The owner of this site is an attorney specializing in the use of statistics with respect to employment discrimination litigation and compliance, and a number of items made available on this page relate to employment discrimination issues or statistical issues related to employment discrimination. But materials made available through this site also address a range of other matters. The pages that address particular subjects are described below. The discussions below of sub-pages involve pages apart from any “Summary” or “Outline and Guide” sub-pages discussed in the note above.
The first four pages discussed below – Measuring Health Disparities, Scanlan’s Rule, Mortality and Survival, and Measures of Association – address the problematic nature of standard measures of differences between rates at which demographic groups experience an outcome, given that, for reasons inherent in the shapes of the underlying distributions of factors associated with an outcome, each measure tends to be affected by the overall prevalence of the outcome. Most notably, 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. Thus, among many things of comparable significance, when an outcome like mortality declines in overall prevalence (or an outcome like adequate hemodialysis, mammography, or obesity increases in overall prevalence), whether health or healthcare disparities are deemed to have increased or decreased often will depend on whether one examines relative differences in the favorable outcome or in the (opposite) adverse outcome. Absolute differences and odds ratios are also systematically affected by the overall prevalence of an outcome, though in a more complicated way (as explained in detail in the introduction to the Scanlan’s Rule page). Roughly, as uncommon outcomes (those with rates of less than 50% for both groups) become more common, absolute differences between rates tend to increase; as common outcomes (those with rates of greater than 50% for both groups) become even more common, absolute differences tend to decrease. Differences measured by odds ratios tend to change in the opposite direction of absolute differences between rates. As a result of the failure to understand the way that ( or even the fact that) these and other common measures of differences between outcome rates tend to be affected by the overall prevalence of an outcome, very little that has been said to date regarding the comparative size of two or more differences between outcome rates or regarding whether a difference between two outcome rates should be deemed large or small – either in the law and the social and medical sciences or in any other setting where the size of a difference between rates is a matter of consequence – has had a sound foundation.
Measuring Health Disparities Page
The Measuring Health Disparities page (MHD) lists, and in all but a few instances provides links to, about 140 references describing the implications of the above-described tendencies either generally or with reference to studies or commentaries that interpret data on group differences without an understanding of the effects of overall prevalence on the measure employed. As of the most recent updating of this summary, the references are comprised of approximately 13 published articles, 23 conference presentations, and about 120 on-line comments on articles in medical or health policy journals, and several unpublished articles. The most important published articles are Can We Actually Measure Health Disparities? (Chance, Spring 2006), Race and Mortality (Society, Jan/Feb 2000), Divining Difference (Chance, Fall 1994), The Perils of Provocative Statistics (Public Interest, Winter 1991), and The “Feminization of Poverty” is Misunderstood (Plain Dealer, Nov. 11, 1987).
The most important conference presentation are Measuring Health Inequalities by an Approach Unaffected by the Overall Prevalence of the Outcomes at Issue (pp) (Royal Statistical Society, 2009) Interpreting Differential Effects in Light of Fundamental Statistical Tendencies (pp) (Oral) (Joint Statistical Meetings, 2009), Can We Actually Measure Health Disparities ?(pp)? (Oral) (7th International Conference on Health Policy Statistics, 2008) Measurement Problems in the National Healthcare Disparities Report (pp) (Oral) (Addendum) (American Public Health Association, 2007), The Misinterpretation of Health Inequalities in the United Kingdom (British Society for Population Studies, 2006)
After an introduction of about 1150 words, MHD sets out in Sections A through D the above-described references (with Sections A, B, and D, also made directly accessible as sub-pages): A – Published Articles, B – Conference Presentation, C – Unpublished Papers, D – On-line Comments. Sections E.1 to E.6 provide brief summaries of, and list pertinent references regarding, six issues: (1) the misinterpretation of health inequalities in the United Kingdom and/or the Whitehall Studies; (2) the misinterpretation of health inequalities in Nordic Countries; (3) absolute differences between rates as a measure of disparities; (4) the approaches to disparities measurement of the National Center for Health Statistics and the Agency for Healthcare Research and Quality; (5) issues regarding health disparities and pay-for-performance; (6) approaches to the measurement of disparities that are unaffected by the overall prevalence of an outcome.
