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Disease Occurrence II Main Points to be Covered. Incidence rates (person-time incidence) “Average” incidence rate Calculating “average” incidence rate Uses of incidence rates STATA commands Instantaneous incidence (hazard) rate

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Disease Occurrence II Main Points to be Covered

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Disease Occurrence IIMain Points to be Covered

  • Incidence rates (person-time incidence)

  • “Average” incidence rate

    • Calculating “average” incidence rate

    • Uses of incidence rates

    • STATA commands

  • Instantaneous incidence (hazard) rate

  • Cumulative incidence and incidence rate: different but related

  • Assumptions of survival and person-time analyses

Rate versus Risk

  • Two basic measures of the occurrence of new events (disease)

    • Cumulative incidence=Risk=Probability of event in a given time period

    • Incidence rate=Rate=events per unit time

  • Last week we discussed the concept of cumulative incidence

    • Commonly calculated by the Kaplan-Meier method when different follow-up times exist

  • Incidence rate of disease is somewhat less intuitive but is the more fundamental measure

The Three Elements in Measures of Disease Incidence

  • E = an event = a disease diagnosis or death

  • N = number of at-risk persons in the population under study

  • T = time period during which the events are observed

Measures of Incidence

  • The proportion of individuals who experience the event in a defined time period (E/N during some time T) = cumulative incidence

  • The number of events per amount of person-time observed (E/NT) = incidencerate.

    • Average incidence rate (“incidence rate”)

    • Instantaneous incidence rate (“hazard” or “hazard rate”)

“Average” Incidence Rates

  • The numerator is the same as incidence based on proportion of persons = events (E)

  • The denominator is the sum of the follow-up times for each individual

  • The resulting ratio of E/NT is not a proportion--may be greater than 1

  • Value depends on unit of time used

Incidence rate value depends on the time units used

Incidence rate of 8 cases per 100 person-years:

  • 0.67 cases per 100 person-months

  • 0.15 cases per 100 person-weeks

Assumptions of Average Incidence Rate Estimation

  • “A” time units of follow-up on “B” persons is the same as “B” time units on “A” persons

  • E.g. Observing 20 deaths in 200 persons followed for 50 years gives the same incidence rate as 20 deaths in 10,000 persons followed 1 year

  • The rate is constant for the time period during which it is calculated

    • Rates calculated over long time periods may be less meaningful

When is the rate not constant?

  • Event rate may change with follow-up time (e.g. age effect, cumulative exposure effect)

    • Example from text: risk of bronchitis for 3 smokers followed 30 years is not the same as the risk for 90 smokers followed 1 year. Cumulative effects of exposure.

  • Event rate may change with calendar time (cohort or period effect)

Survival changing over calendar time

Note on Average Incidence Rates

  • Person-time concept may seem unfamiliar because often described as “annual rate” or “annual rate per 100,000 persons” or “per 100,000 persons” (i.e., person-time denominator is not made explicit)

  • Example: “The incidence of Pediatric Cardiomyopathy in two regions of the United States” (NEJM, 2003)

    • 467 cases of cardiomyopathy in registry of 38 centers (New England, Southwest) 1996 - 1999

    • denominator “population estimates…1990 census with an in- and out-migration algorithm” ages 1 - 18

    • “overall annual incidence of 1.13 per 100,000 children”

  • Better to make person-time explicit: “incidence among children was 1.13 per 100,000 person-years”

How to Calculate an Average Incidence Rate: Obtaining the Denominator

  • Method 1: If have exact entry, censoring, and event times for each person, can sum person-time for each person for denominator

  • Method 2: If no individual data but have the time interval and average population size, can take their product as denominator

    • Some datasets may only have the average population size at risk


Rate: 6/9.583 = 0.626 per person-year = 62.6 per 100 person-years

Method 2: Using average number of persons at risk during time interval

10 persons at baseline; 1 person at end of 2 years

(6 deaths + 3 censored before 2 years = 9 losses)

Formula: Average number of persons at risk =

N baseline + N end / 2 = 11 / 2 = 5.5

Rate = 6/5.5 over 2 years = 0.545 per person-year

or 54.5 per 100 person-years

OR: 1 person with 2 years of follow-up and 9 with “some” follow-up. Assume 1(2) + 9 (2)(1/2) = 11 person-years

Average incidence rate based on grouped vs. individual data

  • Szklo and Nieto use incidence rate when based on group data (average population at risk) and incidence density when based on individual data

  • This terminology distinction is not followed by most

  • Average population method assumes uniform occurrence of events and of censoring during the interval (like life table)

Waiting Time Property of Incidence Rates

  • Waiting time to an event is reciprocal of the incidence rate (1/rate)

    • Eg, if rate 300 per 100 person-years, reciprocal is 1

      (300/100 person-years)

      = (1/3) person-year

    • Average waiting time between events is 0.33 person-year = 4 person-months

Why Use Average Incidence Rates?

