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Revista Brasileira de Medicina do Esporte

Print version ISSN 1517-8692

Rev Bras Med Esporte vol.3 no.4 Niterói Oct./Dec. 1997

http://dx.doi.org/10.1590/S1517-86921997000400002 

ARTIGO DE OPINIÃO

 

On combining samples that differ allometrically with size

 

 

William D. Ross, Ph.D.

Scientific Director, Rosscraft, Canada. E-mail: billross@netcom.ca

 

 


ABSTRACT

An anecdotal introduction to dimensional theory is used to focus on the issue of combining samples that differ allometrically with size. None of the ideas are original with the author. Indeed, the principles are well-known. Proponents of the body mass index who propose a common scale for individuals 20 to 65 years old blatantly ignore them. The error of combining samples is demonstrated using log log regressions of weight on one predictor height for men, women, and combined samples of men and women in five-year age increments from age 20 to 65 years. This paper serves as an invitation to examine the assumption of biological constancy illustrated by the so-called body mass index that was never true and now in the microcomputer age is unnecessary.

Key words: Allometry. BMI. Conceptual error. Height. Weight. Meredith. Quetelet. Ratios. Tanner. Cross-sectional samples. Men. Women.


 

 

My introduction to ratios and combining samples was in lectures by the late Howard V. Meredith. I had the good fortune to be a student when he was a Professor at the University of Oregon. Lindsay Carter had a similar influence when Meredith returned to the University of Iowa and the Iowa Child Welfare Research Station.

I always associate the word "integrity" with the man. James Tanner, who was also inspired by Howard Meredith, called him "the greatest anthropometrist of his generation".

A shy, modest man, with a pronounced stuttering, Meredith's countenance masked his great passion for research. I recall him discussing combined samples and emphasizing that ratios are neither as simple or as informative as they seem.

This theme was developed by Tanner (1949) in an important but often ignored paper entitled "The fallacy of per-weight and per-surface area standards and their relation to spurious correlation". J. Appl. Physiol. 2, 1-15. Other papers relevant to the theory of allometry that provide a background appreciation for scaling are: Bonner (1961) abridged edition of D'arcy Thompson, On Growth and Form, McMahon and Bonner (1983), Packard and Bordman (1987), Zeger and Harlow (1987), Heusner (1991), Albrecht, et al. (1993), Bingham (1993), Ross (1996).

 

RATIOS AND DIMENSIONS

The most famous ratio in human biology is the Quetelet Index, the so-called Body Mass Index or BMI. It is generally expressed as mass in kg divided by height in meters squared (m/h2). To my knowledge, Adolphe Quetelet (1776-1874) never proposed his index to assess adiposity or "fatness". Indeed, he was adamant about the uniqueness of physique and was inspiration to Alphonse Bertillion (1853-1914), who developed a system of criminal indentation based on Quetelet's view than no two people are alike in all respects. The odds of finding exact matches in seven or more variables was infinitesimal (Thorwald, 1965).

One hundred and sixty-four years ago, Adolph Quetelet, philosopher, mathematician, astronomer, artist, and father of anthropometry, understood principles of proportionality. He knew historically in scaling sculpture, when shape and composition were constant, an increase in mass was proportional to the cube of an increase in the linear measurements. Not so in humans because they change in proportion with increasing size in a systematic way.

Quetelet reported one of his observations as follows:

7. Aprés le développment complet des individus des deux sexes, les poids sont à peu près commes les carrés des talles. On déduit des deux relations precedentes, que l'accroissement en hauter est les plus grand que the l'accroissement transversal comprenand la largeur et l'éppaiseur. (Presented in 1832 at the 5 May and 2 June meetings of L'Academie Royale, published the following year _ Quetelet, 1833).

His observation was that in adults body weight was approximately a function of the square of stature (i.e. [wt] = h2). From this, one deduces that the increase in length is greater than widths and that shortness is associated with greater transverse measurements. This systematic difference between tall and short men and between tall and short women was demonstrated by the Phantom proportionality stratagem (Ross, et al., 1987). The human architectural design, with normal biological variance, is for tallness to be associated with relative lightness and smallness with relative heaviness or ponderosity.

To my knowledge, Quetelet never proposed any index to assess adiposity or "fatness". Bray (1978) is often cited for classification of obesity using the relationships observed by Quetelet in the so-called Body Mass Index or BMI, where body mass in kilograms is divided by stature expressed as square meters.

In recent years there has been a bandwagon effect. Many human biologists and health professionals have endorsed the BMI as a basic tool to assess adiposity. Grant committees, editors and referees of scientific journals have followed, indeed, abetted the parade. It does not seem to be waning in advocacy as one can surmise by a net search.

Perhaps the most blatant and persistent endorsement of the BMI is by the Health Promotion Branch of Health and Welfare Canada (1988a, 1988b). They advocate a common scale for men and women and recommend a healthy body weight BMI range of 20 to 27, and an ideal range of 20 to 25 for men and women age 20 to 65 years old. Other ostensibly reputable agencies elsewhere have a similar approach.

