SEMANTIC ISSUES
(Feb. 23, 2009; rev. May 22, 2011)
A. Implications of Choice of Numerator
In a number of places, particularly with regard to the disparities measurement approach of the Agency for Healthcare Research and Quality (AHRQ), I have discussed the way that the absolute difference tends to change in the opposite direction the larger of the two relative differences (when the relative difference is measured by means of a rate ratio where either the larger rate or the smaller rate is used as the numerator). As qualified by the introductory on the Scanlan’s Rule page, that point would be better said to hold for situations where the differences between means of the underlying distributions are larger than .49 and in other situations where the rates of the two groups are outside the range defined by a rate of 50% for each group.
I usually measure relative differences by means of a rate ratio where the larger rate is always used as the numerator (Approach A). Among the advantages of this approach, as well as the approach whereby the smaller rate is used as the numerator in both ratios (Approach B), is that the point where the rate ratio for the favorable outcome and the rate ratio for the adverse outcome are equal is also the point where the two relative differences are equal.
But one can also measure relative differences by means of rate ratios where the disadvantaged group’s rate is used as the numerator for both the favorable and the adverse outcome rate ratios (Approach C)) (which is a very common approach and that AHRQ) or by means of rate ratios where the advantaged group’s rate is used as the numerator for both the favorable and the adverse outcome rate ratios (Approach D).
Under the latter two approach, the rate ratios for the favorable outcome and the rate ratio for the adverse outcome will be the reciprocals of one another – and hence may be said to be equal – at the same point where the rate ratios for the adverse outcome and for the favorable outcome derived with either the higher rate or lower rates the numerator in both ratios are equal to one another. But the relative differences derived from those ratios will not be the same. The reason for this is that, for example, when the rates examined are 80% and 100%, in which case we would regard the difference to be the same whether measured by a rate ratio of .8 or 1.25, the relatives would be 20% or 25% depending on the numerator in the rate ratio.
I illustrate the point below from a situation where the differences between means is approximately half a standard deviation and where the relative differences in the favorable and the adverse outcomes approximately equal each other under Approaches A and B. Such point occurs with an adverse outcome rate of 40.1% for AG and 59.5% for DG. All figures are rounded somewhat.
Approach A
Adverse rate ratio 1.48
Favorable rate ratio 1.48
Adverse Relative difference .48
Favorable Relative difference .48
Approach B
Adverse rate ratio .67
Favorable rate ratio .67
Adverse relative difference .33
Favorable relative difference .33
Approach C
Adverse rate ratio 1.48
Favorable rate ratio .67
Adverse Relative difference .48
Favorable Relative difference .33
Approach D
Adverse rate ratio .67
Favorable rate ratio 1.48
Adverse Relative difference .33
Favorable Relative difference .48
If we envision the matter in terms of a favorable outcome increasing in overall prevalence, under Approach C, the two relative differences will be approximately equal at the point where the AG’s favorable outcome rate reaches 50%. At that point, DG’s favorable outcome rate will be about 30.9% and both relative differences will be about 38.3%.
Under Approach D, the two relative differences will be approximately equal at the point where the DG’s favorable outcome rate reaches 50%. At that point, AG’s favorable outcome rate will be about 68.8% and both relative differences will be about 37.6%
The implication of AHRQ’s reliance on Approach C, given that it measures disparities in terms of the larger of the two relative differences, is that it will more often rely on the increasing relative difference in the adverse outcome than the decreasing relative difference in the favorable outcome than if it employed Approaches A, B, or D.
[Note added June 27, 2020. See the June 27, 2020 material added to the Addendum to the 2007 American Public Health Association presentation regarding my confusion about the way AHRQ was measuring disparities in the National Healthcare Disparities Report. Specifically, rather than relying on the larger of the two relative differences, as suggested above, the NHDR always measured health and healthcare disparities in terms of relative differences in the adverse outcomes. Thus, I think the above discussion of ways choice of numerator in a ratio affect relative differences, it does not really bear on the National Healthcare Disparities Report.]
B. Sizes of Odds Ratios or the Differences they Reflect
It is often noted that an attractive feature of the odds ratio (OR) is that the size of the disparity it reflects is unaffected by whether one examines the favorable or the adverse outcome. But that is not the same thing as saying that the odds ratio is the same whether one examines the favorable or the adverse outcome.
Actually, there are four possible ways to calculate the OR, depending on whether the ratio is of the odds of experiencing the favorable outcome or the odds of experiencing the adverse outcome and depending on which group’s odds is used as the numerator. Two of these calculations will result in one value and two will result in a value that is the reciprocal of the first value. And two will be greater than 1 and two less than 1. As with RR, if OR is above 1, the larger the OR the larger the difference; if OR is less than 1, the smaller the OR, the smaller the difference.
It is for that reason that in discussing the OR, I refer to the size of difference measured or reflected by the OR not the size of the OR.