Misreading Racial Disparities - Beware Of Ratios of Percentages 

 It's fascinating how far you can get by taking a second look at the simplest statistics - in this case percentages and ratios.  Case in point, James Scanlan's clever and unjustly ignored observation that African Americans will necessarily appear to be losing ground relative to whites even as their standing improves in absolute terms.  (Actually, the argument holds for any inter-group comparisons, not just race.)  Scanlan shows that this is an artifact of measuring progress by focusing exclusively on ratios of percentages from dissimilar distributions.  This insight begs the question of how best to measure progress.  Here are some of Scanlan's examples. 

 Black-white differences in infant mortality: In 1983, 19.2 black infants but only 9.7 white infants died per 1000 births in each group. The resulting black-white ratio was 1.98.  In 1997, infant mortality had decreased quite a bit, to 14.2 for blacks and 6.0 for whites.  Note that in raw percentage terms, infant mortality had improved more for blacks than for whites.  That should be good news, no?  But, lo, now look at the black-white ratio in 1997 - it increased from 1.98 to 2.4.  How can infant mortality have improved more for blacks than for whites in absolute terms at the same time as the relative position of 
blacks to whites has worsened? 

 Here's another example for the same underlying statistical phenomenon:  Moving the income distributions of blacks and whites up by the same dollar amount relative to the poverty threshold would increase the racial disparity in poverty (because relatively more blacks suffer extreme poverty than whites)! Except for extreme circumstances, this will be true even if we boost black real incomes more than white real incomes. How can it be that helping blacks more than whites in absolute terms would worsen blacks' relative economic position? 

 Here's my favorite example - racial disparities in college acceptance rates. Suppose that college admissions are solely a function of SAT scores (as I'm told they essentially are for some large, selective state schools) and that the SAT distribution of black test takers equals that of whites except it's shifted to the left (as it is).  Let the cut-off point for college acceptance be the same for blacks and whites (i.e. no affirmative action). Lowering the admission standard (for everybody) would then reduce the racial disparity in admission rates. That's good, no? But at the same time - and necessarily so - the lowering of admission standards would increase the racial disparity in rejection rates. That's bad, no? Huh? 

 It turns out that seemingly straightforward comparisons of ratios of percentages may hide more than they tell (in these examples, with important policy implications).  Interestingly, all three examples draw on the same statistical phenomenon.  The secret lies in the funny shape of cdf-ratios from density functions that are shifted against each other. I plan to provide an intuitive explanation for this point once we've figured out how to post graphics on this blog.  Until then, read  James P. Scanlan's "Race and Mortality" in the Jan/Feb 2000 issue of Society .