1.5.4 measures incidence+incidence density

Measures - incidenceIncidence time• Not sufficient to just record proportion of population

affected by disease• Necessary to account for the time elapsed before

disease occurs and the period of time during which the disease events take place

Measures - incidenceA

P(DA) = 0.6

B

P(DB) = 0.6

Measures - incidenceIncidence time• Incidence time is time from referent or zero time (e.g.,

birth, start of treatment or exposure, start of measurement period) until the time at which the outcome event occurs

• Also called event time, failure time, occurrence time

Measures - incidenceIncidence time• “Censoring” occurs if the time of event is not known

because something happens before the outcome occurs– Examples: lost to follow-up, death, surgery to make outcome

impossible like hysterectomy, end of measurement period

ime = average time until an event• Average incidence t occurs

Measures – incidence densityIncidence density (ID) - aka incidence rate (IR)• The rate of occurrence of new cases of disease during

person-time of observation in a population at risk of developing disease

• Numerator: number of new cases of disease– Only count cases in the numerator that are contributing to

person-time in the denominator• Denominator: person-time of observation in population

at risk– Only count contributions to the denominator that could yield

cases for the numerator• A rate• Units are “inverse time” (1/time, time-1)• Range is 0-infinity

Measures – incidence densityIncidence density• What is “person-time”?• Person-time at risk: length of time for each individual

that they are in the population at risk– Sum over population is total person time at risk

• When a person is no longer “at risk” they cease contributing to person-time, this includes when they get the outcome of interest

• One person year could be 2 people x 6 months each, 1 person x 12 months, 3 people x 4 months, etc.

• Helps account for censoring and different observation periods

Measures – incidence density“Figure 2 suggests that ID may be viewed as the concentration or 'density' of new case occurrencesin a sea of population time. The more dots per unit area under the curve, the greater is the ID.”

Morgenstern et al. 1980

Measures – incidence densityPerson-time calculations for individual level data

1) If exact time contribution of each individual is known:– Sum the disease-free observation time

Measures – incidence densityPerson-time calculations for individual level data

2) If data on each individual is collected at regular intervals:– Estimate the disease-free observation time in each

interval

– Note: variants of this formula also subtract Ij/2 from N’0j

Measures – incidence densityPerson-time estimation from group level data1) If the population is in steady state can estimate based

on population size (N’) and duration of follow-up (Δt)

2) If the population is not in steady state can estimate based on mid-interval population (N’1/2) and duration of follow-up (Δt)

– Note: mid-interval population size can be estimated as: (Nt0 + Nt1)/2

Measures – incidence densityUses and limitations of incidence density• Appropriate for fixed or dynamic populations; does not

assume that everyone is followed for specified time period

• Does not distinguish between people who do not contribute to disease incidence because they were not in the study population long enough for disease to develop and those who do not contribute because they never got the disease (relates to next point)

Measures – incidence densityUses and limitations of incidence density• 100 person-years could come from following 100 people

for one year or two people for 50 years – no way to tell the difference without knowing the incidence time– Have to consider whether study design allowed appropriate time

to elapse to plausibly consider an exposure disease relation– Disease process is important to consider in developing

appropriate study design and disease measures– Example: disease free cohort of 50 exposed and 50 unexposed

followed for 1 year might not allow sufficient time to elapse for exposure to cause disease

Measures – incidence density• In class exercise

– Study population observed monthly for 6 months– What is the person-time contributed by this

population?– What is the incidence density?

Measures – incidenceHazard rate• The instantaneous potential for change in disease

status per unit of time at time t relative to the size of the candidate (i.e., disease-free) population at time t

• Instantaneous rate in contrast to incidence density which is an average rate

• Cannot be directly calculated because it is defined for an infinitely small time interval

• Hazard function over time can be estimated using modeling techniques (more in the analyzing epidemiologic data section)

Measures – incidenceHazard rate

Measures – incidenceSurvival function

Measures of disease outline– Big picture– Illustration/discussion of measuring disease in time– Populations– Time scales affecting disease in populations– Epidemiologic measures

• Basic concepts• Measuring diseases• Prevalence• Incidence density (incidence rate)• Cumulative incidence (risk)• Relations among measures

– Standardization– Summary– Appendix: specific measures of disease

Measures – cumulative incidenceCumulative incidence (CI) – aka risk, incidence

proportion (IP – Rothman)• The proportion of a closed population at risk that

becomes diseased within a given period of time• Numerator: number of new cases of a disease or a

condition (Rothman calls this A)• Denominator: number of persons in population at risk

(Rothman calls this N)• A proportion• Range is 0-1 – dimensionless

Measures – cumulative incidenceCumulative incidence• Calculated for a fixed time period

– Only interpretable with information on time period over which it was measured

• Population measure that translates most readily to individual– Interpreted as capturing individual risk of disease

• Different methods for calculating– Variations depending on how time at risk is handled– Option for calculating from rate measure

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)


– N’0 – number at risk of disease at t0 (baseline)– N’0j – number at risk of disease at the beginning of

interval j


– 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


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


• 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


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


• 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)– Product 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

1.5.4 measures incidence+incidence density

Healthcare