Relative Versus Absolute Differences in School Discipline
(Apr. 19, 2015)
Prefatory note added February 24, 2020: This page concerns the failure of the education research community to understand that it is even possible for relative and absolute differences to yield opposite conclusions about whether a racial disparity has increased or decreased over time of is larger in one setting than another or that, in the school discipline context, this be the usual case. Of special pertinence to this page are the Massachusetts Disparities, Virginia Disparities, Allegheny County (PA) Disparities, Denver Disparities, Oakland (CA) Disparities subpages of the of the Discipline Disparities page. These subpages discuss the way that on the basis of the fact that the absolute racial differences in suspensions decreased during a period of overall decreases in suspensions, observers (including in peer reviewed journals) stated or implied that relative racial differences decreased (though the relative differences had in fact increased). See the Appendix to my “Usual, But Wholly Misunderstood, Effects of Policies on Measures of Racial Disparity Now Being Seen in Ferguson and the UK and Soon to Be Seen in Baltimore,” Federalist Society Blog (Dec. 4, 2019). See also the Spurious Contradictions subpage of Measuring Health Disparities page, which involves a situation where observers called for further research into why two studies had shown opposite results as to the effects of increases in medical procedure rates on racial differences in procedure rates when in fact the studies had shown essentially the same results (but one had measured disparities in terms or relative differences and the other had measured disparities in terms of absolute differences). That situation, however, involved health and healthcare disparities research where at least a few researchers had recognized that it was possible for the absolute difference and the relative difference in the outcome they happened to be examining (that is, the favorable/adverse outcome dichotomy) to change in opposite directions. Those few researchers, however, have not been able to do anything useful with that recognition. See “Race and Mortality Revisited,” Society (July/Aug. 2014) (at 335-336) and “The Mismeasure of Health Disparities,” Journal of Public Health Management and Practice (July/Aug. 2016). And none has ever discussed the existence of a relative difference for the opposite outcome even though anytime the absolute difference and the relative difference they happen to be examining have changed in opposite directions, the relative difference for the opposite outcome will necessarily have changed in the opposite direction of the relative difference being examined and the same direction as the absolute difference. But see the prefatory note to the Richmond Disparities subpage of the Discipline Disparities page regarding the 2019 study by Girvan et al. that may be the first instance where a peer-reviewed study by education researchers recognized that it was possible for relative and absolute differences to change in opposite directions. The study also recognized that this was the usual case when suspensions generally decreased.
This subpage is related to the Relative Versus Absolute and the Spurious Contradictions subpage of the Measuring Health Disparities page of jpscanlan.com. The former subpage discusses the fallacy that relative differences and absolute (percentage point) differences can both provide useful information about the strength of the forces causing a pair of outcome rates to differ even when the two measures yield opposite conclusions about such things as the direction of changes over time. That subject is treated more recently in the “Absolute Differences and the Value Judgment Fallacy” section of “Race and Mortality Revisited,” Society (July/Aug. 2014) (at 335-36). The latter subpage discusses a study where the authors opine about the reasons why two earlier studies yielded different results as to directions of changes in certain healthcare disparities over time, while not recognizing that the earlier studies showed essentially the same results and that the only difference was that one measured disparities in terms of relative differences in the favorable outcome (the larger relative difference) and the other measured disparities in terms of absolute differences. This subpage is also related to the NEPC National Study subpage of the Discipline Disparities, which discusses an October 20, 2011 study by Daniel Losen that measured discipline disparities in terms of absolute differences.
In order to fully understand this subpage, readers should be aware that, as discussed in "Race and Mortality Revisited,” as the frequency of an outcome changes relative differences in experiencing it and relative differences in avoiding it tend to change in opposite directions, and the absolute difference tends to change in the same direction as the smaller relative difference (while the odds ratio tends to change in the opposite of the absolute difference). Since observers who measure disparities in terms of relative difference tend to examine the larger of the two relative differences (as in the discipline disparities context where relative differences in rates of experiencing adverse discipline outcomes are generally much larger than relative differences in avoiding those outcome), they tend to reach opposite conclusions about the comparative size of disparities from observers who rely on absolute differences. To date, however, researchers discussing discipline disparities largely do so without evident awareness that it is even possible for one to reach different conclusions about the comparative size of disparities depending on whether one examines relative or absolute differences. (They also universally fail to understand that one tends to reach opposite conclusions depending on whether one examines relative differences in the favorable outcome or relative differences in the corresponding adverse outcome, but that is a different issue).