  • To calculate incidence from population-based disease registries - where the persons at risk cannot all be individually followed

(1) Calculating a rate from population-based registry of diagnoses

  • Research question: What is the incidence rate for first diagnoses of breast cancer in Marin County and how does it compare with rates from other counties?

  • Nearly all new breast cancer diagnoses are reported to the SEER cancer registry

  • How to obtain a denominator for a rate?

Large Population Incidence Rates

“Since the production of stable rates for cancers at most

individual sites requires a population of at least one million

subjects, the logistic and financial problems of attempting

to maintain a constant surveillance system [of everyone in

the population] are usually prohibitive.”

Breslow and Day, Statistical Methods in Cancer Research

Solution: Do surveillance of all the cancer diagnoses and

estimate the population denominator to get person-time at risk.

To get an incidence rate person-time denominator by the group method requires only an estimate of the average population size during the year (=the population at mid-year).

Average Population (Group data) rates versus individual data rates

  • If losses are perfectly uniform, total person-time calculation for the denominator (and thus the rate) is the same whether based on average population size or individual follow-up

  • For large populations the rate will be nearly identical calculated by either method

Potential Weakness of Using Census Data

  • Calculating rates from census population data is very useful but caution is required as a full census is only done every 10 years

  • Interim estimates of population change are made by the Census but over 10 years denominators may become inaccurate

Invasive Breast Cancer Incidence Rates for Marin County versus Other California, 1995-2000

Rates per 100,000 person-years

*Excluding 5 Bay Area Counties

Census Denominators for Incidence

Rates are Estimates

The estimates of breast cancer incidence (number of new cancers per year) most recently reported for Marin and other areas of the country were based on 1990 census information. Data from Census 2000 have enabled researchers to recalculate rates for Marin. Preliminary results show that revised incidence rates for Marin County based on the 2000 census are substantially lower than the rates calculated using 1990 census information. The discrepancy between using the 1990 and 2000 census data is due to projected population growth differing considerably from actual population growth.

Why Use Average Incidence Rates?

  • To calculate incidence from population-based disease registries

  • To compare disease incidence in a cohort (individual-level data) with rate from the general population OR to compare incidences between 2 or more general populations

(2) Comparing a rate from a cohort to the rate in the general population

  • A cohort study of petroleum refinery workers followed up subjects for mortality for 36 years and found 765 deaths.

  • Research question: Was the cohort mortality incidence high, low, or just average for those calendar years?

  • How would you calculate the mortality incidence in the cohort?

Example of Using Incidence Rates for Cohort Comparisons

  • Cohort of petrochemical workers

    • 6,588 white male employees of Texas plant

    • Mortality determined from 1941-1977

    • 137,745 person-years of follow-up time

    • 765 deaths

  • Overall death rate = 765 / 137,745 person-years =

    5.6 per 1000 person-years

  • Question: Is this a high death rate?

    Austin SG, et al., J Occupat Med, 1983

Cohort of petrochemical workers

  • Could calculate KM estimate of cumulative incidence (for 36 years of follow-up), but what is the comparison group?

  • Using the incidence rate, the observed rate can be compared to the rate that would be expected if the rate from a reference population (eg, U.S. population) is applied to the cohort

Standardized Mortality Ratio

  • If U.S. death rates for age-sex-race-calendar period groups applied to the cohort, 924 deaths were expected in the cohort versus the 765 observed.

  • Ratio of 765 observed/924 expected = 0.83. This is called a Standardized Mortality Ratio (SMR).

Obtaining an expected rate for comparison

Cause Specific SMR’s

Austin SG, et al., J Occupat Med, 1983

Example of using both cumulative incidence and incidence rates in the same analysis for different purposes

End stage renal disease:

Cumulative incidence

(survival) within cohorts

defined by age at diagnosis

Ratios of mortality

incidence rates in

renal disease children

compared with national

child mortality rates

McDonald et al., NEJM 2004

Another example of SMR: Is mortality higher after a fracture?

Bluic et al. JAMA 2009

(2b) Comparing hip fracture incidence in different populations

Per 100,000 person-years

e Standardized to 1990 non-Hispanic white US population

Why Use Average Incidence Rates?

  • To calculate incidence from population-based disease registries

  • To compare disease incidence in a cohort with a rate from the general population OR to compare incidence in 2 or more populations

  • To compare incidence from a time-varying exposure in persons while exposed and unexposed

(3) To compare incidence from a time-varying exposure in persons while exposed and unexposed

  • Research question: In a Medicaid database is there an association between use of non-aspirin non-steroidal anti-inflammatory drugs (NSAID) and coronary artery disease (CAD)?

  • How would you study the relationship between NSAID use and CAD?

