Subgroup Effects/Interaction – Nonclinical
(Jan. 4, 2014; rev. May 21, 2014)
Prefatory note added May 21, 2014: This subpage is fairly complicated, or so it has seemed to me on my occasionally reviews of it. Readers can better understand it if they are familiar with the discussions of large relative differences in adverse outcomes among advantaged populations/subpopulations in “Race and Mortality (Society, Jan./Feb. 2000), “Race and Mortality Revisited” (Society, July/Aug. 2014) (in press) or “The Perils of Provocative Statistics” (Public Interest, Winter 1991), or, more briefly, Comment on Kawachi Health Affairs 2005. I think, however, that the page would remain useful to persons familiar with such materials because of its focus on perceptions about interactive effects.
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This subpage is related to various other subpages of the Scanlan’s Rule page that pertain to misinterpretations of subgroup effects/interaction arising from the failure to recognize that a factor that similarly affects to two groups with different baseline outcome will tend to cause a larger proportionate change in the outcome for the group with the lower baseline rate while causing a larger proportionate change in the opposite outcome for the other group. Said pattern is a corollary to the pattern whereby the rarer an outcome the greater tends to be the relative difference in experiencing it and the smaller tends to be the relative difference in avoiding it. The subpage is also related to subpages addressing the illogic of the rate ratio and is corresponding relative difference as measures of association. Related subpages include the Subgroup Effects subpage (addressing these issues with respect to clinical settings and from which this subpage principally differs in that this subpage focuses on nonclinical settings); the Illogical Premises subpage (addressing the illogic of the expectation that a factor generally will cause equal proportionate changes in different baseline rates for an outcome, given that a factor cannot cause equal proportionate changes in rates of experiencing the outcome while at the same time causing equal proportionate changes in rates of experiencing the opposite outcome); the Illogical Premises II subpage (more broadly addressing the illogic of the rate ratio as a measure of association, given that whenever rate ratios are the same for different baseline rates of experiencing an outcome the rate ratios for the opposite outcome are necessarily different); the Inevitability of Interaction subpage (addressing that the fact that in circumstances where two groups have different baseline rates for an outcome a factor that affects the outcome rate will necessarily show different proportionate effects for the two groups either as to that outcome or as to the opposite outcome); and the Interactions by Age subpage (addressing that a factor that affects an outcome rate will tend to show opposite interactions by age group depending on whether one examines the favorable or adverse outcome).
This subpage page will largely be devoted to showing common examples of the above-described pattern by which a factor that affects an outcome rate will tend to cause a larger proportionate change in the outcome rate of the group with the lower baseline rate while causing a larger proportionate change in the opposite outcome for the other group. Initially I note that the patterns discussed here can always be addressed from two perspectives, given that groups that the advantaged and disadvantage group can be identified either (a) on the basis of their baseline adverse (or favorable) outcome rates or (b) on the basis of whether have the factor or not. I clarify that matter with the first example in Tables 1 and 2 (which are based on data discussed in the “The Perils of Provocative Statistics” (Public Interest, Winter 1991)) concerning the comparative size of differences between the poverty rates of blacks and whites in married-couple families and female-headed families. With regard to the instant matter, which involves comparisons of effects of factors (rather than comparisons of sizes of differences), the matter can be examined either in terms of (a) the effect of race on poverty of married-couple families (the advantaged group) and female-headed families (the disadvantaged group) or (b) the effect of family status on the poverty of whites (the advantage group) and blacks (the disadvantaged group). Where one observes the above-described pattern of proportional effects from one perspective, as one commonly does whenever baseline rates differ substantially, one will necessarily observe the pattern from the other perspective.
