PERCENTS AND PERCENTAGE POINTS
(Dec. 9, 2009)
Note: This item concerns the way that researchers use the term “percent” in discussing absolute differences between rates, – i.e., 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 alone provide useful information about the comparative size of two differences between rates, as discussed on the Measuring Health Disparities and Scanlan’s Rule pages of this site. With regard to understanding the degree to which a factor increases or decreases an individual’s risk of an adverse or favorable health outcome – and with respect to which I share the view of many that the absolute difference between rates where the factor is or is not present is the crucial criterion for decision-making – the need for care in distinguishing percents from percentage points may be greatest. But the pages on this site largely deal with other issues.
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 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 note (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. 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 the usage of “percentage point” simply as a means of clarifying an ambiguity.
But, while references that discuss the usage issue are generally of the views just described, one will find many instances on the Internet (and no doubt elsewhere) where, even in the discussion of the difference between relative and absolute differences, both differences will be described as percent differences, as, for example, in the American College of Physicians’ online guide concerning the difference between relative and absolute differences. And one will find many scientific articles where the authors will use “percent” where 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 “percents” pertain to percentage points without reading the underlying articles.[ii]
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, where the issue involves change over time. But they 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);[iii] 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
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Year
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AG Rate
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DG Rate
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Relative Difference
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Absolute Difference
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One
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20%
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37%
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85%
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17 percentage points
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Two
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10%
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22%
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120%
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12 percentage points
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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 2006 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, 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 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 and SR (especially the introduction to SR) 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 that is the subject of reference D40 on the Measuring Health Disparities page.
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% (P<.001 for both race-year interactions). 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 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 D40, see the articles that are the subjects of references D25, D27, D64, and D74. Compare with the article that is the subject of D73, which is very careful in its usage.
One noteworthy example of the use of percent changes when discussing percentage point changes involves the National Healthcare Disparities Report, published yearly by the Agency for Healthcare Research and Quality (AHRQ). The report measure disparities in terms of relative differences between rates.[iv] 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 used to measure health and healthcare disparities and with respect 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 it 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 disparity 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.[v] 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 National Healthcare Disparities Report illustrates the extent of the size of differences in characterization that might result depending on whether on 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][vi] 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.[vii]
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% 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 do so. 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] 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] See Sehgal AR. Universal health care a health disparity intervention/ Ann Int Med 2009;!50:561-562, which comments on, inter alia, the subject of item D74 on MHD.
[iii] 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 MHD and SR 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. Further with regard to semantic aspects of relative differences, many refer to the figures 1.85 and 2.20, which are ratios by which the relative differences are determined, as the “relative differences.” But these figures, which are appropriately termed “risk ratios,” and which may or may not be appropriately termed the “relative risk” (I remain uncertain on the point), is different from the “relative difference.”
[iv] I discuss varied measurement problems with the report in a number of places, including the APHA 2007 presentation and A.6 of the Scanlan’s Rule page. Certain technical issues are addressed in the NHDR Technical Issues sub-page of MHD.
[v] For purposes of the Health 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, the use of absolute differences between relative differences. However, the Midcourse Review is careful to term these differences “percentage point” differences.
[vi] While AHRQ usually frames what is calls “core measures” in terms of favorable outcome, it relies on whichever relative difference (in the favorable or the adverse outcome is larger) is larger. This is usually the adverse outcome. In the case of this highlighted item, it 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.
[vii] 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 the 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 changed. 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% increase over three years; a 20% yearly decrease would translate into a 49% decrease over three years. See discussion in the NHDR Technical Issues sub-page of SR regarding an error in my communication to AHRQ on this point.
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