This is one of several subpages to the Vignettes page of jpscanlan.com principally addressing presentation issues.Other such subpages include the Times Higher and the Journalists and Statistics subpages.The former addresses the way that scientists and other observers incorrectly describe a rate such as 3% as three times higher than a rate of 1% and certain related issues, including that most dictionary definitions of multiplication are incorrect.The latter 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 Mortality and Survival page of the site, which treats the way that medical journals discuss relative differences in mortality and relative differences in survival interchangeably without recognizing that the two tend to change systematically in opposite directions as survival generally increases or decreases, involves a presentation issue as well.But that page principally concerns the more substantive matter of the failure generally to recognize that relative differences in experiencing an outcome and relative differences in avoiding the outcome tend to change in opposite directions as the prevalence of an outcome changes or that neither relative difference is a useful measure of the size of a disparity.
This item concerns the way that researchers use the term “percent” or the percent sign (“%” in discussing absolute differences between rates, – that is, when they should be using the term “percentage point.”Such usage can lead to misunderstandings about the size of difference between rates and the size of changes in those differences, as well as, in the common circumstances where relative and absolute differences change in opposite directions, misunderstandings as to the directions of changes over time.Despite the attention I give to this point here, however, the discussion should not be read to suggest that either relative differences or absolute differences can aloneprovide useful information about the comparative size of two differences between the circumstances of an advantaged and a disadvantaged group reflected by their rates of experiencing or avoiding some outcome, as discussed on scores of pages on this site and most comprehensively in “Race and Mortality Revisited,” Society (July/Aug. 2014), the November 2013 Federal Committee on Statistical Methodology 2013 Research Conference page “Measuring Health and Healthcare Disparities” and the September 2013 University of Kansas School of Law faculty workshop paper “The Mismeasure of Discrimination.”
This page is referenced at page 16 n. 24 of a July 1, 2015 letter to the Director of the Agency for Healthcare Research and Quality that discusses the substantive issues addressed in the references just mentioned as well as certain anomalies in recent National Healthcare Disparities Reports (NHDRs) arising from modification of methodology beginning the 2010 report.Those anomalies may result from the reports use of “%” in discussing both percent (relative) changes or differences and percentage point (absolute) changes or differences.But the following, not mentioned in the letter, also warrants note.The text below discusses the possibility that the use of percent or % to refer to percentage point changes or differences is so widespread that one may no longer employ the correct usage to refer to a relative change or difference and expect to be properly interpreted.As it happens one of the instances where the NHDR uses % with respect to a relative change or difference – that is, the correct usage – is found where the report explains that a healthcare disparity will be deemed meaningful where there is a 10 percent (relative) difference between rate rates of the groups being compared.In doing so, for some reason, the report added the word “absolute value” in a place where the term can have no meaning of which I am aware, as at page 28 (emphasis added): “Second, the relative difference between the comparison group and the reference group must have an absolute value of at least 10%.”Whether or not the reason lies in the insertion of the term “absolute value,” the Center for Medicare & Medicaid Services (CMMS), in its 2015 National Impact Assessment (at 167-68), erroneously read the 2013 NHDR as identifying meaningful disparities on the basis of 10 percentage point differences in rates (while CMMS adopted a standard of 5 percentage points).
When one compares an outcome rate of 20 percent with one of 22 percent,[i] most observers who are careful about scientific usage would say either that the latter is 10 percent greater than the former (when discussing relative differences) or that the latter is 2 percentage points greater than the former (when discussing absolute differences between rates). The American Medical Association Style Manual is clear on this point. After explaining the meaning of a percent difference and a percentage point difference, the manual notes (Section 19.7.2 (at 831): “The two terms are not interchangeable.” (Original emphasis). Discussions of the distinction on the Internet seem generally to express views either that a percent difference (or change) means one thing and a percentage point difference (or change) means another, or that, given the ambiguity, one should use the term “percentage point” difference (or change) when one is discussing the absolute difference between rates. See, e.g., Link1 and Link 2. The recent book Know Your Chances: Understanding Health Statistics, by Steven Woloshin, Lisa M. Schwartz, and H. Gilbert Welch, is very careful in explaining the difference and in its use of percentage points to describe absolute differences between outcome rates, though (at 48) it refers to the usage of “percentage point” simply as a means of clarifying an ambiguity.
