Incidence Rate versus Incidence Density: My Attempt to Dispel (some of) the Confusion Bruce Weaver, Northern Ontario School of Medicine Email: [email protected]

Introduction Medical students learning basic epidemiology often struggle to understand some of the key terms. They are not helped by textbook authors who may appear to offer contradictory definitions of terms such as incidence rate. For example, Greenberg et al. (2005, p. 21) use the following formula to compute what they call the incidence rate (IR).

IR 

A PT

(1)

In equation 1, A = the number of new cases of disease in some population, and PT = person-time at risk for developing the disease. When the denominator is person-time, as in equation 1, some authors (e.g., Shah, 2003, p. 42) use the term incidence density rather than incidence rate. Shah’s formula for what he calls the incidence rate is as follows (see Shah’s Table 3.3, p. 78):

Incidence rate 

number of new cases of disease in a time interval  1000 population at risk

(2)

The denominator for equation 2 is “the number of people in the defined population who do not have the disease or event at the beginning of the follow-up period” (Shah, 2003, p. 42). So what is the medical student (or other student of epidemiology) to make of this? The books by Greenberg et al. and Shah are both resources recommended by the Medical Council of Canada (MCC – see http://mcc.ca/pdf/ListofSuggestedReferenceBooks_e.pdf), and yet, they appear to disagree on something as basic as the definition of incidence rate. In an effort to dispel this confusion over terminology, let us consult a more complete and authoritative source, specifically, A Dictionary of Epidemiology (5th Ed.), edited by Miguel Porta.1

Annotated Definitions from the Dictionary of Epidemiology This section contains key definitions from Porta’s dictionary of epidemiology. The definitions are not listed alphabetically; rather they are ordered in terms of conceptual complexity. Porta’s 1

This is dictionary sponsored by the International Epidemiological Association. The first four editions were edited by John Last.

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definitions appear with light blue shading in the background. My comments are italicized, or enclosed in square brackets if they appear within one of Porta’s definitions. RATE A measure of the frequency of occurrence of a phenomenon. In epidemiology, demography, and vital statistics, a rate is an expression of the frequency with which an event occurs in a defined population, usually in a specified period of time. Physical units other than time may be used for constructing rates, however; for example, in accident epidemiology, deaths per passenger-mile is a more meaningful way of comparing modes of transportation. The use of rates rather than raw numbers is essential for comparison of experience between populations at different times, different places, or among different classes of persons. The components of a rate are the numerator, the denominator, the specified time in which events occur, and usually a multiplier, a power of 10, that converts the rate from an awkward fraction or decimal to a whole number. In vital statistics,

Rate 

number of events in specified period  10n average population during this period

In epidemiology, the denominator is usually person-time. [emphasis added] All rates are ratios, calculated by dividing the numerator (e.g., the number of deaths or newly occurring cases of a disease in a given period) by a denominator (e.g., the average population during that period). Some rates are proportions, where the numerator is contained within the denominator. Rate has several different usages in epidemiology: 1. As a wrong synonym for ratio, it refers to proportions as rates, as in the terms cumulative incidence rate or survival rate. Proportion and ratio are not synonyms for rate. 2. In other situations, rate refers only to ratios representing relative changes (actual or potential) in two quantities. This accords with the Oxford English Dictionary, which gives “relative amount of variation” among its definitions for rate. 3. Sometimes rate is further restricted to refer only to ratios representing changes over time. In this sense, the term prevalence rate is to be avoided, because PREVALENCE cannot (and does not need to) be expressed as a change in time; of course, different prevalence estimates may vary, change, and be compared. In contrast, the force of mortality and force of morbidity (hazard rate) are proper rates, for they can be expressed as the number of cases developing per unit time divided by the total size of the population at risk. This definition illustrates quite clearly why there is so much confusion over rates. As it says, “Rate has several usages in epidemiology”, and some of them are considered more appropriate than others. For purists, the term rate denotes per unit of time (or person-time), or per some other physical measure such as passenger-miles. INCIDENCE The number of instances of illness commencing, or of persons falling ill, during a given period in a specified population. More generally, the number of new health-related events

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in a defined population, within a specified period of time. It may be measured as a frequency count, a rate or a proportion. The 4th edition of the dictionary listed incident number as a synonym for incidence, and stated that strictly speaking, incidence (or incident number) is a simple count. INCIDENCE PROPORTION (Syn: cumulative incidence) Incidence expressed as a proportion of the population at risk. A measure of risk. The time duration must be specified for it to be meaningful. It approximates to the INCIDENCE RATE multiplied by the time duration when that product is small. The proportion of a CLOSED POPULATION at risk for a disease that develops the disease during a specified interval. CUMULATIVE INCIDENCE, CUMULATIVE INCIDENCE RATE (Syn: incidence proportion, average risk) The number or proportion of a group (cohort) of people who experience the onset of a health-related event during a specified time interval; this interval is generally the same for all members of the group, but, as in lifetime incidence, it may vary from person to person without reference to age. I dislike the terms “cumulative incidence” and “cumulative incidence rate” for two reasons: 1) One must rely on the context to know whether it refers to a count or a proportion; and 2) it is either a count or a proportion, not a true rate. For it to become a true rate, it must have “per some period of time” appended to it. CLOSED POPULATION A population that gains no new members and loses members only to death. Compare CLOSED COHORT. [Contrast DYNAMIC POPULATION.] CLOSED COHORT A population in which membership begins at a defined time or with a defined event and ends only through occurrence of the study outcome or the end of eligibility for membership. An example is a population of women in labor being studied to determine the vital status of their offspring (i.e., whether live or stillborn). DYNAMIC POPULATION A population that gains and loses members; all natural populations are dynamic—a fact recognized by the term population dynamics, which is used by demographers to denote changing composition. See also POPULATION DYNAMICS; STABLE POPULATION; FIXED COHORT. INCIDENCE RATE The rate at which new events occur in a population. The numerator is the number of new events that occur in a defined period or other physical span. The denominator is the population at risk of experiencing the event during this period, sometimes expressed as person-time; it may instead be in other units, such as passenger-miles. The incidence rate most frequently used in public health practice is calculated from the formula

