James P. Scanlan, Attorney at Law

This page principally addresses the extent to which observers carelessly describe a situation where one figure is X times as high as another as if the first figure is X times higher than the other.  Section A address the matter generally.  Section B addresses the matter with regard to medical, health policy, and scientific journals.  Section C addresses two related issues.  One involves situations where a rate that is X percent greater than another is described as being 100 + X percent greater than the other.  The other involves the standard definition for multiplication, which, if implemented, would always cause the multiplicand to be increased by an additional 100% (that is, for example, to cause 3 times 3 to equal 12 rather than 9).

A.  Times Higher Versus Times as High - General

When Group A’s rate of experiencing some outcome is 3% and Group B’s rate of experiencing the outcome is 9%, it could properly be said that Group B’s rate is 3 times as high, or 3 times as great, as Group A’s rate.  It could also properly be said that Group B is 3 times as likely to experience the outcome as Group A.  And it could properly be said that Group B’s rate is 200% greater than Group A’s rate. 

Few people would seriously contend that Group B’s rate is in fact 3 times higher than Group A’s rate or that Group B is 3 times more likely than Group A to experience the outcome, just as few would seriously contend that Group B is 300%, rather than 200%, more likely to experience the outcome.[i]  Thoughtful observers would point out that 3 times higher means the same thing as 4 times as high. 

Nevertheless, in the circumstances of the figures just given, observers commonly state that Group B’s rate is 3 times higher or greater than Group A’s rate or that Group B is 3 times more likely to experience the outcome. 

Since 1991, this issue has been addressed in Philip Meyer’s classic Precision Journalism, though Professor Meyer notes that the Associated Press had defended the “times greater” usage as a “’harmless ‘mathematical colloquialism.’”  At least since 1999, the issue has been addressed in The New York Times Manual of Styles and Usage (1999 edition at 335), which note that precise readers would regard a usage such as “three times more” to mean the same thing as“four times as.”  The manual therefore advises against the “times more” usage.  The issue has also been addressed in the National Geographic Style Manual, which notes in the 2006 online version that “three times larger” is ambiguous, since some read it to mean “three times as large as” and others read it to mean “four times as large as.”  Like the New York Times manual, the National Geographic manual therefore states that the “times larger” usage should be avoided.    

The issue also receives occasional attention online, where observers generally insist that usages like “times higher” are either incorrect or ambiguous.  See Link1, Link2, Link3, Link4, Link5, Link 6, Link7.

Link1 is interesting both for the clarifier’s explaining that “three times larger than N” means “four times N” – “but only when you think about it” – and its recording of the unfortunate circumstance where a student’s answer was found incorrect because the student, though not the teacher, interpreted the question in the way that a precise reader would.

Despite the consistency with which authorities discourage usages like “times higher,” such usages are far more common than usages like “times as high.” 

Table 1 below sets out counts of web hits for various pairs of phrases akin to “times higher” and “times as high,” which are based on Google searches for exact phrases in late March 2009.   In the table, the former phrase is termed the “incorrect usage” and the latter the “correct usage” (though the reader should keep in mind that the former usage is only incorrect when, for example, 9 is described as 3 times higher than 3, not when 9 is described as 2 times higher than 3).  The Ratio 1 field shows the ratio of the incorrect usage to the correct usage.  The Ratio 2 field, which is populated only in the infrequent situations where the correct usage is more common, shows the ratio of the correct usage to the incorrect usage.[ii] 

An effort to explore the reasons for the degree of predominance of the incorrect usage with regard to different phrases (and in some cases the predominance of the correct usage), as well as the effect of the inclusion of additional words in the search – as, for example, the word “quiz,” which tends to limit the search to situations where the phrases are employed in a quiz question – is not yet complete.

It should be recognized that this search undoubtedly captures a fair number cases that do not actually involve the usage at issue (as in “circulation of Times higher than Post”).  Possibly some of the usages like “times higher” in fact involve situations where, for example, 9% is described correctly as 2 times higher than 3%, though probably these cases are few.  Despite some inexactness, however, the table very likely gives a pretty fair picture of relative rates of the usages.      

An interesting aspect of this issue is that it does seem that a usage like “times higher” better captures the matter of interest.  That is, one is interested in how much greater than zero one number is than another, and a usage like “times higher” or “times greater” seems to convey that information better than something like “times as high” or “times as great.”  The problem is that, as commonly used, “times higher,” “times greater,” and the like inform the reader or listener that one figure is one more time higher than another than it actually is.  The careful writer, however, will avoid the “times higher” usage to state, for example, that one group’s rate is two times higher than another when the rates are 9% and 3%, because too many people will regard the usage as indicating that one figure is only two times as high as the other.

Some might maintain that characterizing, for example, a rate of 3% as three times greater than a rate of 1% is simply a convention.  A problem with the argument, however, is that not everyone will recognize the convention.  The thoughtful student mentioned in Link1was either unaware of any putative convention or believed that on an examination logic ought to trump convention.   See also note i.

