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1.5.4 measures incidence+incidence density
1. Measures - incidence
Incidence 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
3. Measures - incidence
Incidence 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
4. Measures - incidence
Incidence 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
5. Measures – incidence density
Incidence 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
6. Measures – incidence density
Incidence 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
7. Measures – incidence density
“Figure 2 suggests
that ID may be
viewed as the
concentration or
'density' of new case
occurrences
in a sea of
population time. The
more dots per unit
area under the
curve, the greater is
the ID.”
Morgenstern et al. 1980
8. Measures – incidence density
Person-time calculations for individual level data
1) If exact time contribution of each individual is known:
– Sum the disease-free observation time
9. Measures – incidence density
Person-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
10. Measures – incidence density
Person-time estimation from group level data
1) 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
11. Measures – incidence density
Uses 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)
12. Measures – incidence density
Uses 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
13. 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?
14. Measures – incidence
Hazard 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)
17. 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
18. Measures – cumulative incidence
Cumulative 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
19. Measures – cumulative incidence
Cumulative 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
20. Measures – cumulative incidence
• Different methods for calculating
– Simple cumulative
– Actuarial
– Kaplan-Meier
– Density
21. 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)
22. 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
23. Measures – cumulative incidence
• Subscript notation
– Ij – incident cases during the interval j
– Wj – withdrawals during the interval j
24. 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
25. Measures – cumulative incidence
Simple cumulative method:
• Example: incidence of a foodborne illness if all those
potentially exposed are identified
26. 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
27. Measures – cumulative incidence
Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)
28. • 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
33. • 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
34. 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
