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1.5.5 measures – cumulative incidence
- 1. Measures – cumulativeincidence • Different methods for calculating – Simple cumulative – Actuarial – Kaplan-Meier – Density
- 2. Measures – cumulativeincidence • Subscript notation – R(t0,tj) – risk of disease over the time interval t0 (baseline) to tj (time j) – R(tj-1,tj) – risk of disease over the time interval tj-1 (time before time j) to tj (time j)
- 3. Measures – cumulativeincidence • Subscript notation – N’0 – number at risk of disease at t0 (baseline) – N’0j – number at risk of disease at the beginning of interval j
- 4. Measures – cumulativeincidence • Subscript notation – Ij – incident cases during the interval j – Wj – withdrawals during the interval j
- 5. Measures – cumulativeincidence Simple cumulative method: R(t0,tj) = CI(t0,tj) = I N'0 • Risk calculated across entire study period assuming all study participants followed for the entire study period, or until disease onset – Assumes no death from competing causes, no withdrawals • Only appropriate for short time frame
- 6. Measures – cumulativeincidence Simple cumulative method: • Example: incidence of a foodborne illness if all those potentially exposed are identified
- 7. Measures – cumulativeincidence Actuarial method: R(tj-1, tj) = CI(tj-1, tj) = Ij N'0j - Wj/2 • Risk calculated accounting for fact that some observations will be censored or will withdraw • Assume withdrawals occur halfway through each observation period on average • Can be calculated over an entire study period – R(t0,tj) = CI(t0, tj) = I/(N’0-W/2) • Typically calculated over shorter time frames and risks accumulated
- 8. Measures – cumulativeincidence Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)
- 9. • Where tostart – set up table with time intervals • Fill incident disease cases and withdrawals into appropriate intervals • Fill in population at risk Measures – cumulative incidence Actuarial Method
- 10. • Calculate intervalrisk • R(tj-1, tj) = Ij/(N’0j-(Wj/2)) • R(0,2)=1/(10-(1/2)) = 0.11 Measures – cumulative incidence Actuarial Method
- 11. • Calculate intervalsurvival • next step: S(tj-1,tj) = 1-R(tj-1,tj) Measures – cumulative incidence Actuarial Method
- 12. • Calculate cumulativerisk – example of time 0 to 10 • R(t0, tj) = 1 - Π (1 – R(tj-1,tj)) = 1 - Π (S(tj-1,tj)) • R(0, 10) = 1 – (0.89 x 0.88 x 1.0 x 1.0 x 0.85) = 0.34 Measures – cumulative incidence Actuarial Method
- 13. • Calculate cumulativesurvival • S(t0,tj) = 1-R(t0,tj) Measures – cumulative incidence Actuarial Method
- 14. • Intuition forwhy R(t0, tj) = 1 - Π (Sj) using conditional probabilities • Example of 5 time intervals: – Π (Sj) = P(S1)*P(S2|S1)*P(S3|S2)*P(S4|S3)*P(S5|S4) = P(S5) – Multiply first two terms: P(S2|S1)*P(S1) = P(S2) – Multiplying conditional probabilities gives you unconditional probability of surviving up to any given time point – the value (1 - survival) up to (or at) a given time point is then the probability of not surviving up to that time point Measures – cumulative incidence
- 15. Measures – cumulativeincidence • Exercise for home (discuss in lab) – Study population observed monthly for 6 months – Calculate the cumulative incidence of disease from month 0 to 6
- 16. Measures – cumulativeincidence Kaplan-Meier method: Ij Nj Rj = CIj = • Risk calculated at the time each disease event occurs – Accounts for withdrawals in that Nj only includes those at risk at each time j point – Result differs from actuarial approach in that the time of a withdrawal (in Kaplan-Meier analysis) coincides with time of an incident disease • Risks at each onset time j accumulated
- 17. • Where tostart – set up table with times of incident cases • Fill in population at risk – anyone who has withdrawn by a time j is no longer at risk at that time Measures – cumulative incidence Kaplan-Meier Method JC: discuss withdrawals
- 18. • Calculate riskat time j • Rj = Ij/Nj • R2=1/10 = 0.10 • R4=1/8 = 0.125 Measures – cumulative incidence Kaplan-Meier Method
- 19. • Survival calculatedas in actuarial method • Cumulative risk calculated as in actuarial method – R(t0, tj) = 1 - Π (1 – Rj) = 1 - Π (Sj) • Cumulative survival calculated as in actuarial method Measures – cumulative incidence Kaplan-Meier Method JC: mention product-limit
- 20. Measures – cumulativeincidence Density method: R (-ID*Δt) (t0,t) = 1 – S(t0,t) = 1- e • Depends on functional relationship between a risk and a rate • Can be calculated over an entire study period if the rate is constant • Can also be calculated over shorter time frames and risks accumulated JC: Mention Elandt-Johnson article
- 21. Where to start– set up table with time intervals • Fill incident disease cases, withdrawals and population at risk by interval • Calculate person time (for example used formula PTj=(N’0j-(Wj/2)) Δtj) • Calculate IDj = Ij/PTj Measures – cumulative incidence Density Method
- 22. R(t0,t) = 1– S(t0,t) = 1- e (-ID*Δt) • Calculate interval risk • • R(0,2) = 1-e(-0.05*2) = 0.10 Measures – cumulative incidence Density Method
- 23. R(t0,t) = 1-e • Calculate cumulative risk – example of time 0 to 10 • Accumulate interval risks as in actuarial method • Or calculate cumulative risk directly • (-∑ID*Δt) • R(0,10) = 1-e(-(0.05*2+0.06*2+0*2+0*2+0.08*2) = 0.32 Measures – cumulative incidence Density Method
- 24. • Cumulative survivalcalculated as in actuarial method Measures – cumulative incidence Density Method
- 25. Measures – cumulativeincidence Cumulative incidence • Summary of methods for calculating and basis of choosing – Simple cumulative – complete follow-up – Actuarial – incomplete follow-up – Kaplan-Meier – incomplete follow-up – Density – converting incidence density to cumulative incidence
- 26. Choosing among theCI methods • Do you only have rate data? Generally you will choose incidence density. • • Do you have zero withdrawals and a short time period of interest? If so, simple CI usually OK. • • Do you have fairly exact data on time of incidence and time of withdrawal? If so, density preferable. • • Do you have fairly exact data on time of incidence but only interval data on withdrawals? If so, KM most common choice; actuarial or density may not be too different depending on withdrawal timing. • • Do you have interval data for incidence and withdrawal? If so, actuarial most common choice, KM and density may not be too different depending on withdrawal timing.
- 27. • Assumptions – Uniformityof events and losses within each interval (the W/2) – Independence between censoring and survival – otherwise biased/not accurate (also true for ID) – Lack of secular trends Measures – cumulative incidence
- 28.