One will also find many scientific articles where the authors use “percent” when referring to absolute differences, and do so, moreover, without evidencing an awareness of the ambiguity and leaving the reader to pore over the article to find out what the authors mean. In commentaries that do not go deeply into the data in the articles on which they are commenting it may be impossible to know that their references to “percent” pertain to percentage points without reading the underlying articles.[iv]
Indeed, the use of the term “percent” to describe percentage points is common enough that use of “percent” to describe a relative difference or change – arguably the only correct usage – itself may often involve an element of ambiguity. And in circumstances where the context fails to make clear that percent is used to describe a relative difference or change, persons using the term in that manner would be wise to take steps to make sure that their meaning is understood.
The example above involved the simple comparison of two rates. But most of the discussions of differences between rates on this site, particularly on the Measuring Health Disparities page (MHD) and Scanlan’s Rule page (SR), involve comparison of the size of differences between outcome rates in different settings. Those settings are often differentiated temporally, as where the issue examined involves change over time. But the settings are also differentiated many other ways, for example, where the issue is whether some difference is larger in one country than another, among one population subgroup than another, or with respect to one outcome than another.
So suppose that in Year One an advantaged group’s (AG’s) rate of experiencing some adverse outcome is 20 percent and a disadvantaged group’s (DG’s) rate of experiencing the outcome is 37 percent and in Year Two AG’s rate is 10 percent and DG’s rate is 22 percent. In Year One, in relative terms, DG’s rate is 85 percent greater than AG’s ((37/20)-1);[v] in absolute terms, DG’s rate is 17 percentage points greater than AG’s (37-20). In Year Two, in relative terms, DG’s rate is 120% greater than AG’s rate ((22/10)-1); in absolute terms DG’s rate is 12 percentage points greater than AG’s.
Table A: Example of changes in relative and absolute differences between outcome rates
17 percentage points
12 percentage points
The figures are set out in Table A above. For purposes of verisimilitude, they are based on Table 1 of a 2006 British Society for Population Studies presentation and reflect a pattern along the lines of what one might expect in circumstances where, given the rates for both groups in Year One, AG’s rate declined by half – that is, a pattern where the means of the underlying distribution of factors associated with an outcome (or its absence) differ by half a standard deviation.
The table shows that the size of the difference between rates changes in different directions from Year One to Year Two depending on whether one examines the relative difference or the absolute difference. I note also that one can derive from the figures the fact that the relative difference in failing to experience the outcome changed in the opposite direction from the relative difference in experiencing the outcome (and hence in the same direction as the absolute difference, which difference is always the same whether one examines one outcome or its opposite). MHD (especially its Relative Versus Absolute subpage) and SR (especially its introduction) and the references they make available explain why such patterns will commonly occur, as well as why, given the rate ranges at issue, the relative difference in experiencing the outcome and absolute difference tend to change in opposite directions. But the issue of concern here involves clarity of expression. And the fact that directions of changes over time may vary depending on whether one relies on relative or absolute differences is a reason why it is important to be clear as to what measure is being used.
Despite the importance of distinguishing percent differences from percentage point differences, and notwithstanding a great deal of literature emphasizing the importance of using the term “percentage point” rather than “percent” when one is describing an absolute difference, one still will find in scientific journals (as noted above) a great deal of discussion of group differences and changes in those differences over time that use the term “percent” when discussing percentage points. The following is an example from the article[vi] that is the subject of reference Comment on Sequist Arch Int Med 2006.
