This document discusses different methods for calculating cumulative incidence (CI) when studying disease occurrence over time, including simple CI, actuarial CI, Kaplan-Meier CI, and density CI. It provides details on the notation, formulas, and steps used for each method. The key differences between the methods relate to whether and how they account for withdrawals/censoring over time. The appropriate method depends on the type of data available and study time frame.

1. Measures – cumulative incidence
• Different methods for calculating
– Simple cumulative
– Actuarial
– Kaplan-Meier
– Density

2. Measures – cumulative incidence
• 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 – cumulative incidence
• 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 – cumulative incidence
• Subscript notation
– Ij – incident cases during the interval j
– Wj – withdrawals during the interval j

5. Measures – cumulative incidence
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 – cumulative incidence
Simple cumulative method:
• Example: incidence of a foodborne illness if all those
potentially exposed are identified

7. Measures – cumulative incidence
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 – cumulative incidence
Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)

9. • Where to start – 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 interval risk
• 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 interval survival
• next step: S(tj-1,tj) = 1-R(tj-1,tj)
Measures – cumulative incidence
Actuarial Method

12. • Calculate cumulative risk – 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 cumulative survival
• S(t0,tj) = 1-R(t0,tj)
Measures – cumulative incidence
Actuarial Method

14. • Intuition for why 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 – cumulative incidence
• 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 – cumulative incidence
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 to start – 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 risk at time j
• Rj = Ij/Nj
• R2=1/10 = 0.10
• R4=1/8 = 0.125
Measures – cumulative incidence
Kaplan-Meier Method

19. • Survival calculated as 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 – cumulative incidence
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 survival calculated as in actuarial method
Measures – cumulative incidence
Density Method

25. Measures – cumulative incidence
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 the CI 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
– Uniformity of 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. Epidemiologic measures
Szklo Exhibit 2-1