I have found various and seemingly related explanations:
- Multiplying by 10,000 [patient days or patients] standardizes the rate so it can be compared to other hospitals / populations that may have fewer or greater number of patient days or patients.
- Multiplying by 10n converts decimal fractions to a standard population size which is a more understandable description of the prevalence within a population.
- Multiplying by 10n is performed because the frequency of the numerator compared to the denominator is usually rather low.
The third explanation makes the most sense to me, since a relatively small numerator and large denominator can result in very small fractions with lots of leading zeros (e.g., 0.000045) and multiplying by 10,000 would give an easier to read/interpret number (e.g., 4.5). Contrastingly, explanation #1 above is not satisfying since it seems legitimate to compare hospital A with a rate of 0.000045 to hospital B with a rate of 0.000067, for example, without the need to compare these same rates only after "standardizing" / multiplying by 10,000 (e.g., 4.5 vs 6.7).
What is the underlying theory for this type of standardization and why is it so common in health and epidemiological measures (e.g., Clostridium diff. infections, SSERs, etc.)?