Section E.7, which is separately accessible as a sub-page, describes the extent of scholarly agreement or disagreement with the views expressed in the listed references. In summary, there is an emerging recognition, more in Europe than in North America and elsewhere around the world, that standard differences between outcome rates tend to be systematically affected by the overall prevalence of an outcome and that it is therefore necessary to take overall prevalence into account in interpreting differences between outcome rates. With rare exception, however, those expressing such recognition have not shown an understanding of the forces underlying observed patterns and have not evidenced the recognition in their subsequent work.
MHD has eleven sub-pages that provide substantive material elaborating on issues addressed in references listed on MHD (and in some cases expanding on issue addressed in Sections E.1 to E.6). The Solutions sub-page addresses an approach to appraising differences between rates of experiencing an outcome that is not affected by the overall prevalence of the outcome and the Solutions Database sub-page provides a downloadable database that allows one to conveniently implement that approach. (As discussed in the introductory material to the Solutions sub-page, the approach described there, and implemented mechanically in the database provided, yields the same result as that derived formulaically in the probit analysis developed by Chester Ittner Bliss in 1934.) As discussed in a number of places, the approach has certain weaknesses. But it nevertheless is far superior to reliance on standard measures of difference without regard to the way such measures tend to be affected by the overall prevalence of an outcome.
The Irreducible Minimums sub-page of MHD addresses the implications, with regard to the measurement approach of the Solutions and Solutions Database sub-pages, of a situation where an advantaged group’s adverse outcome rate reaches a level where it is difficult or impossible to further reduce the rate given the current state of medical knowledge and related factors (a concept termed “minimum achievable level” by other authors). The item explains a modification to the Solutions Database to address the issue. The Cohort Considerations sub-page addresses limitations of the solution in circumstances where outcome rates are calculated from among persons who have not yet experienced the outcome, as distinguished from outcome rates for the entire cohort that may experience the outcome. These issues are related to those addressed on the Truncation Issues and Life Tables Illustrations sub-pages of the Scanlan’s Rule page. The Relative Versus Absolute page, using as an example a situation where the subject at issue is the degree of employer bias against a particular group, discusses why it is unreasonable to consider opposite conclusions as to the comparative size of disparities based on relative and absolute differences between outcome rates both to be valid.
The Pay-for-Performance sub-page discusses issues related to the perceived impact of pay-for-performance on health or healthcare disparities. In the main, in the United States such perception, based on the observed increasing absolute differences in procedure rates for relatively uncommon procedures, is that pay-for-performance will tend to increase healthcare disparities, and that it may be necessary to address such impact by making changes in healthcare disparities an element of any pay-for-performance program. In the United Kingdom, however, the perception, based on observed declining absolute difference between rates of advantaged and disadvantaged groups for relatively common procedures/favorable outcomes, is that pay-for-performance programs will tend to reduce healthcare disparities. Neither perception has a sound statistical foundation since both involve attributing significance to patterns of changes in absolute differences between rates that, solely for statistical reasons, are generally to be expected during periods of increases in rates that are in the ranges at issue in the studies. The Between Group Variance discusses the way that the measure used to measure healthcare disparities in a Massachusetts program that includes effects on disparities as a performance criterion may tend to increase healthcare disparities.
The Concentration Index sub-page and the Gini Coefficient sub-pages address the way those measures tend to be affected by the overall prevalence of an outcome. The Reporting Heterogeneity sub-page addresses the way perceptions of reporting heterogeneity fail to consider the extent to which observed patterns are functions of the underlying distributions. The issues are related to those addressed on the Illogical Premises and Subgroup Effects sub-pages of the Scanlan’s Rule page.
The NHDR Technical Issues sub-page addresses certain technical issues in the National Healthcare Disparities report that are unrelated to the central criticism of the measurement approach in the report that I have discussed in various places – i.e., measuring health and healthcare disparities in terms of relative differences between rates without recognizing the way relative differences are affected by the overall prevalence of an outcome.
The Institutional Correspondence sub-page discusses the roles of governmental and nongovernmental institutions in promoting flawed research and serves as a repository of correspondence to institutions that are involved in some manner with the appraisal of differences in outcome rates. Such correspondence addresses with those institutions the problems with standard approaches to such appraisals.