Number of new events in specified period  10n Average number of persons exposed to risk during this period

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Strictly speaking, this ratio is neither a rate nor a proportion but is instead the rate multiplied by the length of the specified period. If the period is a year, the ratio is nonetheless often called the annual incidence rate. The average size of the population is often the estimated population size at midperiod. The ratio divided by the length of the period is an estimate of the person-time incidence rate (i.e., the rate per 10n person-years). If the ratio is small, as with many chronic diseases, it is also a good estimate of the cumulative incidence rate over the period (e.g., a year). If the number of new cases during a specified period is divided by the sum of the persontime units at risk for all persons during the period, the result is the person-time incidence rate. [Emphasis added] One of the key points here is that true rates include an element of time. For example, an annual rate might be described as X events per 10n per year. The definition in the 4th edition of the dictionary (edited by John Last) indicated that the incidence rate “most frequently used in public health practice” (computed via the formula shown above) estimates the person-time incidence rate. This implies that the person-time incident rate is preferred when it can be computed (i.e., when the time at risk for each individual is known, and person-time can be computed). Note too that when the denominator is person-time, Porta uses the term person-time incidence rate rather than incidence density. However, if one follows the trail of cross-references between various definitions, it appears that Porta also views incidence density as an acceptable label (see below). INCIDENCE DENSITY The average person-time incidence rate; sometimes used to describe the hazard rate. See FORCE OF MORBIDITY; INCIDENCE RATE. FORCE OF MORBIDITY (Syn: hazard rate, instantaneous incidence rate, person-time incidence rate) A theoretical measure of the number of new cases that occur per unit of population-time (e.g., person-years at risk). This is a measure of the occurrence of disease at a point in time, t, defined mathematically as the limit, as t approaches zero, of

Probability that a person well at time t will develop the disease in the interval t  t t The average value of this quantity over the interval t to (t  t ) can be estimated as Incident cases observed from t to (t  t ) Number of person-time units of experience observed from t to (t  t)

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In his definition of INCIDENCE RATE, Porta said, “If the number of new cases during a specified period is divided by the sum of the person-time units at risk for all persons during the period, the result is the person-time incidence rate.” This describes exactly the second formula in the definition for FORCE OF MORBIDITY. So the person-time incidence rate for a specified interval is the same thing as the average force of morbidity for that same interval. HAZARD RATE (Syn: force of morbidity, instantaneous incidence rate) A theoretical measure of the risk of occurrence of an event, e.g., death or new disease, at a point in time, t, defined mathematically as the limit, as t approaches zero, of the probability than in individual well at time t will experience the event by t  t , divided by t . The 5th edition of the dictionary does not include a separate entry for HAZARD RATE. The definition shown here is from the 4th Edition.

Summary Incidence (or incidence number) is the number of new cases (or incident cases) of some disease or event in a specified time period. Strictly speaking, it is a simple count. Incidence proportion is incidence expressed as a proportion of the population at risk for the event of interest. (You may also see it reported as a percentage, or as X per 1000 or X per 100,000.) As Miguel Porta notes, the associated time duration must be included for it to be meaningful. (It is also referred to as cumulative incidence or cumulative incidence rate, although I do not like these terms for reasons outlined above.) Incidence rate is a general term for the rate at which events are happening. There are various methods for estimating the incidence rate. Which method one uses to estimate the incidence rate may vary by discipline (e.g., epidemiology versus demography or vital statistics), and may also be dictated to some degree by the nature of the available information. For example: 

In a cohort study where one has precise information about the length of time each individual was at risk, the person-time incidence rate (i.e., the number of new cases in the entire period of observation divided by the total person-time) would probably be the preferred measure. It gives the number of new events per unit of person-time at risk (e.g., X events per person-year at risk).



If one lacks detailed information about length of observation for each individual, another method must be used. For a closed population, one would likely divide the number of new cases by the population initially at risk, and then multiply by 10n. (The dictionary describes this as “the incidence rate most frequently used in public health practice”, and says it can be treated as an estimate of the person-time incidence rate.) For a dynamic population, the denominator is an estimate of the average population for the period rather than the number initially at risk. The estimated size of the population at the midpoint of the interval is often used. This form of incidence rate is reported as X per 10n per time period (e.g., X events per 1000 per year for an annual incidence rate).

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I hope the reader finds these comments helpful in resolving the apparent disagreement between Greenberg et al. (2005) and Shah (2003) concerning the definition of incidence rate.

References Greenberg RS, Daniels SR, Flanders WD, Eley JW, Boring III JR. (2005). Medical Epidemiology (4th Ed.). New York: McGraw-Hill (http://www.accessmedicine.com/resourceTOC.aspx?resourceID=12). Last, JM., Editor. (2001). A Dictionary of Epidemiology (4th Ed.). New York: Oxford University Press. Porta, M., Editor. (2008). A Dictionary of Epidemiology (5th Ed.). New York: Oxford University Press. Shah CP. (2003). Public Health and Preventive Medicine in Canada (5th Ed.). Elsevier, Canada.

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