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EPI 5240: Introduction to Epidemiology Incidence and survival December 7, 2009

EPI 5240: Introduction to Epidemiology Incidence and survival December 7, 2009. Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa. Survival curve (1). Previous graph has a problem What if some people were lost to follow-up?

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EPI 5240: Introduction to Epidemiology Incidence and survival December 7, 2009

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  1. EPI 5240:Introduction to EpidemiologyIncidence and survivalDecember 7, 2009 Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

  2. 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.

  3. 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

  4. 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.

  5. Survival Curves (1) • Primary outcome is ‘time to event’ • Also need to know ‘type of event’

  6. 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

  7. Histogram of death time • Skewed to right • pdf or f(t) • CDF or F(t) • Area under pdf from ‘0’ to ‘t’ t

  8. 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

  9. ‘Rate’ of dying • Consider these 2 survival curves • Which has the better survival profile? • Both have S(3) = 0

  10. 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

  11. 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

  12. 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 Mortality Incidence rate Incidence density Range: 0 ∞

  13. Hazard curves (1)

  14. Hazard curves (2)

  15. Hazard curves (3)

  16. 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

  17. 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 

  18. 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

  19. What is CI over 3 years? 100+90+81 Standard 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 and • Don’t die in year 2 and • Don’t die in year 3 Basis for alternate analysis method

  20. AND SO ON

  21. 1990-1 10,000 6,750 2,025 6,625 0.306 0.694 0.694 0.306

  22. 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

  23. 1990-4 10,000 7,652 2,296 5,026 25,130 0.091 0.366 Computes ID for each year of follow-up - 0.22 in each year  ID is constant at 0.22

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