A further thing for readers to bear in mind (unaddressed in "Race and Mortality Revisited") is that discipline disparities are often discussed in terms of “disproportionality” or some variation on that word, or “overrepresentation.” Sometimes such terms simply refer to relative differences. But sometimes they refer to the difference between the proportion a group comprises of enrolled students (pool) and the proportion the groups comprises of students experiencing some adverse discipline outcome (though often without discussing how one would measure the disproportionality). Because the proportions groups most susceptible to an outcome comprise of persons experiencing an adverse outcome is a function of relative differences in experiencing the outcome, both the relative difference and absolute differences between the proportion a group comprises of the pool and the proportion it comprises of persons disciplined will change in the same direction as the larger relative difference. Hence, as the frequency of an outcome changes, disproportionality and overrepresentation (whether measured in relative or absolute terms) will tend to change in the opposite direction of the absolute difference between rates. See discussion in the IDEA Data Center Disproportionality Guide subpage of the Discipline Disparities page and Section I.B of my amicus curiae brief in Texas Department of Housing and Community Development, et al. v. The Inclusive Communities Project, Inc., Supreme Court No. 13-1731 (Nov. 17, 2014). See also Table 11(d) (slide 59) of my “The Mismeasure of Discrimination,” Methods Workshop, Center for Demographic and Social Analysis, University of California, Irvine (Jan. 20, 2015).[i]
“Similarly, although absolute rates of suspension and expulsion are higher at the secondary school level for both black and white students, the discrepancy between black and white rates of suspension is greater at the elementary school level (Rausch & Skiba, 2006;
Wallace et al., year). Finally, Wallace et al. reported that, although males of all racial and ethnic groups were more likely to be disciplined, disparities between African American and White students were greater among female students.”
It also states (at 5, original emphasis): “Noltemeyer and Mcloughlin (2010a), in a multivariate analysis of variables contributing to suspension across a single state, reported poverty was a significant predictor of a school's rate of suspension, but not of disproportionality in suspension.”
Both statements involve relative differences. Typically, absolute differences between rates would show opposite results as to the comparative size of disparities, as shown on the main Discipline Disparities page.
Later, however, the Skiba paper makes the following statement (at 12):
“Gregory, Cornell, and Fan (2011) explored the relationships between factors of school climate, student perceptions of teacher support and academic expectations, and discipline rates. Using a sample of 9th grade students from 199 schools across the state of Virginia, multivariate analyses were used to examine the relationships between suspension rates and academic expectations and support in high school climate. Schools rated as having the lowest levels of support and academic expectations were associated with the highest rates of suspension, as well as the largest Black-White suspension gap.”
In this case, the Skiba paper cited a study that relied on absolute differences between rates and that would likely have shown that the relative difference between black and white suspension rates tended to be smaller at schools with the lowest levels of support.
The paper referenced by Skiba et al. was Gregory, A., Cornell, D., & Fan, X. (2011). The relationship of school structure and support to suspension rates for black and white high school students. American Educational Research Journal, 1-31. Gregory, like Losen, tends to rely on absolute differences to measure discipline disparities.
The Gregory 2011 paper begins (at 2-3) by discussing “racial disproportionality in school discipline,” and when doing so mentions that “Black students are 2 to 3 times as likely to be suspended as White students (Wallace et al., 2008).” But the paper later explains that it will be relying on absolute differences, stating (at 11) that “[w]e judged that the difference in rates would be more useful and informative than the risk ratio in this study. A 13% difference in suspension is more readily understood and comparable than a risk ratio of 2.18.”[ii] The authors give no indication of recognition that relative and absolute differences could yield opposite conclusions about the comparative size of disparities, much less that would tend to do so systematically, or that disproportionality would tend to vary inversely with absolute differences.
A February 2015 study by Losen et al., for the Civil Rights Project, “Are We Closing the Discipline Gap,” also relied on absolute differences between rates, as is the group’s and Losen’s usual practice. In this case, the study included in an appendix (at 48-49) an explanation of why it relied on absolute differences. But the explanation gave no indication of recognition that relative differences in discipline rates and absolute differences between discipline rates would tend to give opposite conclusions about the comparative size of disparities.
In the summary (at 3) the report discusses a modest decline in disparity nationally between the 2010-11 and 2012 school years, where black and Hispanic rates remained at about 16% and 7% respectively, while white rates rose from 4% to 5%. In a situation where the disadvantaged groups’ rates are unchanged while the advantaged group rates increase, all measures of disparities will decrease. But in the case of the subsequent statement on the same page that many large districts reduced dramatically reduced suspension rates and reduced disparities the report was referring to absolute differences between rates (which typically will decrease when discipline rates generally decrease). Relative differences, however, will tend to increase.
The report also noted (at 3) that discipline disparities were much larger in secondary school (where rates are generally higher) than in elementary school. Relative differences usually are higher in elementary school than secondary school.
The 2013, the Rhode Island Chapter of the American Civil Liberties Union study titled “Blacklisted: Racial Bias in School Suspensions in Rhode Island” that is the subject of the Rhode Island Disparities subpage of the Discipline Disparities page makes a particular point of the larger relative racial differences in elementary school than secondary schools, noting (at 2) that “[w] While black high school students are twice as likely as white high school students to be suspended, a black elementary school student is six times as likely as a white elementary school student to be suspended from school.
See Table 8 of "Race and Mortality Revisited" (at 342) regarding the way that relative racial differences in multiple suspensions are larger in preschool than K-12, though the absolute differences are larger in K-12.
[i] It is important to distinguish between (a) the absolute difference between the more susceptible groups comprise of persons experiencing an outcome and the proportion such groups comprise of persons experiencing an outcome and (b) absolute differences between outcome rates. As the frequency of an outcome changes, the former tends to change in the same direction as the larger relative difference while the latter tends to change in the opposite direction of the larger relative difference.
[ii] The use of the percent sign (%) when referring to percentage point differences raises of different problem, which I address on the Percentage Points subpage of the Vignettes page of jpscanlan.com. But it adds to the confusion.