I add here two points of clarification. First, whereas in the discussion in the paragraph immediately above, I contrasted comparisons of the size of the difference between two groups with comparisons of the effects of a factor, any analysis of group difference in outcome rates is actually examining the effect of group membership on outcome rates. So, while the distinction I draw above seems useful in the instant context, it is not a matter of substance. Second, partly because this page, like its clinical counterpart, is framed in terms of effects, and partly because I may be characterizing other analyses that speak of effects, I discuss patterns in terms of causes and results. In reality, however, I am discussing patterns of association which may or may not involve causation. But such issues are unimportant to the points I make about patterns of relative effects/association that I make here.
Table 1 illustrates the contrasting pattern of comparative sizes of effects from the perspective whereby the subgroups are defined by marital status and the effect is a function of race. It shows that being black, which increases the likelihood of being poor, does so to a larger proportionate degree for married-couple families (the group with the lower baseline rate) than female-headed families, while being black decreases the likelihood of avoiding poverty more for female-headed families (the group with the lower baseline rate for avoiding poverty) than for married-couple families.
Table 1: Effects of black race on likelihood of being in poverty and likelihood of avoiding poverty for married-couple and female-headed families [ref N2/b4901a1]
|
|
Comparison
|
Base Status
|
Group
|
Wh Pov Rate
|
Black Pov Rate
|
Adv Increase
|
Fav Decrease
|
|
Effect of black race
|
A
|
MC
|
5.20%
|
12.30%
|
136.54%
|
7.49%
|
|
Effect of black race
|
D
|
FH
|
26.70%
|
51.80%
|
94.01%
|
34.24%
|
Table 2 then shows the alternative perspective. The table treats whites as the advantaged group and blacks as the disadvantaged group, and shows the effect of being in a female-headed family rather than a married-couple family on rates of being in poverty and rates of avoiding poverty for whites and blacks. Again, the proportional increase on the likelihood of experiencing poverty as a result of being in a female-headed is greater for whites (the group with the lower baseline) than for blacks, while the proportional decrease in rates of avoiding poverty is greater for blacks (the group with the lower baseline rate for avoiding poverty) than for whites.
Table 2: Effects of being in a female-headed family rather than a married-couple family on likelihood of being in poverty and likelihood of avoiding poverty for whites and blacks [ref N2/b4901a1]
|
|
Comparison
|
Base Status
|
Group
|
MC Adv Rate
|
FH Adv Rate
|
Adv Increase
|
Fav Decrease
|
|
Effect of being female-headed
|
A
|
W
|
5.20%
|
26.70%
|
413.46%
|
22.68%
|
|
Effect of being female-headed
|
D
|
B
|
12.30%
|
51.80%
|
321.14%
|
45.04%
|
This matter, of course, could be further recast in terms of the effects of being in the advantaged group (that is, being white rather than black and being in a married-couple family rather than a female-headed family). But while doing so would necessarily illustrate the same patterns, it would not add anything useful to the discussion. I note, however, that the discussion below concerning differences in lending outcomes is cast in terms of the effects of being in the advantaged situation (whether defined by race or income). I further note that most of my other illustrations of patterns of relative differences/effect are based on rate ratios with the higher rate in the numerator (for reasons discussed in note 2 of my September 20, 2013 University of Kansas School of Law Faculty Workshop paper “The Mismeasure of Discrimination”). In such circumstances, while the quantification of the relative effect, if viewed in terms of the shown rate ratio minus one, is indeed the effect of the disadvantaging factor for the adverse outcome; but it is actually the relative effect of the advantaging factor for the favorable outcome. That is, if Table 2 were based on rate ratios with the larger figure in the numerator as to both outcomes, the favorable rate decrease would be 29.33% for the whites and 34.24% for blacks (rather than the 22.68% and 45.0% figures in the final column of Table 2). But these nuances have no bearing on the essentials of the patterns I describe here.