“Rates of annual low-density lipoprotein cholesterol level testing increased from 39% to 64%, while the white-black disparity decreased from 14% to 4%; rates of low-density lipoprotein cholesterol level control increased from 15% to 43%, while the white-black disparity decreased from 9% to 6% … Statin therapy rates increased from 20% to 37%; however, black patients remained less likely than white patients to receive therapy. The 1997 rates of annual glycosylated hemoglobin level testing (76%) and annual eye examinations (74%) were high, and there was no white-black disparity over time. Rates of glycosylated hemoglobin level control remained low (31%), and the white-black disparity remained constant at 10%.”
The above is a useful example because the overall prevalence figures are in ranges where it would be plausible for the disparity figures to reflect either relative or absolute differences. It would eventually be clarified in the text of the article that all references to “%” differences actually involve absolute differences (that is, for example, the first sentence concerns a situation where a 14 percentage point disparity declined to a 4 percentage point disparity). It warrants note that this article was from Archives of Internal Medicine, an AMA publication, which one would expect to follow the AMA style manual. Such usage in any case is common among researchers who rely on absolute differences between rates. In addition to the article that is the subject of reference Comment on Sequist noted above, see the articles that are the subjects of Comment on Sehgal JAMA 2003, Comment on Trivedi NEJM 2005, Comment on Ashworth BMJ 2008, Comment on McWilliams Ann Int Med 2009. Compare those articles with the article that is the subject of Comment on Kanjilat Arch Int Med 2006, which is very careful in its usage.
In November 2013, I created the Spurious Contradictions subpage of MHD, which discusses a situation where a study opines about the seemingly contrary findings of two healthcare disparities studies without recognizing that one study relied on relative differences as a measures of disparity while the other relied on absolute differences and that the results of the two studies were in fact very similar. The study addressing the seeming contradiction refers to its own findings of disparities in outcome in “%” terms without providing any information that would allow one to determine whether it meant a percentage difference or a percentage point difference (though I believe it likely that it meant the latter). The failure of the study to distinguish between percents and percentage points with respect to its own findings may be related to the study’s failure to recognize the difference between the measures used in the two studies that it opined about.
One noteworthy example of the use of percent changes when discussing percentage point changes involves the National Healthcare Disparities Report (NHDR), published yearly by the Agency for Healthcare Research and Quality (AHRQ). The report measure disparities in terms of relative differences between rates.[vii] But it measures changes in relative differences in terms of absolute differences. That is, in the example set out in Table A, AHRQ would appraise the change in terms of an 85 percent relative difference in Year One and the 120 percent relative difference in Year Two. But it would appraise the size of that change, not in terms of the 47 percent increase in the relative difference ((125/85)-1), but the 35 percentage point increase in the relative difference.
Invariably, however, the report uses the term “%” in referring both to the relative differences between rates that it uses to measure health and healthcare disparities and to the absolute differences (between relative differences) that is uses to measure changes over time. Thus, when it refers to a “10%” difference from a reference group’s rate as a criterion for identifying an important disparity, it means a relative difference (page 22, 2006 Report). But when the report discusses that it treats as a change over time only those situations where a disparity increased by “1% per year,” it in fact means situations where the relative difference between rates increased by 1 percentage point per year. The same holds where the report describes changes as being between 1% and 5% or greater than 5%, as in Figure 4.21 of the 2006 report.[viii] In fact, if rather than increasing, the disparity had completely disappeared, as where, say, in Year Two of Table A, the rates of both groups were 10 percent, the report would describe that as an 85 percent decrease in the disparity rather than a 100 percent decrease in the disparity.
The 2006 NHDR illustrates the extent of the size of differences in characterization that might result depending on whether one relies on percent reductions or percentage point reductions in the disparities. The report highlights the following information (at 6):
“From 2000 to 2003, the proportion of adults who [failed to receive][ix] care for illness or injury as soon as wanted decreased for Whites (from 16.2% to 13.4%) but increased for Blacks (from 17.5% to 18.4%). This corresponds to an increase of 9.8% per year in this disparity. However, from 2000 to 2004, the rate of new AIDS cases remained about the same for Whites (from 7.2 to 7.1 per 100,000 population age 13 and over) but decreased for Blacks (from 75.4 to72.1 per 100,000 population), corresponding to a decrease of 7.9% per year in this disparity.”
