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101/2014
EPI 5344:Survival Analysis in
EpidemiologyEpi Methods: why does ID involve person-time?
March 13, 2014
Dr. N. Birkett,Department of Epidemiology & Community
Medicine,University of Ottawa
The Issue (1)
• Epidemiology focuses on:– Incidence Proportion or Cumulative Incidence (CI)– Incidence Density or Incidence Rate (ID).
• Standard formulae are:
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The Issue (2)
• How do these measures relate to survival analysis?
• Why does ID involve person-time?
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Incidence Density (rate)
• Rate of getting disease.– A number with units (time-1)– Ranges from 0 ∞
• Often measured from time ‘0’ (recruitment)• Can be measured for any time interval
– Separate ID’s for each year of follow-up• If the time units get smaller, we approach
the ‘instantaneous ID’
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Incidence Density (rate)
• Rate of getting disease (outcome) at time ‘t’ given (conditional on) on having survived to time ‘t’
• Instantaneous ID is the same as the hazard
• Average ID is more common in epidemiology
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• Epidemiology formulae ignore ID variability over time and compute average ID (ID`)
• Actuarial method (density method) lets each interval have a different ID• Linked to piecewise exponential model
Why does ID relate to person-time?
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• Let’s look at a simple situation (assumption):• No losses (i.e. no censoring)• A constant ID over time (I)
• Then, we have:
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Area under S(t) from 0 to 1
Actually a curve but we assume it’s a straight line
Graph of S(t)
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In general, area under S(t) from ‘0’ to ‘t’ is given by:
How does this help? In the formula we derived for ID, multiply top and bottom by ‘N’ (the initial # of people at risk)
Now, CI(t) * N = # new cases by time ‘t’.
• Person-time approach to ID assumes that ID (hazard) is constant– Can be seen as estimating an average ID
• BUT, constant hazard gives the exponential survival model which does not reflect real-world S(t)’s.
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• Why does epidemiology ignore this and use a constant ID?– Lack of data– Lack of measurement precision– Tradition– ”teaching”
• Old fashioned methods or learning by rote
• What can we do?– Piece-wise constant hazard approach is
better– Density methods– Survival methods
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Density method (1)GOAL: to estimate CI for outcome by year ‘t*’
1. Select a time interval (usually 1 year)
2. Divide follow-up time into intervals of this size
3. Within each interval, compute the ID of surviving the
interval given you are disease-free at start:
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Density method (3)
• Very similar to the methods based on H(t).
• When h(t) is piecewise constant, we have:
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