## Thursday, March 03, 2011

### Incidence Rate (IR) – How could this be wrongly calculated?

I am very surprised to see how a simple concept of ‘incidence rate’ can be wrongly calculated in documents  submitted to regulatory agencies (such as FDA). In a briefing document titled “Tiotropium (SPIRIVA): Pulmonary Allergy Drug Advisory Meeting – November 2009” submitted by a sponsor, there were wrong statements every where about the calculation of the incidence rate for safety variables.

For example, on page 50, it says “Incidence rates of adverse events were computed as the number of patients experiencing an event divided by the person-years at risk”; In Section 8.1.5 (Statistical methods), it says “For each event, an incidence rate (IR) was calculated from the number of patients with an event divided by the cumulative time at risk within a treatment group and expressed as patient-years.”  In their summary tables, they footnoted “the number of patients with an event” (instead of the number of total events) was used in calculating the incidence rate. They never listed the total number of patient year (the denominator) for their Incidence rate calculation. In ‘Statistical method’ section, they even tried to justify the use of “the difference in incidence rate” because “most Tiotropium trials have significantly greater number of patients in the placebo group discontinuing the trial early compared to tiotropium treated patients.”
“Incidence Rate” is a basic concept from epidemiology studies and is calculated as the number of events divided by the number of patient years. According to free medical dictionary, “incidence rate is the probability of developing a particular disease during a given period of time; the numerator is the number of new cases during the specified time period and the denominator is the population at risk during the period. “   According to Wikipedia, “The incidence rate is the number of new cases per population in a given time period. When the denominator is the sum of the person-time of the at risk population, it is also known as the incidence density rate or person-time incidence rate. In the same example as above, the incidence rate is 14 cases per 1000 person-years, because the incidence proportion (28 per 1,000) is divided by the number of years (two). Using person-time rather than just time handles situations where the amount of observation time differs between people, or when the population at risk varies with time. Use of this measure implicitly implies the assumption that the incidence rate is constant over different periods of time, such that for an incidence rate of 14 per 1000 persons-years, 14 cases would be expected for 1000 persons observed for 1 year or 50 persons observed for 20 years.”
In an article by Marco et al “a Cohort of Young Adults According to the Presence of Chronic Cough and Phlegm”, the incidence rate is correctly defined for calculation.
“Incidence rates of COPD were estimated as the ratio of the number of new cases and the number of person-years at risk (per 1,000), which were considered equal to the length of the follow-up for each member of the cohort.”

The key is that if you calculate the ‘incidence rate’, your numerator must be ‘number of events’, not ‘number of patients with an event’. For events that can only occur once in a lifetime for a specific patient (such as cancer), there may not be much difference between ‘number of events” and “number of patients with an event”. However, for events occurr more than one time for a specific patient, “number of events” and “number of patients with an event” are very different concepts.

In Tiotropium briefing document, the correct calculation for incidence rate should be ‘number of events (AEs or COPDs)’ divided by ‘the patient year’. It was simply wrong when they used ‘number of patients with an event’ as the numerator in their calculation of incidence rate. Their justification for using the difference in incidence rate is just the opposite of their statement. If placebo group has more dropouts, their way of calculating the incidence rate will overestimate the rate for placebo group and underestimate the rate for Tiotropium group. This can be easily illustrated using an example below:

Assuming 10 patients in Tiotropium and 10 subjects in Placebo group, 5 patients in Tiotropium group and 5 patients in Placebo group had at least one COPD during the study. The incidence of COPD will be 5/10 = 50% in both groups. Suppose it is a one-year trial, all patients in Tiotropium group completed the one-year and all patients in Placebo group completed only 6 months. The patient year will be 10X1 = 10 for Tiotropium group and 10x0.5 = 5 for Placebo group. The incidence rates now become 5/10 = 50% in Tiotropium group and 5/5 = 100% in Placebo group – this is just simply wrong. In this case, when the patient year (or person year) is used as denominator, the numerator used in the calculation should be the number of events, not the number of patients with an event.

It is unfortunate this simple concept of ‘incidence rate’ has been wrongly calculated in Tiotropium studies. This wrong calculation may have been embedded in their paper published in prestigious New England Journal of Medicine.

If ‘number of patients with an event’ is used in the numerator, the denominator has to be the total number of patients (not the number of patient year). ‘Number of patients with an event’ divided by ‘number of total patients’ is called ‘incidence of events’ – this is a typical way when we summarize the adverse events in clinical trials.

wei said...

but the wiki mentioned that 'the number of NEW CASES per population in a given time period' and that incidence should be differentiated from prevalence. So if we only consider NEW CASES, you won't have multiple events for a subject, and number of NEW cases would equal to number of subjects with one or more of certain AE.

Web blog from Dr. Deng said...

you are right. If you count New Cases, it would be fine. In clinical trials, we are counting the number of adverse events or the number of exacerbations (in the example I mentioned). We are counting multiple events. If the denominator is duration (patient year), the numerator must be # of events; if the denominator is total # of patients, the numerator is the # of patients with at least one event. mismatch would be wrong.

maomao said...

I would like to ask if it is still valid to use Poisson distribution to calculate the confidence interval if you only count new cases. If we are only counting new cases, time at risk is up to the time of event for the subjects with events. I don't think Poisson distribution applies here any more. Could you please comment? Thank you!

maomao said...

I would like to ask if it is still valid to use Poisson distribution to calculate the confidence interval if you only count new cases. If we are only counting new cases, time at risk is up to the time of event for the subjects with events. I don't think Poisson distribution applies here any more. Could you please comment? Thank you!

Web blog from Dr. Deng said...

who not? if it is the count data, you can use poisson distribution. However, depending on the actual data, there may be other distribution fitting the data better.