The discussion of the first point involves an initial black-white ratio of 1.08 (17.5/16.2) and a final black-white ratio of 1.37 (18.4/13.4). Thus, an 8% greater black rate increased to a 37% greater black rate. That translates into a 29 percentage point increase (37 - 8) or proximately 9.8 percentage points per year over three years. But it translates in a 363% relative increase (29/8), which is approximately a 65% yearly increase.[x]
The discussion of the second point involves an initial black-white ratio of 10.47 (75.4/7.2) and a final black-white ratio of 10.15 (72.1/7.1). Thus, a 947% greater black rate decreased to a 915% greater black rate. That translates into a 32 percentage point decrease or approximately 7.9 percentage points per year over four years. But it translates into a 3.3% relative decrease (32/947), which is just a little less than 1% per year.
In this instance the underlying figures are set out, so an observer can divine that the changes being discussed are actually percentage point changes. But not everyone who reviews the information, including those who review the information and report on it, will necessarily look at those figures and think through their implications. And in circumstances where, say, the 2004 black-white ratio of 10.15 either increased to 11.15 or decreased to 9.15, many who learned that the relative difference increased by 100 percent or decreased by 100 percent would derive understandings of the changes substantially different from the reality.
I used the above example from the 2006 NHDR in an early version of this page. The 2009 NHDR provides a like example, though in this case without setting the numbers from which one might recognize that the “%” references were to percentage points. At page 6, the report states:
“The largest disparities for Blacks, AI/ANs, and Hispanics included the rate of new AIDS cases. The rate for Blacks was almost 10 times as high as the rate for Whites, for Hispanics more than 3 times as high, and for AI/ANs 1.4 times as high. However, from 2000 to 2007, for Blacks, AI/ANs, and Hispanics, this measure was among those with the greatest reduction in disparities for each group (10.2% per year, 2.7% per year, and 4.2% per year, respectively; Table H.1).”
The underlying figures actually show that between 2000 and 2007 the black rate went from being 943% higher than the white rate to 871% higher than the white rates (a 72 percentage point and 7.6% decrease over the seven years); the AI/AN rate went from being 51% greater than the white rate to 41% greater than the white rate (a 10 percentage point and a 19.6% decrease over the seven years); and the Hispanic rates went from being 277% greater than the white rate to 234% greater than the white rates (a 43 percentage point decreases or a 11.4 percent increase over the seven years). The 2.7 AI/AN figure in the report is apparently simply an error, since the 10 percentage point decrease divided by 7 years is 1.4 See note x regarding the reasons why the yearly percentage point change will simply be the total percentage point change divided by the number of years while the yearly percent change will not simply be the total percent change divided by the number of years and why the percent change figure will be smaller than the percentage point change figure when the percent difference is greater than 100 (as in the situation for differences between rates of whites and blacks or whites and Hispanics) and will be larger than the percentage point change figure when the percent different is less than 100 (as in the situation for differences between the rates of whites and AI/ANs).
[i] In medical journals “percent” is almost always presented as “%.” Nevertheless, for purposes of drawing the distinctions here, I think it more useful to spell out the term and hence do so save when quoting a usage.