Scanlan’s Rule Page
The Scanlan’s Rule page[i] (about 11,500 words) addresses various nuances of the statistical patterns discussed in the references on the Measuring Health Disparities page. See Scanlan’s Rule Outline and Guide. It has about twenty sub-page pages addressing particular issues.
The Framingham Illustrations, NHANES Illustrations, and Life Tables Illustrations sub-pages use the data referenced in their titles to illustrate the patterns of correlation between measures of differences between outcome rates and the prevalence of an outcome described in the introduction to the Scanlan’s Rule page.
The Employment Tests sub-page explores whether, given the theories generally expressed on the Measuring Health Disparities and Scanlan’s Rule pages, lowering a cutoff in fact reduces the disparate impact of a test in a meaningful way and explains why it does (assuming selection among persons who pass the test is not correlated with test scores). The Case Study sub-page uses a case study approach to illustrating some of the issues raised on the Scanlan’s Rule page and its sub-pages and the Case Study Answers sub-page provides answers to the questions posed.
The Subgroups Effects sub-page discusses the way observers mistakenly identify subgroup effects on the basis of the way factors are associated with different proportionate changes in the rates of groups with different base rates without recognizing the extent to which the different proportionate changes are functions of the different base rates or that the group with the larger proportionate decrease in an outcome will tend to have the smaller proportionate increase in the opposite outcome. That is, just as lowering a cutoff will tend to decrease the failure rate proportionately more for the higher-scoring group while increasing the pass rate proportionately more for the lower-scoring group, a factor that reduces mortality will tend to reduce mortality proportionately more for the group with the lower base rate while increasing survival proportionately for the other group. The page also explains how the approach described on the Solutions sub-page of the Measuring Health Disparities page can be used to appraise the size of the affect of an intervention as well provide a basis for estimating the absolute reduction in an outcome for a base rate other than that in the study forming the basis for the perception that the intervention reduces the outcome.
The Illogical Premises sub-page, which is related to the Subgroup Effects sub-page, explains why it is illogical to regard it as somehow normal that two groups with different base rates should experience equal proportionate changes in an outcome rate (given that it is not possible for two groups with different base rates to experience equal proportionate changes in such rates while also experiencing equal proportionate changes in rates of experiencing the opposite outcome). The point is addressed from a slightly different perspective in the Inevitability of Interaction sub-page, which explains why equal proportionate changes will never be observed as to either rate save on the rare occasion when a meaningful differential effect, by happenstance, causes the relative changes in some outcome to coincide.
The Comparing Averages sub-page explains why, irrespective of adjustment considerations, the issues discussed generally on the main Scanlan’s Rule page affect comparisons of an average of outcome rates for more than one sub-group with another average of outcome rates for more than one sub-group. The Meta-Analysis sub-page briefly explains that factors that tend generally to undermine subgroup analyses similarly undermine meta-analyses of effects on dichotomous outcomes. The Immunization Disparities sub-page addresses patterns by which various measures of disparities in immunization and other health care procedures tend to reach different conclusions with attention given the to the January 2011 Health Disparities and Inequalities Report of the Centers for Control and Prevention.
The Explanatory Theories sub-page addresses the way that researchers who believe they have identified a larger difference between rates in one setting than another may devise explanations for the perceived larger difference, usually without a sound basis for the perception that the difference is larger. The Truncation Issues sub-page, which is related to the Cohort Considerations sub-page of MHD, discusses why the patterns described in the introduction to the Scanlan’s Rule page may vary when the populations examined are truncated portions of larger populations, as well as reasons why the Solutions approach on MHD is unsuitable in such circumstances. The Representational Disparities sub-page explains why it is not possible to appraise the size of a disparity solely on the basis of the proportions a group comprises of persons eligible to experience an outcome and of persons who experience the outcome. The Case Control Studies sub-page addresses a fundamental problem with case control studies in that, while one may be able to derive an approximation of the relative risk from such study, one cannot derive the actual rates. The issue is related to that addressed in the Representational Disparities sub-page.