Aghast at the conceptual error in using the BMI and the implications, Ross et al. (1988) published results from two major studies. We showed the BMI had different meaning in men and women, that it was less than fifteen percent better than pure chance in predicting sum of five skinfolds, and often would grossly misclassify individuals (Ross et al., 1988). The paper has never been challenged or refuted. Indeed, it is selectively ignored. I suspect this is a matter of grantsmanship where contrary evidence is discredited or when this is not possible, just ignored.

There already is a netsite that deals with junk science by Steve Milloy that is worth bookmarking: http://www.junkscience.com/. It is inevitable that other similar sites and links will soon evolve from expanded e-mail interaction. Journals will change. They will become increasingly more responsive and seek critical comment. Eventually they will come out in electronic form and may well set as an additional acceptance criterion, a willingness of the author or authors to make available the basic data in spreadsheet format.

 

PURPOSE

The purpose of this paper is to develop the theme that one cannot with impunity combine samples that differ allometrically with size. The resulting exponent in the combined sample described the true relationship in neither of the subsamples. Our intent is to show the common BMI scale for men and women is a mathematical artifact.

An ancillary purpose is to alert young scientists and health professionals to the need to use new technology to become participants and witnesses in the advance of science. This means they must challenge assumptions and propositions in the literature, especially those that claim validity by virtue of "expert committee opinion".

 

SUBJECTS

Two large databases were available: The YMCA Lifestyle Inventory and Fitness Evaluation project and the Canada Fitness Survey. The YMCA LIFE database consisted of 5,039 women and 12,192 men who were studied using a comprehensive test battery in fifty Canadian Cities (Bailey, Carter and Mirwald, 1982). This was given to me by my long-time friend and colleague, Donald A. Bailey, of the University of Saskatchewan. The Canada Fitness Survey database consisted of 5,137 women and 4,742 men studied comprehensively in a geographical sampling plan for the entire country. Despite being the criterion anthropometrist responsible for training thirteen regional teams, I had to purchase the CFS database.

 

METHOD

Female and male samples in each study were treated independently and as a combined female and male sample.

All of the above were isolated in five year increments age 20 to 65 years old.

Allometric relationships for female, male, and combined incremental samples for each study were described by changing the data into logs and computing the slope as illustrated in the arithmetic example in the spreadsheet shown as Fig.1.

 

 

The obtained slopes for male, female, and combined age incremental samples represent the allometric regression of weight on one predictor height.

 

RESULTS

According to BMI proponents, the allometric equations should be 2.0. Clearly, as shown in Tables 1 and 2, females and males had exponents generally less than 2.0 with females values lower still than that of males. In the combined samples, the exponent was greater than in either of the subsamples.

 

 

 

As illustrated in Fig. 2 and 3, one cannot with impunity combine samples that differ allometrically with size. The combined sample does not describe the relationships in the subsamples. Thus, the common scale for men and women is easily exposed as a mathematical artifact, not a biological phenomenon.

 

 

 

CONCLUSIONS

1)A study group devised a series of icons to illustrate salient concerns for data analyses in the microcomputer age (Ross, DeRose, Savage, in preparation). The phenomenon of combining samples that differ allometrically with size was illustrated in Fig. 4. Three ascending log-log regression lines perhaps similar to those for the regression of weight on height for adult females (1.5) and males (1.7) is shown to have a combined slope of (2.0+). The icon is not to scale, its purpose is to illustrate the principle and explain didactically the conclusion of this paper. One cannot with impunity combine samples that differ allometrically with size.

 

 

2)The findings are consistent with the view that there are systematic differences between men and women. Notwithstanding the common scale for men and women, for every incremental increase in stature there is about a three-kilogram greater difference in body weight for men compared to women.

 

REPRISE

Heretofore those who challenged the BMI did so on the basis of commonsense. Two hundred men and women the same height and weight age 20 to 65 have the same BMI. To assert they have a common level of adiposity and health risk is patently false. As we have shown in this paper, the common BMI for men and women scale is easy to disavow with simple algorithms and access to data.

Nevertheless, there is an oldboys' network that is amazingly resilient in propagating nonsense. There is, of course, an eventual accounting. Perhaps the BMI will server as example and object lesson for the Mismeasure of Man, as the IQ did for Steven Jay Gould in his thought-provoking book with that title (1983).

The late Albert Behnke was mentor to many kinanthropometrist. He often closed his letters or punctuated an observation with the enjoinder _"check it out". Nowadays with our microcomputers this is becoming increasingly easier to do.

 

REFERENCES

1.Albrecht GH, Cleven BR, Hartman SE. Ratios as size adjustment in morphometrics. American Journal of Physical Anthropology 1993;91:441-58.

