Basic question - we have data representing the average performance of two sets of hospitals on a quality measure. The denominator for each set is different. Is it acceptable to calculate percentage difference between one average and another? I was always under the impression that such a measure is misleading and you would need the raw data (numerator and denominator) that were used to calculate the average performance.
There is not much detail here, but in the absence of other information let us assume for simplicity that the measure is bounded in principle by 0 and 100%. In any case, the question could apply much more broadly to almost anything measured as a percentage.
The restriction is not tautologous, as measures can be imagined such as actual performance / target performance in which case a high-performing hospital might exceed its target, and so the measure might exceed 100%.
Quirk warning: There is small lobbying here for the simpler word
percent rather than
percentage. Apologies if you think that illiterate or ungracious writing, which in general I try hard to avoid.
The issue is not whether percents of percents are defined, as measures such as
(percent$_1 - $ percent$_2$) / percent$_1$
are perfectly computable so long as the denominator is not zero. The question is more
(a) precisely what you want to measure
(b) how you expect percents to behave?
Colloquially, numbers should correspond to ideas of what is a 'big deal' and what is a 'small deal'.
Consider an improvement of 5% from 25% to 30% and one from 75% to 80%. If the improvements are considered equivalent, then you can just cite the two percents and the difference, sometimes worded as 5% points, although the full context clearly requires knowing the original percents too.
Often, however, improvements are easier in the middle of the scale than at one end or the other. Moving from say 50 to 51% perfection is then likely to be much easier than moving from 98% to 99% or from 99% to 100%. The latter two require halving the problems or inefficiency from 2% to 1%, or removing all remaining problems!
When this is true, using percents of percents is changing the measure in entirely the wrong direction, as 1/50 change is really less of an achievement than 1/98 or 1/99.
There is a connection here between elementary (but also fundamental) questions about what you want to measure and more formal questions of appropriate transformations or link functions for measures that arrive as proportions between 0 and 1, which are just percents with different presentation conventions.
It is common with long-term study of adoption of innovations (e.g. how many people are literate, use cell (mobile) phones, have inside toilets) to find that a logistic curve is a good first approximation to the pattern of change over time. (Sometimes the logistic curve has negative slope, as with % illiterate or those attributing illness to witchcraft or demonic possession.) Such time curves imply that a logit or logistic scale is the appropriate way to view proportions. In practice, that might not be adopted for either or both of two quite different reasons, if values of 0 and 100% do actually occur, or if this is regarded as too complicated for people who would be expected to read a report.
I'll add more technical cross-references to folded power transformations.