Editorial   |    
The Use of Numbers and Percentages in Scientific Writing
Henry R. Cowell, M.D., Ph.D.
J Bone Joint Surg Am, 1998 Aug 01;80(8):1095-96
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Numbers, which play an integral part in the presentation of scientific data, are often used to create percentages in order to provide an easy way to express the findings of a study and to compare the results of one study with those of another. Over time, the percentage becomes the important figure, and authors frequently report the percentage without bothering to report the numbers. Regrettably, it is apparent from reviewing the materials submitted for publication in The Journal that most authors do not question the clinical validity or mathematical accuracy of the percentages that they report. For this reason, and for a number of others, The Journal reports numbers in addition to percentages and sometimes does not report percentages, even for certain data for which it is customary to do so.
As a case in point, the reader is referred to the article entitled "Ultrasound Surveillance for Asymptomatic Deep Venous Thrombosis after Total Joint Replacement" by Ciccone et al. in this issue of The Journal. Typically, sensitivity, specificity, the positive predictive value, and the negative predictive value are reported as percentages. Yet, the percentages that would have been recorded for the sensitivity and the positive predictive value in this article, although mathematically accurate, would not have led to clinically valid conclusions. Consider, for example, the positive predictive value for proximal thrombi reported on page 1170. The positive predictive value is defined as the number of patients with a positive test who have the disease, divided by the number of all patients with a positive test. Here, the numerator is two, as two patients who had a deep venous thrombus as confirmed with venography had a positive ultrasound study. There was one false-positive result; therefore, the denominator is three. Two divided by three is 67 per cent, but data based on three patients should not be reported as a percentage for several reasons. Although the data are mathematically accurate, it is clinically invalid to draw conclusions from an evaluation of three patients. Suppose, for example, that one more patient had had a false-positive test; the positive predictive value would have been reported as 50 per cent. If, on the other hand, the one patient with a false-positive test had been missed, then the positive predictive value would have been reported as 100 per cent, on the basis of two patients. It makes little sense to report percentages when the addition or loss of one patient can create such a large variation in the value.
Similarly, reporting percentages for the sensitivities in the article by Ciccone et al. would also have been misleading. The one proximal thrombus that was identified with venography after total hip arthroplasty was also detected with ultrasound; thus, the sensitivity could have been reported as 100 per cent (one of one). One of two proximal thrombi following total knee arthroplasty was detected with ultrasound, a sensitivity that would have been reported as 50 per cent. Thus, in general, the reader of any article must be suspicious about the true-positive rate, false-negative rate, and sensitivity when there were few patients who had the disease, and he or she must be suspicious about the positive predictive value when there were few patients who had a positive test. On the other hand, the true-negative rate, false-positive rate, and specificity are suspect when most of the patients in the study had the disease, and the negative predictive value is suspect when most patients had a positive test.
Regrettably, problems with percentages are not limited to such examples. Let me mention several errors that were detected during the editing of materials for publication in The Journal in the last several months.
Frequently, authors incorrectly reported percentages that they had obtained from other articles. These errors were detected when the percentages in the submitted manuscript were checked against those in the original source or when the author was requested to supply numbers, rather than percentages, from the original reference. For example, the originally reported 20 per cent in one submitted manuscript became 24 per cent when the numbers were requested. In another submitted manuscript, 37 per cent became 73 per cent when the numbers turned out to be forty-four of sixty. The original error was typographical. The only plausible explanation for these errors is either that the author failed to read and understand the source or that he or she copied the percentages incorrectly. There is no way to detect such errors when reading articles unless the numbers are given.
Many authors also did not round off percentages properly or check to see if the percentage had been reported properly in the original source. For example, seventy-six divided by ninety-three equals 81.7 per cent, which rounds off to 82 per cent. However, the author who reported that value rounded down to 81 per cent. In another instance, an author reported 92 per cent of forty-five patients. Forty-one of forty-five patients is 91 per cent. Forty-two of forty-five patients is 93 per cent. When such errors are found in the literature, it is apparent that either the author did not know how to calculate percentages or the percentages were calculated from a different subset of patients.
Another problem arose when authors reported percentages from several series. For example, one author stated that the prevalence of satisfactory results ranged from 43 to 100 per cent, as reported in the literature. As it is apparent that it is difficult to obtain 100 per cent good results with almost any procedure, the reference was checked. The reported 100 per cent was mathematically correct; it represented a successful procedure in one patient in a case report.
Unfortunately, once percentages are reported in the literature, readers and future authors frequently accept them as valid and accurate without checking the original source. Readers should question percentages any time that the numbers are not given, and they should be aware that reported percentages may represent a subset of the data and thus relate to a very small number, even if the series is large. This is particularly true with regard to sensitivity and specificity, which are reported as percentages by convention.
Although the issue of percentages might seem minor to some, it must be remembered that percentages—even misleading ones—are often the basis of clinical decisions.
Henry R. Cowell, M.D., Ph.D.

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