B.  Times Higher versus Times as High - Scientific

Materials on this site are frequently critical of statistical analyses in scientific journals because of the near universal failure to recognize the ways differences between rates are affected by the overall prevalence of an outcome (as discussed on the Measuring Health Disparities, Scanlan’s Rule, and Mortality and Survival pages of this site).  But irrespective of the validity of such criticism, one still would expect such analyses to reflect precision in the expression of mathematical relationships.  Writers of articles in scientific journal generally have advanced degrees and generally are of far above average intelligence.  Moreover, the great majority of scientific articles have multiple authors, most of whom would be expected at least to review the manuscript.  And one would expect some number of them to find a description of 9 as three times higher than 3 to be jarring in the same way that they would find a grammatical error jarring.  The same holds for peer reviewers and the editors of the journal itself.  And, assuming anyone grasps that a usage like “times higher” is incorrect and may mislead even a small number or readers (or mislead journalists who report on findings in an article) it is hard to understand the thinking that would allow one to continue to employ that usage. 

Nevertheless, with the exception of the New England Journal of Medicine (NEJM), the predominance of incorrect usage with regard to the quantification of a risk ratio is striking.  Table 2 sets out information similar to that in Table 1, but with the counts derived by means of the search functions of the individual journals.  As with Table 1, where the counts are of web pages rather than individual usages, the counts in Table 2 reflect numbers of articles containing the phrases rather than numbers of times the phrases are used. 

The counts are mainly limited to the three most common pairs of phrases of interest (times higher/times as high; times more likely/times as likely; times greater/times as great). 

The counts for the first two pairs are from March 2009.  Based on the low rates of usage for either “times greater” or “times as great” in the NEJM I initially did not include that pair in the table, except with respect to the NEJM.   Later review of the matter revealed that other journals employ this usage much more commonly (typically in the “times greater” form).  So the “times greater/times as great” pair was added in October 2009 and counts for that pair are from October 2009.

Efforts to determine whether these patterns might have changed over time did not reveal anything apparent in that regard except in the case of the NEJM.  Material regarding that issue was previously included in this item but has been moved to here.  The thrust of that material is that apparently the NEJM began to give serious attention to this issue in the late 1990s, but has not always been as careful as it might be.  A question is why, having apparently recognized the issue and having decided to address it, the journal would ever permit the incorrect usage.

The Journal of the American Medical Association (JAMA) warrants mention with regard to a different matter.  The American Medical Association has its own extensive style manual, The AMA Manual of Style, which endeavors to treat most issues that JAMA’s contributors might face.  It does not seem to intend to address the particular usage issue discussed here and in at least one place seems to employ the correct usage.  In the discussion of the use of numerals to express numbers (at 822 (2007 ed.)), it states as an example:  “The relative risk of exposed individuals was nearly 3 times that of the controls.”[iii]  It warrants note, however, that what the manual in this instance refers to as “relative risk” is actually the “risk.” 

But in another place (at 891 (2007 ed.)), after defining “relative risk (RR)” as  “probability of developing an outcome within a specified period if a risk factor is present, divided by the probability of developing the outcome in the same period if that risk factor is absent. …,” the manual provides as a clarification the following language:

“Example: The individuals with untreated mild hypertension had a relative risk of 2.4 (95% confidence interval, 1.0 to 3.0) for stroke or transient ischemic attack.   [In this example, individuals with mild hypertension were 2.4 times more likely than were individuals in the comparison group to have a stroke of transient ischemic attack.]”

Emphasis added; brackets in the original.[iv]

It is doubtful that such guidance plays a significant role in the predominance of the incorrect usage in JAMA (where the journal is the second worst offender on “times higher,” though near the median on “times more likely”).  But the fact that a putatively clarifying entry in the journal’s style manual employs the usage incorrectly suggests that the editors have given the matter little attention.

As in the case of the NEJM, it seems that the Agency for Healthcare Research and Quality has given the issue some attention in the yearly National Healthcare Disparities Report.  After commonly employing the incorrect usage through the 2005 report, the agency appeared to have largely corrected the problem in the 2006 report.  This suggested that the agency had given the issue some attention.  Nevertheless, in the 2007 report, while “times as likely” and “times more likely” are both used some number of times, in all 30 instances where the report quantitatively describes one rate as being some number of times as higher or higher than another, it uses the phrase “times higher.”

C.  Related Issues

Subsections 1 and 2 below discuss two other situations where in some manner advice is given that erroneously increases a figure by an additional 100% of an underlying value.

