# Incidence Rate

To address the problem of competing risks, epidemiologists often resort to a different measure of disease occurrence, the *incidence rate.* This measure is similar to incidence proportion in that the numerator is the same. It is the number of cases,

A, that occur in a population. The denominator is different. Instead of dividing the number of cases by the number of people who were initially being followed, the incidence rate divides the number of cases by a measure of time. This time measure is the summation across all individuals of the time experienced by the population being followed.

One way to obtain this measure is to sum the time that each person is followed for every member of the group being followed. If a population was followed for 30 years and a given person died after 5 years of follow-up, that person would have contributed only 5 years to the sum for the group. Others might have contributed more or fewer years, up to a maximum of the full 30 years of follow-up.

For people who do not die during follow-up, there are two methods of counting the time during follow-up. These methods depend on whether the disease or event can recur. Suppose that the disease is an upper respiratory tract infection, which can occur more than once in the same person. Because the numerator of an incidence rate could contain more than one occurrence of an upper respiratory tract infection from a single person, the denominator should include all the time during which each person was at risk for getting any of these bouts of infection. In this situation, the time of follow-up for each person continues after that person recovers from an upper respiratory tract infection. On the other hand, if the event were death from leukemia, a person would be counted as a case only once. For someone who dies of leukemia, the time that would count in the denominator of an incidence rate would be the interval that begins at the start of follow-up and ends at death from leukemia. If a person can experience an event only once, the person ceases to contribute follow-up time after the event occurs.

In many situations, epidemiologists study events that can occur more than once in an individual, but they count only the first occurrence of the event. For example, researchers may count the occurrence of the first heart attack in an individual and ignore (or study separately) second or later heart attacks. If only the first occurrence of a disease is of interest, the time contribution of a person to the denominator of an incidence rate will end when the disease occurs. The unifying concept in regard to tallying the time for the denominator of an incidence rate is simple: The time that goes into the denominator corresponds to the time experienced by the people being followed during which the disease or event being studied could have occurred. For this reason, the time tallied in the denominator of an incidence rate is often referred to as the time at risk for disease. The time in the denominator of an incidence rate should include every moment during which a person being followed is at risk for an event that would get tallied in the numerator of the rate. For events that cannot recur, after a person experiences the event, he or she will have no more time at risk for the disease, and therefore the follow-up for that person ends with the disease occurrence. The same is true of a person who dies from a competing risk.

**Figure 4-2 illustrates the time at risk for five hypothetical people being followed to measure the mortality rate of leukemia. A ***mortality rate*** is an incidence**

Figure 4-2 Time at risk for leukemia death for five people.

rate in which the event being measured is death. Only the first of the five people died of leukemia during the follow-up period. This person's time at risk ended with his or her death from leukemia. The second person died in an automobile crash, after which he or she was no longer at risk for dying of leukemia. The third person was lost to follow-up early during the follow-up period. After a person is lost, even if that person dies of leukemia, the death will not be counted in the numerator of the rate because the researcher would not know about it. Therefore the time at risk to be counted as a case in the numerator of the rate ends when a person becomes lost to follow-up. The last two people were followed for the complete follow-up period. The total time tallied in the denominator of the mortality rate for leukemia for these five people corresponds to the sum of the lengths of the five line segments in Figure 4-2.

Incidence rates treat one unit of time as equivalent to another, regardless of whether these time units come from the same person or from different people. The incidence rate is the ratio of cases to the total time at risk for disease. This ratio does not have the same simple interpretability as the risk measure.

A comparison of the risk and incidence rate measures (Table 4-1) shows that, whereas the incidence proportion, or risk, can be interpreted as a probability, the incidence rate cannot. Unlike a probability, the incidence rate does not have the range of [0,1]. Instead, it can theoretically become extremely large without numeric limit. It may at first seem puzzling that a measure of disease occurrence can exceed 1; how can more than 100% of a population be affected? The answer is that the incidence rate does not measure the proportion of the population that is affected. It measures the ratio of the number of cases to the time at risk for disease. Because the denominator is measured in time units, we can always imagine that the denominator of an incidence rate could be smaller, making the rate larger. The numeric value of the incidence rate depends on what time unit is chosen.

