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**Rate should be reserved for measuring incidence**.

- In general a rate is a
**change in one measure with respect to change in a 2nd.**

- Here we are pointing out that rate should be restricted to use as a term for
**incidence but not prevalence.**

- Some epidemiologists also call cumulative incidence a rate, but that is incorrect. Both are ways to measure 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**

**A person-time 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)

- The longer the time period, the more the cumulative incidence will differ from the rate.
- The higher the rate the more quickly this difference appears.

The cumulative incidence can only be constant as long as no new event occurs, but every time an event occurs, the cumulative incidence has to increase.

- Remember, cumulative incidence is among a closed population of persons for a specified time period.

**If time period is short,** incidence rate and cumulative incidence will be close.

**If rate is low,** incidence rate and cumulative incidence will be close (unless study period is long).

Since cumulative incidence is the result of applying a rate to a closed, finite number of persons, the number of persons at risk decreases over time as they are removed by experiencing the event. Thus applying the same rate over a long period of time to a closed cohort of persons will result in steadily decreasing numbers of new events and a greater and greater divergence between the rate and the cumulative incidence. For a very short time there will be little difference between the rate and the cumulative incidence.

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

**If you know the instantaneous incidence rate, you can derive the cumulative incidence/survival function or vice-versa**

There is a formal mathematical relationship between rate and cumulative incidence. It is represented by this formula for a constant rate. For rates that are changing over time, there are other mathematical formulas that express the relationship. Rates that increase or decrease can be handled by a family of mathematical distributions known as gamma distributions.

Applying the formula above, gives the values in this table for cumulative incidence for a low and a high incidence rate over 4 time periods.

NOTE: At one-year neither cumulative incidence differs much from the rate (when the rate is expressed per 100 person-years)

- The high rate rapidly gives a quite different cumulative incidence while the low rate takes longer to diverge.

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

- For a short time period, the rate (expressed per 100 person-years) and the cumulative incidence will be very close, but not identical.
- For a 1 year period mortality rate of 0.855 per 100 person-years produces a cumulative incidence = 0.851%, very slightly different from the rate.
- But if the U.S. mortality rate of 0.855 per 100 person-years applies to everyone alive at this moment and stays constant for five years, the cumulative incidence of mortality at five years would be 4.2%.
- At 10 years the cumulative incidence would be 8.2%, both quite different from the rate.
- And if the rate is high the cumulative incidence will differ significantly faster.
- A rate of 30 per 100 person-years gives a 1-year cumulative incidence of 25.9%.

**Increasing rate would cause the cumulative incidence to increase faster.**

Szklo, M., & Nieto, F. (2007). *Epidemiology: Beyond the Basics* (2nd Edition ed.). Boston: Jones and Bartlett Publishers.

Topic revision: r4 - 05 Jun 2009 - 18:41:03 - MaryB?

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