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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?

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The Issue (1)

• Epidemiological analysis 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)• an exponential model

• Then, we have:

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Area under S(t) from 0 to ‘t’

Graph of S(t)

Why does ID relate to person-time?

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• So, how can we figure out the area under S(t)?

• Let’s look at the next slide

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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’.

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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.

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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

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What can we do different

• 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 (2)4. Compute:

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5. Then, we have:

Density method (3)

• Very similar to the methods based on H(t).

• When h(t) is piecewise constant, we have:

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