12/20091 EPI 5240: Introduction to Epidemiology Incidence and survival December 7, 2009 Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

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4 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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6 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

12/20097 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 1 st exposure) Recruitment long after 1 st exposure –Immortal person time –Lack of info on early events.

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10 Survival Curves (1) Primary outcome is ‘time to event’ Also need to know ‘type of event’ PersonTypeTime 1Death100 2Alive200 3Lost150 4death65 And so on

12/ 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

12/ Histogram of death time -Skewed to right -pdf or f(t) -CDF or F(t) -Area under pdf from ‘0’ to ‘t’ t

12/ 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

12/ 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

12/ Survival Curves (5) Conditional Probability h(t 0 ) = rate of failing at ‘t 0 ’ conditional on surviving to t 0 Requires the ‘conditional survival curve’ S(t|survive to t 0 ) = 1 if t ≤ t 0 = P(survival ≥ t | survive to t 0 ) Essentially, you are re-scaling S(t) so that S * (t 0 ) = 1.0

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12/ S * (t) = survival curve conditional on surviving to ‘t 0 ‘ CI * (t) = failure/death/cumulative incidence at ‘t’ conditional on surviving to ‘t 0 ‘ = 1 - S * (t) Hazard at ‘t 0 ‘is defined as: ‘the slope of CI * (t) at t 0 Hazard (instantaneous) Force of Mortality Incidence rate Incidence density Range: 0  ∞

12/ Hazard curves (1)

12/ Hazard curves (2)

12/ Hazard curves (3)

12/ 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

12/ 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

12/ 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 What is CI over 3 years? Standard Epi formula: = 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 and  Don’t die in year 2 and  Don’t die in year 3 Basis for alternate analysis method

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12/ AND SO ON

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12/ ,000 6,750 2,025 6,

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12/ 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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12/ Computes ID for each year of follow-up in each year  ID is constant at ,000 7,652 2,296 5,026 25,

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12/ 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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