The Feminization of Poverty sub-page addresses the way that increases in the proportion of the population comprised by members of female-headed families are interpreted without recognition that decreases in the prevalence of an outcome will tend to cause groups particularly susceptible to the outcome to comprise a larger proportion of the population experiencing the outcome than they did previously (as well as a larger proportion of the population failing to experience the outcome). The subject is also treated in the narrative portion of the Scanlan’s Rule page (Sections B.1 and B.2) and many of the articles discussing the pattern whereby the rarer an outcome the larger tends to be the relative difference in experiencing it and the smaller tends to be the relative difference in avoiding it. The Statistical Significance SR sub-page explores whether, given that the same properties of normal distributions that underlie the patterns described on the Scanlan’s Rule page underlie methods for calculating statistical significance, a test of statistical significance given unchanged population size would meet the key criterion for an effective measure of the size of difference between outcome rates (i.e., that the measure remain unchanged when there occurs a change in overall prevalence akin to that effected by the lowering of a test cutoff) and shows why it does not.
(The Mortality and Survival sub-page had been an earlier version of what is now the Mortality and Survival page discussed below. It is retained solely to refer users of old links to the new page.)
The Semantic Issues sub-page discusses certain technical semantic issues that have some bearing on patterns described in the main Scanlan’s Rule page.
Mortality and Survival Page
The Mortality and Survival page addresses the way that, especially in cancer journals, researchers discuss disparities in mortality and disparities in survival interchangeably without recognizing that the two disparities tend to change in opposite directions as survival rates change generally or that where survival is high relative differences in survival tend to be small while relative differences in mortality tend to be large.
Measures of Association Page
The Measures of Association page briefly explains that the issues addressed with regard to the measures of health disparities and other subjects addressed on various pages of the site are involved in any effort to measure the strength of an association. The page is included as a reminder that fundamentals of epidemiology and every other discipline that is concerned with measures of association, however such matter is characterized, need to be reconsidered with an eye toward the manner in which accepted methods of measuring association are affected by the overall prevalence of an outcome.
Nuclear Deterrence Page
The Nuclear Deterrence page lists several published or unpublished articles on nuclear deterrence issues, most of which were written during the Cold War. The most important of these is Facing the Paradox of Deterrence (Midwest Quarterly, 1987). It addresses the problem with mutual assured destruction doctrine arising from the fact that once one side unleashes a massive first strike it would be irrational for the other side to retaliate. It proposes a solution to such problem whereby the United States would deny itself the option of declining to retaliate. The article also addresses the particulars of such solution.
Employment Discrimination Page
The Employment Discrimination page is divided into two parts. The first part addresses the fundamental flaw with claims of initial assignment discrimination or studies of job segregation at the firm level – i.e., that the observed patterns are of a nature that would exist whether or not the employer discriminated against applicants of any group. Links are provided to six articles on this subject. The most comprehensive of these is Illusions of Job Segregation (Public Interest, Fall 1988). The others mainly involve cases litigated in the 1990s involving initial assignment claims where plaintiffs were quite successful despite the flaws in such claims. The second part of the page provides links to seven articles on varied other employment discrimination issues, some of which involves the statistical issues addressed on the Measuring Health Disparities, Scanlan’s Rule, Mortality and Survival, and Measures of Association pages.
Affirmative Action Page
The Affirmative Action page has three sub-pages. The first provides links to three articles discussing reasons why certain important justifications for affirmative action for minorities do not apply to affirmative action for women. The most important of these articles is The Curious Case of Affirmative Action for Women (Society, Jan/Feb 1992). The second provides links to nine articles published in legal newspapers, and several article from the book Affirmative Action: An Encyclopedia, on varied other affirmative action issues. The third provides link to three articles on Justice John Paul Stevens and the evolution of his views on affirmative action issues, the most important of which is John Paul Stevens (Affirmative Action: An Encyclopedia, 2005).
Statistical Reasoning Page
The Statistical Reasoning page is divided into four parts. The first part contains links to articles addressing issues also covered on the Measuring Health Disparities, Scanlan’s Rule, Mortality and Survival, and Measuring Association pages. The second page contains links to articles addressing issues regarding perceived job segregation that are also covered in the first part of the Employment Discrimination page. The third part contains links to an article and a number of online comments concerning other statistical issues. The fourth part will eventually contain links to varied statistical items, though as yet it contains only one such link. The page also refers the reader to a number of statistical issues addressed on sub-page of the Vignettes page.