Calculating stratified average incidence rates in cohorts

  • For persons followed in a cohort some potential risk factors may be fixed but some may be variable

    • gender is fixed

    • taking medications or getting regular exercise are behaviors that can change over time

  • Adding up person-time in an exposure category to get a denominator of time at risk is a way to deal with risk factors that change over time

Analysis of changing exposure and disease incidence

  • Tennessee Medicaid data base, 1987-1998: are NSAIDs associated with CAD risk?

  • Same person could both use and not use NSAIDs at different times over the 11 years

  • Can’t do cumulative incidence because would have to define groups by baseline characteristics without accounting for changes in subsequent behavior

Ray, Lancet, 2002

Analysis of changing exposure with average incidence rates

  • Person-time totaled for using and not using NSAIDs; MI or CAD death outcome

  • 181,441 periods of “new” NSAIDS use in 128,002 individuals; 181,441 periods of non-use in 134,642 individuals (matched by age, sex, and calendar date)

  • A person can contribute to the denominator both for use and non-use but only after a 365 day “wash out” period between use and non-use

Analysis of changing exposure with average incidence rates

  • Rate ratio = 1.01

  • Concluded no evidence that NSAIDS reduced risk of CHD events

Ray, Lancet, 2002

Calculating Rates in STATA

Declare data set survival data:

. stset timevar, fail(failvar)

.strate gives person-years rate

.strate groupvar gives rates within groups

Example: Biliary cirrhosis time to death data

.use biliary cirrhosis data, clear

.stset time, fail(d)


D Y Rate Lower Upper

96 747.04 0.1285 0.1052 0.1570

.strate treat

Treat D Y Rate Lower Upper

Placebo 49 355.0 0.138 0.104 0.183

Active 47 392.0 0.120 0.090 0.160

Immediate Commands in STATA

  • STATA has an option to use it like a calculator for

  • various computations without using a data set.

  • Called immediate commands.

  • Example, to calculate the confidence interval

  • around a person-time rate:

  • . cii #person-time units #events, poisson

  • E.g. 6 events occur in 10 person-years of follow-up:

    • . cii 10 6, poisson

    • 95% CI = 0.220 – 1.306

Instantaneous Incidence Rate

  • So far, we have considered the “average” incidence rate for an interval

  • The hazard function h( t) gives the instantaneous potential per unit time for the event to occur, given that the individual has survived up to time t.

Hazard Function

Numerator is a conditional probability:

Denominator is time

Instantaneous probability of failure (event)

Properties of Hazard Function

Hazard function for mortality in general population


Hazard Function in STATA

Results shown previously for calculating average incidence rate in STATA:

Declare data set survival data:

. stset timevar, fail(failvar)

.strate gives person-years rate

Example: Biliary cirrhosis time to death data

.use biliary cirrhosis data, clear

.stset time, fail(d)


D Y Rate Lower Upper

96 747.04 0.1285 0.1052 0.1570

Average incidence rate = 0.1285 deaths per person-year

Hazard function in Stata

  • sts graph, hazard

K-M survival curve for same data

Average incidence rate = 0.13 deaths per person-year

10 yr cum incidence = 0.2375

More information in the plot

Difference between an Incidence Rate and Cumulative Incidence

  • Rate can be thought of as how likely an event is to happen at any moment in time

  • Cumulative incidence is the result of applying that rate to a defined population for a specified period of time

  • Average incidence rate is calculated by using data from a time period, but the rate is assumed constant during that period (i.e., at any moment in time during the period the rate is the same)

Illustration of Incidence Rate versus Cumulative Incidence

  • The mortality rate in the U.S. population in 2001 was 855 per 100,000 person-years (or 0.855 per 100 person-years)

  • If everyone alive at the beginning of the period were followed for 5 years, the cumulative incidence of death (if the rate held constant) would be 4.2% at 5 years; at 10 years it would be 8.2%.

Relationship between Incidence Rate and Cumulative Incidence

  • A constant rate produces an exponential cumulative incidence (or survival) distribution

  • If know the constant incidence rate, can derive the cumulative incidence/survival function or vice-versa

    where F(t) = cumulative incidence and

    1 - F(t) = cumulative survival;

    e= 2.71828;  = rate; t = time units

Constant Rate

Incidence Rate

Cumulative incidence

Effect of high and low constant incidence rates on cumulative incidence

Constant Hazard Rate

Survival and Hazard Functions

Incidence rate

Cumulative incidence

Summary Points

  • Incidence rate (or density)

    • E/NT

    • Not a proportion, time in denominator

  • Average incidence rate can be calculated with individual or average population data

    • Allows incidence estimates in large populations that are not completely enumerated

    • Allows comparison with population reference rates from other data sources

    • Allows accumulation of time at risk for different exposure strata

  • Instantaneous incidence (hazard) rate

    • Hazard function – insight into changes in rate during follow-up

    • Basis for proportional hazards models

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