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Incidence rate
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Incidence rate (synonym: incidence density rate) is expressed as number of events per person-time

In a population that we may study over a predefined period of time, not every member of the population will be at risk of developing the disease for the same amount of time during the study period. Some individuals will develop disease soon and no longer be at risk of disease, some will die, some will be lost to follow-up; some will enter the population half way through the study (birth, immigration), etc. The time contributed by each person is sometimes called “time at risk” (of an event occurring). As a consequence the population contributing to time in the follow up is also called “population at risk

## Denominator

In order to measure the incidence rate of a disease in a population we first need a denominator. The denominator is a measure of the time spent by each individual in the population at risk of developing illness during the study period. We then need to sum up all of the time at risk for each individual person to obtain a time denominator. The time in the denominator includes every instant during which an individual is at risk of developing the disease [1]. All time units in the denominator are equivalent regardless of whether they reflect the time contribution of the same person or of different persons. This way 10 people that have been observed for exactly one year will contribute the same amount of time than 20 persons that have been observed for 6 months. This is why the time at risk is frequently called person-time (e.g. person-years, person-months).

## Numerator

[1]. Thus being part of a population at risk is a dynamic process.

The incidence rate measures the occurrence of disease onsets in a population per unit of time of follow-up. Because of its similarity to population density, in an area, over time, it is sometimes called “incidence density rate”.

The figure illustrates the computation of time contribution of 10 persons of a hypothetical population to the denominator of an incidence rate.

Figure. Graphical example of occurence of disease according to time at risk of developing disease in a hypothetical population of 10 people (D*, Disease onset).

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Table. Summary table of number of years at risk and disease onset in the same hypothetical population of 10 people.

Five disease onsets occurred during a total follow up time of 65 years. This is equivalent to an incidence rate of 7.7 onsets per 100 years of follow up of individuals being at risk of developing disease.

Incidence rate: ( 5/65 persons years)= 0.07 x 100 = 7.7 per 100 persons-years

Alternatively the incidence rate can be written as follows: 7.7 x 100 years ^{-1}

It is very common to multiply the rate per units of 100, 1,000 or 10,000 in order to make comparisons among studies and interpretation easier.

An incidence rate will range from 0 to infinity according to the unit of time used to express the person-time incidence. Among 100 people no more than 100 deaths can occur. But those deaths can occur in 1000 person-years (if on average all 100 die after 10 years), 100 person-years (if on average all 100 die after 1 year) or even 1 person-year (if each of the 100 persons dies on average after 3.65 days). There is therefore no upper limit to an incidence rate. The numerical value of an incidence rate is not by itself interpretable because it depends upon the unit of time chosen. This unit should be chosen in order to make sense. For example 14 deaths per 10 person-year means that a certain number of people (at least 14) were followed for periods of times (quite short) wthe total of which equals 10 years. This rate is better expressed in months or days.

Incidence rate = 14 deaths per 10 person-years

= 14 deaths per (10 x 12) 120 person-months

= 12.3 deaths per 100 person-months

= 14 deaths per ( 10 x 365) 3650 person-days

= 38,3 deaths per 10000 person-days

In an incidence rate the only units involved are time units which appear in the denominator.

Whereas risk (incidence proportion) can be interpreted as a probability, the incidence rate cannot.

## References

1. Rothman KJ; Epidemiology: an introduction. Oxford University Press 2002, p.28-33.

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