Calculation of incidence rate for epidemiological study in hospital

I have kinda puzzled when I heard from the other about the calculation of incidence rate

From Kenneth Rothman's Modern Epidemiology, the incidence rate is calculated as number of cases in a fix period of time divided by person-time at risk at that period of time, which mean if the patient has the disease in the middle of the month, only the first half of patient days will be included in the denominator, but not for the second half. Am I correct?

Problem arises when I saw someone use the following formula for calculation:

number of cases in a fix period of time/total person-time at the period of time

If this is used, then the incidence rate will be underestimated, because the denominator now is bigger than it should be. But this formula seems to be used often, I wonder what this formula does. Thanks!

It is commonly admitted that the denominator for IRs is the "population at risk" (i.e., all individuals in which the studied event(s) may occur). Although your first formula is generally used, I found in The new public health, by Tulchinsky and Varavikova (Elsevier, 2009, 2nd. ed., p. 84) that a distinction is made between ordinary incidence rate, where the average size of the population in the fixed period of time is used in the denominator, and person-time incidence rate, with PT at risk in the denominator.

Obviously, when individuals not at risk of the disease are included in the denominator, the resultant measure of disease frequency will underestimate the true incidence of disease in the population under investigation, but see Numerators, denominators and populations at risk.

Short answer: When in doubt, trust in Rothman.

Long answer: It depends. You only want to include time where you are actually at risk of the outcome in your calculation of the denominator. Time where you aren't at risk (known as immortal person-time) should never be used in the calculation of a rate.

In your case, if the outcome of interest is the onset of a disease, then yes, the moment they have the outcome, they have it. You stop counting their time in the denominator, and they move to the numerator. Doing otherwise underestimates the incidence rate. The answer to what the other formula is doing is well...doing it wrong. Or more likely, doing it with the best data that is available - such as when all a study has is a number of counts per time and a number of person-time for an interval. But it should always be noted that this is a subtle underestimation.

Frequently, one might actually assign times when you only know an interval to try to address this. To use a hospital example:

We have 100 patients, who stay for 1 month. During that time, we get 5 cases.

The underestimation is 5 cases/100 person-months (400 person-weeks). We can however say that, in our experience, most infections occur within the first say, week of hospitalization. So we will assume that all cases occurred in that week. Now it's 5 cases/ 100 person-weeks. Or, if we have reason to believe they all got it in the 3rd week - say, we know the Jello in the cafeteria was contaminated with norovirus, we can assume that all cases occurred then, and now its 5 cases/300-person weeks. Most often, people just pick the middle of the interval for unknown circumstances. The technique you're seeing, where you use the most time possible in the denominator, is the lowest estimation of the rate you can have.

I think Rothman's definition should be used. In the second definition all patients seem to be given the same duration (is this correct?) so that the incidence will not be the incidence rate but will be proportional to the cumulative incidence (cases / total in given time frame)