thus
  uniformly distributed over a given arbitrary
  interval

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Censoring of events

• terminating the study
• close out date
• under the assumption that any occurring event
  during the termination of the study is
  independent of study closure
• losses to follow-up
• withdrawal (administrative or otherwise)

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Two survival analysis methods

• Actuarial
• useful for large sample
• intervals may have to be created
• entry into the cohort is known
• withdrawals may or may not be known
• losses assumed to occur at ½ the interval
• early withdrawals may lead to an underestimation
  (usually) or overestimation (rarely) in the
  magnitude of the association

• Kaplan-Meier
• effective for small sample size
• exact entry and withdrawal from the cohort is
  known
• losses are known at exact times

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Similarities

• non-parametric
• describes individual survival or risk
• comparisons between two subgroups yield
  univariable estimates of the characteristic
• computationally similar cohort is fixed at
  entry

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Actuarial Method

• gives interval specific risks and rates
• can be used with an external standard to get SMRs
  and relative survival
• can be used when exact dates of occurrence not
  known
• computations easy
• intervals are arbitrary
• assumption of mid-interval withdrawals

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Kaplan-Meier

• intervals not arbitrary
• gives more complete survival curves
• computations easy
• requires exact dates
• cumulative risk estimates are weighted towards
  later events

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Kaplan-Meir Analysis

• 10 patients
• Accumulated over time
• 7 died
• 1 lived to end of study time
• 2 lost to follow-up

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From   Meyer Transplantation, Volume
69(8).April 27, 2000.1633-1637
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Figure 1. Five-year lung cancer observed survival
conditioned on already having survived up to 5
years after diagnosis by histology. Data source
SEER cases diagnosed between 1983 and 1992 and
followed through 1995. Estimates with standard
errors gt 5 were not plotted. NOS not otherwise
specified.
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. sts test drug   failure _d died
analysis time _t studytim Log-rank test for
equality of survivor functions Events
Events drug observed
expected ------------------------------- 1
19 7.25 2 6
8.20 3 6
15.56 ------------------------------- Total
31 31.00 chi2(2)
30.19 Prgtchi2 0.0000
. sts test drug, wilcoxon failure _d
died analysis time _t studytim Wilcoxon
(Breslow) test for equality of survivor
functions   Events Events
Sum of drug observed expected
ranks -------------------------------------------
- 1 19 7.25 385 2
6 8.20 -118 3
6 15.56
-267 --------------------------------------------
Total 31 31.00 0
chi2(2) 23.45 Prgtchi2
0.0000
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. sts test drug, tware failure _d
died analysis time _t studytim Tarone-Ware
test for equality of survivor functions
Events Events Sum of drug
observed expected ranks ------------
-------------------------------- 1
19 7.25 66.091042 2 6
8.20 -16.693106 3 6
15.56 -49.397935 --------------------------
------------------ Total 31
31.00 0 chi2(2)
26.71 Prgtchi2 0.0000
. sts test drug, peto failure _d died
analysis time _t studytim Peto-Peto test for
equality of survivor functions Events
Events Sum of drug observed
expected ranks --------------------------
------------------ 1 19
7.25 8.2046993 2 6
8.20 -2.2390101 3 6
15.56 -5.9656892 ------------------------------
-------------- Total 31 31.00
0 chi2(2) 25.27
Prgtchi2 0.0000
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. sts test drug, trend failure _d
died analysis time _t studytim Log-rank test
for equality of survivor functions
Events Events drug observed
expected ------------------------------- 1
19 7.25 2 6
8.20 3 6
15.56 ------------------------------- Total
31 31.00 chi2(2)
30.19 Prgtchi2 0.0000   Test for
trend of survivor functions chi2(1)
27.44 Prgtchi2 0.0000
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. stsum, by(drug)   failure _d died
analysis time _t studytim  
incidence no. of ------ Survival
time ----- drug time at risk rate
subjects 25 50
75 ---------------------------------------------
--------------------------------- 1
180 .1055556 20 4
8 12 2 209 .0287081
14 13 22 23
3 355 .0169014 14
25 33 . -------------------------
--------------------------------------------------
--- total 744 .0416667
48 8 17 33
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. sts list, by(drug) failure compare  
failure _d died analysis time _t
studytim   ------Failure
Function------ drug 1 2
3 ---------------------------------------
------ time 1 0.1000 0.0000
0.0000 5 0.4000 0.0000
0.0000 9 0.5500 0.1488
0.0714 13 0.7750 0.2552
0.1429 17 0.8875 0.3793
0.1429 21 0.8875 0.3793
0.1429 25 . 0.7931
0.3143 29 . 0.7931
0.4122 33 . .
0.5592 37 . .
0.5592 41 . .
. ---------------------------------------------
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• if we are interested in the risk that is
  attributable to the total population - we
  multiply the risk difference by the exposed
  prevalence
• if we are interested in the actual number of
  cases that are attributable - we multiply the
  risk difference by the population size.