Vignettes Page
The Vignettes page is comprised of seven sub-pages addressing particular issues, usually of a statistical nature. The Times Higher Issues sub-page addresses the widespread custom of describing, for example, 3 as being three times greater than 1, rather than three times as great as 1. It provides tables showing the overwhelming predominance of the misusage even in scientific journals, with the notable exception of the New England Journal of Medicine. The sub-page also discusses several related points including the fact that most dictionary definitions of the word “multiplication” are incorrect and, in followed would yield 12 as the result of multiplying 3 times 3. The Gender Differences in DADT Terminations page discusses certain misperceptions in reportage that women are disproportionately discharged for violation of the military don’t-ask-don’t-tell policy, including the bases for the perception that women and disproportionately discharged and the bases for comparisons among the military branches (which involves the issue discussed on the Representational Disparities sub-page of the Scanlan’s Rule page). The Adjustment Issues sub-page addresses several points concerning approaches to adjustment of group differences in outcome rates for group differences in particular outcome-related characteristics, including the confusion between the standard adjustment for different prevalences of a characteristic within different groups and the determining what the differences between rates would be if the characteristic did not exist. The Percentage Points sub-page addresses the way researchers refer to percentage point differences as if they were percent differences. The Journalists and Statistics sub-page addresses a particular instance where, apparently as a result of confusion over the difference between the proportion of a group attending college and the proportion the group comprises of persons attending college, national magazines described the white college attendance rate as almost ten times the black college attendance rate rather than as about a third higher than the black college attendance rate. The Odds Ratios and Statistical Significance Vig sub-pages are as yet only sketches and do not warrant description here.
Miscellaneous Pages
The Sears Case, AT&T Consent Decree, and Cross v. ASPI pages provide information on three cases in which the site owner was involved. The first two of which – one a fully tried nationwide gender discrimination case against what was then the nation’s largest retailer and the other a consent decree establishing an affirmative action program covering the employment practices of what was the nation’s largest private employer –were quite prominent cases in their time and continue to be occasionally discussed. The third raised some interesting issues about the ability of juries to consider complex matters.
The Prosecutorial Misconduct and the Misconduct Profiles pages make available a great volume of narrative and documentary material relating to prosecutorial misconduct by attorneys in the Office of Independent Counsel Arlin M. Adams and Larry D. Thompson in United States v. Deborah Gore Dean, an Independent Counsel case tried in the District of Columbia in 1993. The latter page includes sub-pages on the roles in the Dean prosecution of Robert E. O’Neill (recently appointed United States Attorney for the Middle District of Florida), Bruce C. Swartz (currently Deputy Assistant Attorney General of the Criminal Division in the United States Department of Justice in charge of international issues), Jo Ann Harris (formerly Assistant Attorney General for the Criminal Division and currently Scholar-in-Residence at Pace University Law School), Arlin M. Adams (former United States Circuit Judge, name source of the Arlin M. Adams Center for Law and Society of Susquehanna University and the Arlin M. Adams Constitutional Law chair at the University of Pennsylvania Law School, and recipient in 2011 of the Benjamin Franklin Medal of the American Philosophical Society), Paula A. Sweeney (currently Deputy General Counsel of the Central Intelligence Agency), and Robert J. Meyer (currently partner at Willkie Farr & Gallagher LLP). Most of twenty-two items published on prosecutorial misconduct issues on the editorial blog of the organization Truth in Justice, links to which may be found here, relate to the Dean case and the referenced attorneys who prosecuted it.
The Lantos Hearings page addresses the 1989-1990 hearings of the Employment and Housing Subcommittee of the Committee on Government Operations of the House of Representative, chaired by Congressman Tom Lantos, into abuses of HUD’s moderate rehabilitation program.
Last update of this page: December 29, 2011.
[i] In 2006, I began terming the pattern whereby the rarer an outcome the greater tends to the relative difference in experiencing it and the smaller tends to be the relative difference in avoiding it as “heuristic rule x” or “interpretive rule 1.” In 2008, Bauld et al. (discussed in Section E.7 of MHD) termed the pattern “Scanlan’s rule,” which usage, somewhat modified. I have since employed for certain purposes.
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