            1.         Percent greater

In explaining why, say, 2 times larger is not the same thing as 2 times as large to those to whom the matter is not obvious, it seems useful to point out that when one figure is twice the size of another it should be self-evident that the figure is 100% larger, not 200% larger.  Thus, it seems unlikely that very often anyone would in fact say that a risk ratio of 2.0 means that one figure is 200% greater than another (unless, as was possibly the case with respect to the item discussed in note i, and certainly the case as to the student discussed in Link1, the person is misled by the characterization of a 2.0 risk ratio as indicating that one figure was two times greater than another).  But such things apparently do happen.

For example, the very popular University of Michigan DC-ROM Course Measuring Health Disparities, in the context of a situation where the black death rate from nephritis is 30 per 100,000 and the white rates is 12 per 100,000, the course explains that this means that “blacks are 250% more likely to die as a result of nephritis than whites.”  Rather clearly, however, black are only 150% more likely to die as a result of nephritis than whites.

(In the same paragraph the course also describes a “relative risk” of 2.5 as a “relative difference.”  The confusion of these terms if fairly common (see discussion of Mayo Clinic guidance in December 14, 2013 prefatory note above).  But the relative difference in fact is the  relative risk minus 1, hence, 1.5 (or 1 minus the relative risk when the relative risk is below 1). See the Case Study sub-page of the Scanlan’s Rule page of this site.) 

If such mischaracterization occurs in a document that is intended to explain the meaning of differences between rates, the mischaracterization presumably occurs more commonly in other circumstances.  But such mischaracterizations cannot be identified by an internet search as easily as usages like “time higher” because a search for a phrase like “percent higher” will identify instances where, for example, a rate of 15% is correctly described as 50 percent higher than a rate of 10% and instances where it is incorrectly described as 150% higher. 

            2.         Dictionary Definitions of Multiplication

[An expanded version of the material in Section C.2 of this subpage now appears on a separate Multiplication Definition subpage.] 

On the web site freedictionary.com, one finds three mathematical definitions for the word “multiplication” (from the American Heritage Dictionary of the English Language, Fourth Edition (2000), the Collins Essential English Dictionary 2d Edition (2006), and the American Heritage Science Dictionary, which are set out as items 1-3 below.   Dictionary.com sets out, in addition to two of the three that are found on freedictionary.com, three additional definitions (from the Random House Dictionary, Webster's Revised Unabridged Dictionary (1996, 1998), and MICRA, Inc., WordNet (2006) (of Princeton University)), which are set out as items 4-6 below.

It will be observed that all but the Princeton definition in some manner define multiplication as the process of adding a number to itself a certain number of times.  All at least imply that that number of times is the number by which the original number (multiplicand) is then multiplied (what is commonly called the multiplier, though none of the definitions uses the word).  Items 3 and 4 are explicit in this regard – that is, in stating that to multiply x by y means to add x to itself y  number of times.  The former states, for example, that “multiplying 6 by 3 means adding 6 to itself 3 times.”  Apparently this definition has been around for some time.  A 1970 version of Webster’s Seventh New Collegiate Dictionary also states that multiplication involves adding a number to itself a certain number of times.  

But each definition employing such usage is patently incorrect.  To multiply x by y is either to add x to 0 y number of times or to add x to itself y-1 number of times.  Applying the definitions would, as in the case of the example offered by the American Heritage Science Dictionary, cause 6 times 3 to equal 24 rather than 18.

Definitions

1.  American Heritage Dictionary of the English Language:

            3. Mathematics

            a. The operation that, for positive integers, consists of adding a number (the multiplicand) to itself a certain number of times. The operation is extended to other real numbers according to the rules governing the multiplicational properties of positive integers.

            b. Any of certain analogous operations involving expressions other than real numbers.

2.  Collins Essential English Dictionary

            1. a mathematical operation, equivalent to adding a number to itself a specified number of times. For instance, 4 multiplied by 3 equals 12 (i.e. 4+4+4)

3.  American Heritage Science Dictionary

            1. A mathematical operation performed on a pair of numbers in order to derive a third number called a product. For positive integers, multiplication consists of adding a number (the multiplicand) to itself a specified number of times. Thus multiplying 6 by 3 means adding 6 to itself three times. The operation of multiplication is extended to other real numbers according to the rules governing the multiplicational properties of positive integers.

4.  Random House Dictionary

            2.  Arithmetic. a mathematical operation, symbolized by a × b, a · b, a ∗ b, or ab, and signifying, when a and b are positive integers, that a is to be added to itself as many times as there are units in b; the addition of a number to itself as often as is indicated by another number, as in 2×3 or 5×10.

5.  Webster’s Revised Unabridged Dictionary

            2. (Math.) The process of repeating, or adding to itself, any given number or quantity a certain number of times; commonly, the process of ascertaining by a briefer computation the result of such repeated additions; also, the rule by which the operation is performed; -- the reverse of division.

6.  WordNet® 3.0 (© 2006 by Princeton University)

            3.  an arithmetic operation that is the inverse of division; the product of two numbers is computed; "the multiplication of four by three gives twelve"; "four times three equals twelve."