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EPI 5240:Introduction to Epidemiology

Incidence and survivalDecember 7, 2009

Dr. N. Birkett,Department of Epidemiology & Community

Medicine,University of Ottawa

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Survival curve (1)

• Previous graph has a problem– What if some people were lost to follow-up?– Plotting the number of people still alive would

effectively say that the lost people had all died.

• Instead– True survival curve plots the probability of

surviving.

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Time ‘0’ (1)

• Survival (or incidence) measures time of events from a starting point– Time ‘0’

• No best time ‘0’ for all situations– Depends on study objectives and design

• RCT of Rx– ‘0’ = date of randomization

• Prognostic study– ‘0’ = date of disease onset– Inception cohort– Often use: date of disease diagnosis

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Time ‘0’ (2)

• Effect of ‘point source’ exposure– ‘0’ = Date of event– Hiroshima atomic bomb– Dioxin spill, Seveso, Italy

• Chronic exposure– ‘0’ = date of study entry OR Date of first exposure– Issues

• There often is no first exposure (or no clear data of 1st exposure)

• Recruitment long after 1st exposure– Immortal person time– Lack of info on early events.

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Survival Curves (1)

• Primary outcome is ‘time to event’• Also need to know ‘type of event’

Person Type Time

1 Death 100

2 Alive 200

3 Lost 150

4 death 65

And so on

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Survival Curves (2)

• People who do not have the targeted outcome (death), are called ‘censored’

• For now, assume no censoring

• How do we represent the ‘time’ data.– Histogram of death times - f(t)– Survival curve - S(t)– Hazard curve - h(t)

• To know one is to know them all

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t

dxxftF0

)()(

Histogram of death time-Skewed to right-pdf or f(t)-CDF or F(t)

-Area under pdf from ‘0’ to ‘t’

t

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Survival curves (3)

• Plot % of group still alive (or % dead)

S(t) = survival curve

= % still surviving at time ‘t’

= P(survive to time ‘t’)

Mortality rate = 1 – S(t)

= F(t)

= Cumulative incidence

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‘Rate’ of dying

• Consider these 2 survival curves

• Which has the better survival profile?– Both have S(3) = 0

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Survival curves (4)

• ‘A’ is better.– Death rate is lower in first two years.– Will live longer than in pop ‘B’

• Concept is called:– Hazard: Survival analysis/stats– Force of mortality: demography– Incidence rate/density: Epidemiology

• DEFINITION– h(t) = rate of dying at time ‘t’ GIVEN that you have

survived to time ‘t’• Slight detour and then back to main theme

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Survival Curves (5)Conditional Probability

h(t0) = rate of failing at ‘t0’ conditional on surviving to t0

Requires the ‘conditional survival curve’S(t|survive to t0) = 1 if t ≤ t0

= P(survival ≥ t | survive to t0)

Essentially, you are re-scaling S(t) so that S*(t0) = 1.0

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S*(t) = survival curve conditional on surviving to ‘t0‘

CI*(t) = failure/death/cumulative incidence at ‘t’ conditional on surviving to ‘t0‘

= 1 - S*(t)

Hazard at ‘t0‘is defined as: ‘the slope of CI*(t) at t0

Hazard (instantaneous)Force of MortalityIncidence rateIncidence density

Range: 0 ∞

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Hazard curves (1)

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Hazard curves (2)

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Hazard curves (3)

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Some relationships

h(t) = ‘instantaneous’ incidence density at ‘t’Cumulative hazard = H(t) = = area under h(t) (or ID(t))

from ‘0’ to ‘t’

If the rate of disease is small: CI(t) ≈ H(t)If we assume h(t) is constant (= ID): CI(t)≈ID*t

t

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)(1)( tHetCI

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The real world

• So much for theory• In real world, you can’t measure time to

infinite precision– Often only know year of event– Or, perhaps even just the event happened– Standard Epi formulae make BIG

assumptions• We can do better• More advanced statistics can use discrete survival

models• We won’t go there

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Key Concept to estimate CI

• Divide the follow-up period into smaller time units– Often, use 1 year intervals– Can be: days, months, decades, etc.

• Compute an incidence measure in each year

• Combine these into an overall measure

Year # at start # dying

0 1000 100

1 900 90

2 810 81

What is CI over 3 years? 100+90+81Standard Epi formula: ------------------- = 0.27 1000

Another view:P(die in 3 years) = 1 – P(not dying in 3 years)

How can you still be alive after 3 years?Don’t die in year 1 andDon’t die in year 2 andDon’t die in year 3

Basis for alternate analysis method

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1990-1 10,000 6,750 2,025 6,625 0.306 0.694 0.694 0.306

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• These are the main ways to estimate CI directly

• If cohort is not ‘fixed’, require assumptions about losses.

• Dynamic population– Don’t know who is in cohort– Can not know who dropped out and when.– Can not know who joined population and

when– These formulae don’t work well

• Instead– Estimate ID and the convert to CI

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Computes ID for each year of follow-up- 0.22 in each year ID is constant at 0.22

1990-4 10,000 7,652 2,296 5,026 25,130 0.091 0.366

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• These methods can be applied with dynamic populations– Only need to estimate PY’s

• If rate is small (<0.01), then – CI(1 interval) ≈ ID

EXAMPLE• Apply these methods to the example I’ve been

working with.• Target is to estimate: Prob(dying in 5 years)

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