James P. Scanlan, Attorney at Law

Relative Differences Versus Absolute Differences: 

The Absurdity of Maintaining That One Disparity is Larger from a Relative Perspective While Another Disparity is Larger from an Absolute Perspective

(May 25, 2010; rev. May 30, 2010)

Note:  This page is related to the Illogical Premises sub-page of the Scanlan’s Rule page of this site, which sub-page addresses reasons why it is illogical to regard it as somehow normal that a factor that increases or decreases the likelihood of experiencing an outcome will do so to equal proportionate degrees for groups with different base rates or that different base rates will exhibit equal proportionate changes over time (said reason being that it is impossible for two groups with different base rates of an outcome to experience equal proportionate changes in rates of experiencing an outcome and equal proportionate changes in rates of failing to experience the outcome).   

There is a good deal of literature on differences in outcome rates that considers relative and absolute differences in outcome rates both to be legitimate ways of appraising the size of health, healthcare and other disparities, even when the measures yield opposite conclusions about the directions of changes in disparities over time or about the comparative size of two disparities in settings differentiated other than temporally.  As a rule when illustrating the way that the two measures support different conclusions, those doing so fail to note that one may also reach still different conclusions depending on whether one analyzes the relative difference in experiencing the outcome of the relative difference in avoiding the outcome.  In any case, some will discuss the choice of relative difference versus absolute difference as a measure of disparity as involving a value judgment.

As discussed in many places on this site, neither the relative difference in one outcome, nor the relative difference in the opposite outcome, nor the absolute difference is itself a useful indicator of the comparative size of a disparity because each is affected by the overall prevalence of an outcome.  See, e.g., Measuring Health Disparities (MHD), Scanlan’s Rule, and Mortality and Survival pages.  And while there can be situations where the absolute difference may be a more useful indicator of the importance of a disparity than either relative difference, the only useful indicator of the size of disparity in a meaningful sense is an approach that derives from two rates the difference between the underlying means, as discussed on the Solutions sub-page of MHD. 

The purpose of this page is to illustrate the essential absurdity of maintaining that from a relative perspective one disparity is larger while from an absolute perspective another disparity is larger.   Table 1 below is derived from a table on the Case Study sub-page of the Scanlan’s Rule page where it is used to show the different perspectives from which one may view a situation that reflect the same underlying differences.  (See also Table 1 from the 2008 Joint Statistical Meetings presentation.)  That is, each setting involves a situation where the means of the underlying distributions differ by half a standard deviation, but where success and failure are dichotomized at different points.  The AGFR and DGFR columns present the favorable outcome rates for the advantaged and disadvantaged groups.  The other fields show the relative differences in the favorable outcome rate, the relative difference in the adverse outcome rate, the absolute differences between failure (or success) rates, and the odds ratio.[i]  By way of further explanation with respect to the main point of this item, in Setting A, DG’s favorable outcome rates is 55% less than AG’s favorable outcome rate and 11 percentage points less than AG’s favorable (or adverse) outcome rate; in Setting B, DG’s favorable outcome rate is 43.5% less than AG’s favorable outcome rate and 17 percentage points less than AG’s favorable (or adverse) outcome rate.

Assume that A and B are two different employees, each of whom selects from pools of applicants from the advantaged (AG) and the disadvantaged (DG) racial groups.  Assume also that on average the applicants from each racial group are equally qualified and all difference in selection rates are functions of the bias of the employers.  Now consider which employer is more biased.   Of course, one would reach different conclusions depending on whether one relied on relative differences in selection or relative differences in rejection. 

Here, however, let us simply focus on the relative difference in selection rates and the absolute difference between selection rates.  Focusing on the relative difference, one would conclude that the Employer 1 is more biased; focusing on the absolute difference, one would conclude that Employer 2 is more biased.  I suggest, however, that it would be manifestly absurd to maintain that both the relative and the absolute difference provide a legitimate basis for appraising the situation or to maintain that from a relative perspective Employer A is more biased and from an absolute perspective Employer B is more biased.   Rather, with regard to the crucial issue of the extent to which each employer’s decisions are affected by aversion to a particular race the employers are exactly the same.

The same reasoning holds where employer A and B (or A, B, C, and D) are the same employer during different time periods.  It holds as well where the differences in selection arise not from employer bias, but from differences in the qualifications of the two groups.  And it holds when the decision-making process at issue is entirely objective and the issue concern is the degree to which the selection procedures disadvantage one of the groups (in EEO parlance, which procedure has the greater disparate impact).[ii]  And it holds when the issue is whether one health or healthcare disparity is larger than another or whether a health or healthcare disparity has changed over time.