1.5.5 measures – cumulative incidence

Measures – cumulative incidence• Different methods for calculating

– Simple cumulative– Actuarial– Kaplan-Meier– Density

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)

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

Measures – cumulative incidence• Subscript notation

– Ij – incident cases during the interval j– Wj – withdrawals during the interval j

Measures – cumulative incidenceSimple 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

Measures – cumulative incidenceSimple cumulative method:• Example: incidence of a foodborne illness if all those

potentially exposed are identified

Measures – cumulative incidenceActuarial method:

R(tj-1, tj) = CI(tj-1, tj) = IjN'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

Measures – cumulative incidence

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

• 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 incidenceActuarial Method

• Calculate interval risk• R(tj-1, tj) = Ij/(N’0j-(Wj/2))

• R(0,2)=1/(10-(1/2)) = 0.11

Measures – cumulative incidenceActuarial Method

• Calculate interval survival• next step: S(tj-1,tj) = 1-

R(tj-1,tj)

Measures – cumulative incidenceActuarial Method

• 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 incidenceActuarial Method

• Calculate cumulative survival• S(t0,tj) = 1-R(t0,tj)

Measures – cumulative incidenceActuarial Method

• 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

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

Measures – cumulative incidenceKaplan-Meier method:

IjNj

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

• 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 incidenceKaplan-Meier Method

JC: discuss withdrawals

• Calculate risk at time j• Rj = Ij/Nj

• R2=1/10 = 0.10• R4=1/8 = 0.125

Measures – cumulative incidenceKaplan-Meier Method

• 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 incidenceKaplan-Meier Method

JC: mention product-limit

Measures – cumulative incidenceDensity 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

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 incidenceDensity Method

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 incidenceDensity Method

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 incidenceDensity Method

• Cumulative survival calculated as in actuarial method

Measures – cumulative incidenceDensity Method

Measures – cumulative incidenceCumulative 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

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.

• 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

Epidemiologic measures

Szklo Exhibit 2-1