[ii] The first two entities use essentially the same language. It is not know which borrowed from which. In any case, after explaining that one would derive the absolute difference between a two percent risk of blindness for diabetics treated conventionally and a one percent risk of blindness for diabetics treated intensively by subtracting the latter from the former, the Guide states: “Expressed as an absolute difference, intensive therapy reduces the risk of blindness by 1%.” Although that matter is unrelated to the instant issue, I note that these guides describe a relative difference as “the ratio of two risks.” In explanation, the guides show 1% over 2% and refers to the resulting 50% as the relative difference. But the ratio is actually the “risk ratio” or “relative risk.” The relative difference is the risk ratio minus 1 for ratios above 1 and 1 minus the risk ratio for ratios below 1. By coincidence 50% happens to be both the risk ratio and the relative difference, since 1 minus 50% is 50%. But the presentation problem would have been evident had the risks been 1% and 3%, which would have resulted in a risk ratio of 33% (more often presented as .33) and a relative difference of 67%. The reference to risk ratios or relative risks as relative differences seems to be fairly common. One also finds it in the University of Michigan CD Rom Course Measuring Health Disparities (page 29 of text version).
[iii]The Times Higher subpage discusses some instances where the misusage led to misinterpretations. Using “percent” when describing an absolute difference between rates of course commonly leads to confusion. But this entity’s usage may contribute to its own misinterpretation. After describing a situation whereby a drug would reduce a 10-year heart attack risk from 90 percent to 60 percent, it states that the drug will “save 30% of patients from a heart attack over the next 10 years that would have otherwise occurred.” While the drug will save 30% of the population at risk from a heart attack it will save 33% of the persons who would have had attacks.
[v] With respect to the relative differences, issues arise both as to whether the difference should be based on the rates of experiencing the favorable outcome or rates of experiencing the adverse outcome (which often, as here, may affect whether a disparity is deemed to have increased or decreased over time) and as to which group’s rate should be numerator of the fraction used to calculate the relative difference (which will affect the size of the relative difference but not the direction of change over time). The first point is a subject of much discussion on the MHD and SR pages and the sources they reference and the latter point is the subject of the Semantic Issues sub-page of SR as well as the Addendum to the 2007 APHA presentation. But these issues are not pertinent to the instant subject.
[vi] Sequist TD, Adams AS, Zhang F, Ross-Degnan D, Ayanian JZ. The effect of quality improvement on racial disparities in diabetes care. Arch Intern Med 2006;166:675-681.
[vii] I discuss varied measurement problems with the report in a number of places, including the APHA 2007 presentation and Section A.6 of the Scanlan’s Rule page. Certain technical issues are addressed in the NHDR Technical Issues sub-page of MHD.
[viii] For purposes of the Healthy People 2010 Midcourse Review, the method of appraising changes over time is exactly the same as that used by the AHRQ in the National Healthcare Disparities Report – that is, absolute differences between relative differences. However, the Midcourse Review is careful to term these differences “percentage point” differences.
[ix] While AHRQ usually frames what it calls “core measures” in terms of favorable outcomes, in measuring disparities it relies on whichever relative difference (in the favorable or the adverse outcome) is larger. This is usually the adverse outcome. In the case of this highlighted item, the report mistakenly presented the figures in the terms of the core measure – receipt of care as soon as wanted. But the figures presented were actually the rates for failure to receive care as soon as wanted.
[x] In note xix on page 5 of the 2006 report, AHRQ indicates that is determining what it calls a “percent change” per year by dividing what it terms the “percent change” for the entire period examined by the number of years. This in fact is an appropriate way of determining yearly percentage point changes from an overall percentage point change. So AHRQ’s method is appropriate for determining the yearly percentage point changes that it in fact is examining. When one is speaking of actual relative changes, however, it would not be correct to determine the yearly percent change by dividing the total percent change by the number of years examined since each year the base rate is different. In the case of a yearly increase, as with regard to the change in the relative difference between rates of failing to receive care as soon as needed, the yearly increase is less (and in this case much less) than the figure that would be derived by dividing the all-years change by the number of years. When the change is a decrease, however, the yearly change would be more than the figure that would be derived by dividing the all-years figure by the number of years. By way of illustration, a 20 percent yearly increase would translate into a 73 percent increase over three years; a 20 percent yearly decrease would translate into a 49 percent decrease over three years. See discussion in the NHDR Technical Issues sub-page of MHD regarding an error in my communication to AHRQ on this point.