I may eventually develop this page further with additional examples, possibly with a particular focus on instances where observers draw particular inferences on the basis of the greater proportionate effect of a factor on the particular outcome (favorable or adverse) that the observer happens to be examining for one demographic group than another. It should be recognized, however, that, as noted, the pattern of effects discussed here are corollaries to the pattern by which the rarer an outcome the greater tends to be the relative difference in experiencing it and the smaller tend to be the relative difference in avoiding it. Thus, the many score illustrations of the pattern of changes in the two relative differences, or the size of relative differences in different setting, on the pages and subpages of jpscanlan.com, as well in the conferences presentations made available in Section B and the online comments made available in Section D of the Measuring Health Disparities page, are themselves illustrations of the pattern by which a factor that affects an outcome rate will tend to cause larger proportionate changes for the group with the lower baseline rate for the outcome while causing a larger proportionate change in the opposite outcome for the other group. This includes most of the illustrations in the recent Federal Committee on Statistical Methodology 2013 Research Conference presentation “Measuring Health and Healthcare Disparities.”
For the present I merely note that the matter is specifically treated with respect to effects of having a criminal record on black and white job applicants in the Criminal Record Effects subpage of the Scanlan’s Rule. Table 1on the subpage shows that having a criminal records reduces the chance of a callback proportionately more for a black job applicant than a white job applicant (the pattern for which the author posited an explanation) but reduces the chances of not receiving a callback more for whites than for blacks (a pattern overlooked by the author but which would be the focus in many analyses and which would require a very different explanation than that posited by the author). Table 1a of the subpage shows that being black causes a larger proportionate reduction in callback rates for those with a conviction than those without a conviction (the former having the lower baseline callback rates), but increases rates of not receiving a callback proportionately more for those without a conviction than those with a conviction (the former having the lower baseline rates of not receiving a callback than the latter).
The points made here are implied in the discussion on the Disparities – High Income subpage of the Lending Disparities page of the way that relative differences in adverse lending outcomes tend to be larger among higher-income groups than lower-income groups, while relative racial differences in favorable lending outcomes tend to be larger among lower-income groups than higher-income groups. That is, such patterns could be recast in terms that (a) having higher income decreases adverse lending outcome rates more for whites than blacks, while increasing favorable outcome rates more for blacks than whites; or (2) being white reduces adverse outcome rates more for higher-income groups than lower-income groups, while increasing favorable outcome rates more for lower-income groups than higher-income groups. Note iv of the Disparities -High Income subpage discusses a study that did find that having higher income caused a larger proportionate increase in mortgage approval rates for blacks than white (and in which the authors drew certain inferences from such pattern). But the study did not provide sufficient information to determine whether having higher income caused a larger proportionate reduction in rejection rates of whites than blacks, as commonly would occur.
I also note that, while most studies that discuss comparative effects tend to do so with respect solely to one outcome, sometimes they will do with respect to both outcomes, as in a 2011 Minneapolis Urban League study titled “Racial Disparities in Home Ownership.” In analyzing effects of location on loan denial rates, the study (at 14) noted that the increase in denial rates as a result of being in a particular location was greater for whites than for blacks. But in discussing the effects of location on home ownership (at 24), the study notes that location had a smaller effect on whites than blacks. The patterns are consistent with the above described pattern whereby a factor that affects an outcome rate will have the larger proportionate effect on the group with the lower baseline rate. That the factor has a greater effect on loan denial for whites (the group with the lower baseline denial rates than other groups) but a greater effect on home ownership for other groups (groups with lower baseline home ownership rates than whites). The study did not provide data allowing one to determine whether, in accord with the patterns described above, the effect of location on loan approval would be greater for blacks than for white (since blacks have lower approval rates) and the effect of location on absence of home ownership would be greater for whites than blacks (since whites have lower rates of failing to own a home). See also the Immunization Page, which discusses a study that examined relative differences in a favorable outcome (receipt of full immunization) and relative difference in an adverse outcome (failure to receive any immunization). General increases in immunization rates would tend to reduce the former while increasing the latter. Correspondingly, disadvantaged groups would tend to show larger proportionate increases in full immunization while advantaged groups would tend to show larger proportionate decreases in failure to receive any immunization.