Suppose that we measure an incidence rate in a population as 47 cases occurring in 158 months. To make it clear that the time tallied in the denominator of an incidence rate is the sum of the time contribution from various people,

**we often refer to these time values as ***person-time.*** We can express the incidence rate as**

**We could also restate this same incidence rate using person-years instead of person-months:**

These two expressions measure the same incidence rate; the only difference is the time unit chosen to express the denominator. The different time units affect the numeric values. The situation is much the same as expressing speed in different units of time or distance. For example, 60 miles/hr is the same as 88 ft/sec or 26.84 m/sec. The change in units results in a change in the numeric value.

The analogy between incidence rate and speed is helpful in understanding other aspects of incidence rate as well. One important insight is that the incidence rate, like speed, is an instantaneous concept. Imagine driving along a highway. At any instant, you and your vehicle have a certain speed. The speed can change from moment to moment. The speedometer gives you a continuous measure of the current speed. Suppose that the speed is expressed in terms of kilometers per hour. Although the time unit for the denominator is 1 hour, it does not require an hour to measure the speed of the vehicle. You can observe the speed for a given instant from the speedometer, which continuously calculates the ratio of distance to time over a recent short interval of time. Similarly, an incidence rate is a momentary rate at which cases are occurring within a group of people. Measuring an incidence rate takes a nonzero amount of time, as does measuring speed, but the concepts of speed and incidence rate can be thought of as applying at a given instant. If an incidence rate is measured, as is often the case, with person-years in the denominator, the rate nevertheless may characterize only a short interval, rather than a year. Similarly, speed expressed in kilometers per hour does not necessarily apply to an hour but perhaps to an instant. It may seem impossible to get an instantaneous measure of incidence rate, but in a situation analogous to use of the speedometer, current incidence or mortality for a sufficiently large population can be measured by counting, for example, the cases occurring in 1 day and dividing that number by the person-time at risk during that day. Time units can be measured in days or hours but may be expressed in years by dividing by the number of days or hours in a year. The unit of time in the denominator of an incidence rate is arbitrary and has no implication for the period of time over which the rate is actually measured, nor does it communicate anything about the actual time to which it applies.

Incidence rates commonly are described as annual incidence and expressed in the form of "50 cases per 100,000" This is a clumsy description of an incidence rate, equivalent to describing an instantaneous speed as an "hourly distance" Nevertheless, we can translate this phrasing to correspond with what we have

already described for incidence rates. We can express this rate as 50 cases per 100,000 person-years, or 50/100,000 yr^{-1}. The negative 1 in the exponent means inverse, implying that the denominator of the fraction is measured in units of years.

Whereas the risk measure typically transmits a clear message to epidemiologists and nonepidemiologists alike (provided that a time period for the risk is specified), the incidence rate may not. It is more difficult to conceptualize a measure of occurrence that uses the ratio of events to the total time in which the events occur. Nevertheless, under certain conditions, there is an interpretation that we can give to an incidence rate. The dimensionality of an incidence rate is that of the reciprocal of time, which is another way of saying that in an incidence rate, the only units involved are time units, which appear in the denominator. Suppose we invert the incidence rate. Its reciprocal is measured in units of time. To what time does the reciprocal of an incidence rate correspond?

Under steady-state conditions—a situation in which the rates do not change with time—the reciprocal of the incidence rate equals the average time until an event occurs. This time is referred to as the *waiting time.* Take as an example the incidence rate described earlier, 3.57 cases per person-year. This rate can be written as 3.57 yr^{-1}; the cases in the numerator of an incidence rate do not have units. The reciprocal of this rate is 1/3.57 years = 0.28 years. This value can be interpreted as an average waiting time of 0.28 years until the occurrence of an event.

As another example, consider a mortality rate of 11 deaths per 1000 person- years, which could also be written as 11/1000 yr^{-1}. If this is the total mortality rate for an entire population, the waiting time that corresponds to it will represent the average time until death. The average time until death is also referred to as the *expectation of life* or *expected survival time.* Using the reciprocal of 11/1000 yr^{-1}, we obtain 90.9 years, which can be interpreted as the expectation of life for a population in a steady state that has a mortality rate of 11/1000 yr^{-1}. Because mortality rates typically change with time over the time scales that apply to this example, taking the reciprocal of the mortality rate for a population is not a practical method for estimating the expectation of life. Nevertheless, it is helpful to understand what kind of interpretation we may assign to an incidence rate or a mortality rate, even if the conditions that justify the interpretation are often not applicable.