2.Bailey DA, Carter JEL, Mirwald RL. Somatotype of Canadian men and women. Human Biology 1982;54:113-28.

3.Bingham GP. Perceiving the size of trees: form as information about scale. J Experimental Psychology: Human Perception & Performance. 1993;19,6:1139-61.

4.Bray GA. Definitions, measurements and classification of syndromes of obesity. International Journal of Obesity 1988;2:99-112.

5.Gould SJ. The Mismeasure of Man. Baltimore: Penguin Books, 1983.

6.Health and Welfare Canada. Canadian guidelines for healthy weights. Cat no H39-134/1988E, Ottawa: Minister of Supply and Services, 1988a.

7.Health and Welfare Canada. Promoting Healthy Weights. Cat no H39-131/1988E, Ottawa: Minister of Supply and Services, 1988b.

8.Heusner AA. Size and power in animals. Journal of Experimental Biology 1991;160:25-54.

9.Packard GC, Boardman TJ. The misuse of ratios to scale physiological data that vary allometrically with body size. In: Feder ME, Bennett AF, Buggeren WW, Huey RB, editors. New Directions in Ecological Physiology, London: Cambridge University Press, 1987:216-39.

10.McMahon TA, Bonner JT. On Science and Life. New York: Scientific American Books. New York: WH Freeman & Company, 1983.

11.Quetelet A. Résearches sur le poids de l'hommes aux différents âges. L'Academy Royale. Brussels: M. Hayes, 1833:36-8.

12.Ross WD, Martin AD, Ward R. Body composition and aging: theoretical and methodological implications. Collegium Antropologicum 1987; 11:15-44.

13.Ross WD. On ratios and proportions. In: Essays of Auxology. Hauspie R, Lindgren G, Faulkner F, editors. Welwyn Gardens City: Castelmead, 1996:119-25.

14.Ross WD, DeRose EH, Savage MV. Iconologics; new analyses for the microcomputer age, Rosscraft, in press 1998.

15.Tanner JM. Fallacy of power-weight and per-surface area standards and their relation to spurious correlation. Journal of Applied Physiology 1949;2:1-15.

16.Zeger SL, Soiban DH. Mathematical models from laws of growth to tools for biological analysis: fifty years of growth. Growth 1987;51:1-21.

1.Albrecht GH, Cleven BR, Hartman SE. Ratios as size adjustment in morphometrics. American Journal of Physical Anthropology 1993;91:441-58.         [ Links ]

2.Bailey DA, Carter JEL, Mirwald RL. Somatotype of Canadian men and women. Human Biology 1982;54:113-28.         [ Links ]

3.Bingham GP. Perceiving the size of trees: form as information about scale. J Experimental Psychology: Human Perception & Performance. 1993;19,6:1139-61.         [ Links ]

4.Bray GA. Definitions, measurements and classification of syndromes of obesity. International Journal of Obesity 1988;2:99-112.         [ Links ]

5.Gould SJ. The Mismeasure of Man. Baltimore: Penguin Books, 1983.         [ Links ]

6.Health and Welfare Canada. Canadian guidelines for healthy weights. Cat no H39-134/1988E, Ottawa: Minister of Supply and Services, 1988a.         [ Links ]

7.Health and Welfare Canada. Promoting Healthy Weights. Cat no H39-131/1988E, Ottawa: Minister of Supply and Services, 1988b.         [ Links ]

8.Heusner AA. Size and power in animals. Journal of Experimental Biology 1991;160:25-54.         [ Links ]

9.Packard GC, Boardman TJ. The misuse of ratios to scale physiological data that vary allometrically with body size. In: Feder ME, Bennett AF, Buggeren WW, Huey RB, editors. New Directions in Ecological Physiology, London: Cambridge University Press, 1987:216-39.         [ Links ]

10.McMahon TA, Bonner JT. On Science and Life. New York: Scientific American Books. New York: WH Freeman & Company, 1983.         [ Links ]

11.Quetelet A. Résearches sur le poids de l'hommes aux différents âges. L'Academy Royale. Brussels: M. Hayes, 1833:36-8.         [ Links ]

12.Ross WD, Martin AD, Ward R. Body composition and aging: theoretical and methodological implications. Collegium Antropologicum 1987; 11:15-44.         [ Links ]

13.Ross WD. On ratios and proportions. In: Essays of Auxology. Hauspie R, Lindgren G, Faulkner F, editors. Welwyn Gardens City: Castelmead, 1996:119-25.         [ Links ]

14.Ross WD, DeRose EH, Savage MV. Iconologics; new analyses for the microcomputer age, Rosscraft, in press 1998.         [ Links ]

15.Tanner JM. Fallacy of power-weight and per-surface area standards and their relation to spurious correlation. Journal of Applied Physiology 1949;2:1-15.         [ Links ]

16.Zeger SL, Soiban DH. Mathematical models from laws of growth to tools for biological analysis: fifty years of growth. Growth 1987;51:1-21.         [ Links ]