01/20151 epi 5344: survival analysis in epidemiology epi methods: why does id involve person-time?...
TRANSCRIPT
101/2015
EPI 5344:Survival Analysis in
EpidemiologyEpi Methods: why does ID involve person-time?
March 10, 2015
Dr. N. Birkett,School of Epidemiology, Public Health &
Preventive Medicine,University of Ottawa
Background
• I just showed that we get the same answer from:– Exponential survival model– Person-time epidemiology estimate of ID
• Why?• What is the basis for this link?
01/2015 2
The Issue (1)
• Epidemiological analysis focuses on:– Incidence Proportion or Cumulative Incidence (CI)– Incidence Density or Incidence Rate (ID).
• Standard formulae are:
01/2015 3
The Issue (2)
• How do these measures relate to survival analysis?
• Why does ID involve person-time?
01/2015 4
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’
01/2015 5
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
01/2015 6
01/2015 7
• 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?
01/2015
• Let’s look at a simple situation (assumption):• No losses (i.e. no censoring)• A constant ID over time (I)• an exponential model
• Then, we have:
8
01/2015 9
Area under S(t) from 0 to ‘t’
Graph of S(t)
Why does ID relate to person-time?
01/2015 10
01/2015 11
• So, how can we figure out the area under S(t)?
• Let’s look at the next slide
01/2015 12
Area under S(t) from 0 to 1
Actually a curve but we assume it’s a straight line
Graph of S(t)
01/2015 13
01/2015 14
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’.
01/2015 15
This is the standard Epidemiology definition of ID
• 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.
01/2015 16
Why do we use constant ID
• 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
01/2015 17
What can we do different
• Piece-wise constant hazard approach is better
• Density methods• Survival methods
01/2015 18
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:
01/2015 19
Density method (2)4. Compute:
01/2015 20
5. Then, we have:
Density method (3)
• Very similar to the methods based on H(t).
• When h(t) is piecewise constant, we have:
01/2015 21
01/2015 22