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## Slide 1: Rate versus Risk

• Two basic measures of the occurrence of new events (disease)
• Cumulative incidence=Risk=Probability
• IncidenceMove rate=Rate=events per time units
• The concept of cumulative incidence: Commonly calculated by the Kaplan-Meier method when different follow-up times exist
• CTSpedia.IncidenceMove rate of disease is somewhat less intuitive but is the more fundamental measure

Understanding the difference between these two measures is a central concept in this part of the course. Both are important measures in clinical research that are widely used. Cumulative incidence may be more familiar because of the frequent use of Kaplan-Meier analysis in clinical studies with individual follow-up data, but incidence rates are useful in a variety of other situations where you are comparing two or more populations where at least one population does not have individual-level data or where exposures are changing within subjects over time.

## Slide 2: The Three Elements in Measures of Disease IncidenceMove

• 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

## Slide 3: Two Measures of IncidenceMove

• 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 divided by the amount of person-time observed (E/NT) = incidence rate

Cumulative incidence is a proportion because it has the number of events (E) as the numerator and has as the denominator the number of persons (N) at risk. Because of censoring, we rarely can just plug an N into the denominator. Instead, via Kaplan-Meier or life table techniques, we derive an effective N by taking into account each person’s follow-up until it is censored. In any case, the proportion can be interpreted as the percentage of N persons who develop disease in a given time period. An incidence rate also has the number of events (E) as the numerator but its denominator is different: it is the product of the number of persons x the number of time units they were observed (NT). Both measures have to account for time, but cumulative incidence accounts for it by specifying the time period during which the persons were observed. For example, 40% of subjects experienced the outcome during 3 years of follow-up. IncidenceMove rate accounts for time by making it an element of the denominator.

## Slide 4: Person-Time IncidenceMove 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

If the incidence rate of the event was 35 per 100 person-years, remember that 100 person-years could mean 100 persons followed for 1 year or 50 persons followed for 2 years or 200 persons followed for 6 months or many other combinations of persons and years whose product equals 100. All 3 give 100 person years of time at risk. And in this example the time units are years but years could be replaced by days, months, decades, or any time unit and the rate would change accordingly. So the absolute value of the rate is determined by the units used in the denominator.

## Slide 5: IncidenceMove rate value depends on the time units used

IncidenceMove rate of 8 cases per 100 person-years:
• 0.67 cases per 100 person-months
• 0.15 cases per 100 person-weeks

Note: time period during which rate is measured can differ from the units used (use data from 2 years of follow-up but report a rate per person-months)

An illustration of how the value changes when the time units are changed even though the data remain the same and the interpretation remains the same. Person-years are the most conventionally used person-time denominator and are likely to be familiar, but other time units are possible. Rates are usually presented in units that leave at least one integer to the left of the decimal points. So 9 per 1,000 person-years would usually be preferred over 0.9 per 100 person-years.

Cumulative incidence does not have units since it is always between 0 and 1, and it therefore is not subject to this kind of alternative formulation. But, contrary to incidence rates, the time period of observation has to be specified to make cumulative incidence meaningful (e.g., 34% at two years of follow-up).

## Slide 6: Assumption of Person-Time IncidenceMove Estimation

• “A” time units of follow-up on “B” persons is the same as “B” time units on “A” persons
• Observing 2 deaths in 2 persons followed for 50 years gives the same incidence rate as 2 deaths in 100 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)

The assumption that the denominator of person-time units for a rate does not depend on the proportion of persons and amount of time in the calculation may result in a rate that is hard to interpret if it is based on a long period of follow-up. In the example on the slide, it is obvious that one cannot meaningfully compare the rate from 50 years of follow-up with the rate based on 1 year of follow-up. There is no simple rule about how much time is too long for a period in which to calculate a rate since it depends on how rapidly the outcome being measured may change over time. For something that changes very little year to year, such as the overall mortality rate in the U.S., a period of time of several years might be reasonable. But for this reason one seldom sees rates calculated over, for example, a 20-year period whereas that would not be unusual for a calculation of cumulative incidence.

## Slide 7: Understanding the Difference between a Rate and Cumulative IncidenceMove

• 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 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)

It may help in understanding the difference between an incidence rate and cumulative incidence to think of the rate as how likely an event, such as death or a disease diagnosis, is to occur at any given moment in time (a kind of instantaneous risk) and to think of the cumulative incidence as the result of applying that instantaneous rate to a given number of persons for a longer period of time. 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. So the rate can be thought of as the most fundamental measure of disease incidence, something captured rather metaphorically by older descriptions of a rate as the “force of morbidity” or the “force of mortality.”

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.

## Slide 8: Illustration of Rate versus Cumulative IncidenceMove

• 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 year 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%.

In 2001 the mortality rate was estimated at 855 per 100,000 person-years (or 0.855 per 100 person-years). Part of the confusion around rate is that it is often described with the use of the word “annual”, as the U.S. mortality rate is in “the annual U.S. mortality rate.” The use of the word “annual” suggests that the rate is per one year, but this is misleading. As described above the rate is instantaneous. Annual comes in because we calculate the rate for 2001 by using data from an entire year, but the estimate of the rate is based on the assumption that it was constant throughout the year (a reasonable approximation to reality for something that changes as slowly as the mortality rate). The mortality rate, therefore, was the same at any moment in time in 2001. If the rate remains constant, applying that rate to a given number of persons for a given time period produces a cumulative incidence that follows an exponential distribution. 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%. Of course, in a closed cohort the mortality rate would increase over time as the cohort ages. This does not pose a problem. There are other mathematical formulas for dealing with an increasing rate. Increasing rate would cause the cumulative incidence to increase faster.

## Slide 9: Relationship between IncidenceMove Rate and Cumulative IncidenceMove

• 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

FORMULA: 1 – F(t) = e-yt
• where F(t) = cumulative incidence;
• e= 2.71828; y = rate; t = time units

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.

## Slide 10: Example IncidenceMove Rate versus Cumulative IncidenceMove

http://twiki.library.ucsf.edu/twiki/bin/viewfile/CTSpedia/TICRDisOccurII?rev=1;filename=inc_cumul.JPG

Applying the formula from the previous slide, gives these values for cumulative incidence for a low and a high incidence rate over 4 time periods. Note that at one-year neither cumulative incidence differs that much from the rate (when the rate is expressed per 100 person-years), but that the high rate rapidly gives a quite different cumulative incidence while the low rate takes longer to diverge.