http://informahealthcare.com/ahb ISSN: 0301-4460 (print), 1464-5033 (electronic) Ann Hum Biol, Early Online: 1–5 ! 2014 Informa UK Ltd. DOI: 10.3109/03014460.2014.954615
RESEARCH PAPER
Sitting height as a better predictor of body mass than total height and (body mass)/(sitting height)3 as an index of build Richard F. Burton
Ann Hum Biol Downloaded from informahealthcare.com by University of Ottawa on 11/24/14 For personal use only.
School of Life Sciences, College of Medical, Veterinary and Life Sciences, University of Glasgow, Glasgow, UK
Abstract
Keywords
Background: The Rohrer Index and the ratio of sitting height (SH) to height fall similarly with growth in early childhood, then level off and rise slightly towards adulthood. In adults the BMI correlates with SH/height. The mean cross-sectional areas of the legs of adults are correlated positively with upper body masses and negatively with leg lengths. Aim: To find an index of body build that is less dependent on relative leg length and age in children and adults than are the BMI and the Rohrer Index. Subjects and methods: Published data are analysed to establish the relative importance of SH and leg length as predictors of body mass and to investigate the age dependence of the ratio (body mass)/SH3. Results: SH is a much better predictor of body mass than height, with leg length being barely relevant. Average values of (body mass)/SH3 vary very little over the age range of 1–25 years, despite small non-random fluctuations. Conclusion: The ratio (body mass)/SH3 is proposed as a useful ‘‘sitting-height index of build’’ that is superior to the Rohrer Index and could prove better than the BMI as a predictor of adiposity. Further studies are needed, notably using individual data and fat-free masses.
BMI, children, Cormic Index, Rohrer Index, sitting-height index of build
Introduction Characterizing the dependence of body mass (BM) on age and height in children and adolescents, although long studied (e.g. Quetelet, 1835, 1842) and important to the assessment of adiposity, remains to some extent problematic, with the main complication being the progressive change in the proportions of the body during growth (e.g. Cole, 1991). Relationships between BM and height form the basis of indices such as the BMI, i.e. BM/height2, and the Rohrer Index (RI), BM/height3 (Rohrer, 1908), that are indicators of overweight, underweight and fat content. The BMI’s height exponent of two is the round number that generally minimizes the correlation between BMI and height in adults (Khosla & Lowe, 1967; Cole, 1986). In 1-year age groups of children and adolescents the corresponding round number may be 2 or 3 (Cole, 1986). The height exponent of three of the RI is justified instead by dimensional analysis (see Discussion). Inconveniently, both indices are very dependent on age in young people (see Figure 1). Moreover, at least in adults, the BMI and presumably also the RI tend to increase with the ratio of sitting height (SH) to height (Bogin & Beydoun, 2007; Norgan, 1994a), although the influence of this ratio is not
Correspondence: Dr. Richard F. Burton, West Medical Building, University of Glasgow, Glasgow G12 8QQ, UK. Fax: +44 (0) 330 5481. E-mail: [email protected]
History Received 9 April 2014 Revised 3 July 2014 Accepted 1 August 2014 Published online 17 September 2014
generally taken into account in the interpretation of BMIs in terms of fatness and health risks. This paper explores the possibility that a different index, namely BM/SH3, is much less dependent on both age and leg length (LL). It is called here the ‘‘Sitting-Height Index of Build’’ (SHIB), in keeping with ‘‘height–weight index of build’’, the name used by Bardeen (1920) for the RI. Evidence for the appropriateness of the SHIB comes partly from a comparison of the parallel changes during growth in RI and the ratio SH/height (the Cormic Index). The RI typically falls from the age of 1 year, doing so progressively less steeply towards the age of 7–9 years when a fairly constant ‘‘plateau’’ is reached. Subsequently, mean values of the RI tend to rise a little from 12–16 years. This time course is illustrated graphically by Broman et al. (1942), Burton (2008), Cole (1991) and Gasser et al. (1994), with all these studies being on European males and females. Similar graphs may be derived from other data (Bardeen, 1920, 1923; Brundtland et al., 1975; Lebiedowska et al., 2008; Quetelet, 1835, 1842). During the period of growth when the RI remains fairly constant, the BMI rises with increasing body size, as is mathematically inevitable. This is because BMI ¼ RI height. The ratio SH/height changes with age in a similar manner to the RI, although imperfectly matched to it in timing (Fredriks et al., 2005; Hansman, 1970; Gleiss et al., 2013; Waaler, 1983). These four studies show a gradual decline in the ratio from infancy to a minimum reached at 12–15 years
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Ann Hum Biol, Early Online: 1–5
Table 1. Characteristics of the four population samples. Hansman (1970) Male
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Figure 1. Age dependence of the RI (+), 86(SH/height)3 (), (BM/SH3)/10 (œ) and the BMI (*) in males according to the mean data of Hansman (1970). BM/SH3 is the SHIB.
that is followed by a much smaller rise and a levelling off towards adulthood. This suggests that some function of the ratio SH/height and of RI might vary very little with age and provide a useful index of body build. Dimensional analysis (see Discussion) suggests that an appropriate function might be RI/(SH/height)3. This is numerically equal to BM/SH3. Bardeen (1923) found mean values of 100.BM/SH3 to be nearly constant (8.19–8.50 g/cm3) and uncorrelated with age in girls over the age range 6.5–17.5 years (so that, in terms of the units kg and m, mean BM/SH3 varied from 81.9–85.0 kg/m3). He did not suggest any rationale or significance for this index and it has attracted little subsequent attention, possibly because it ignores the obvious contribution of the legs to BM and because younger children, in which SH/height is most age-dependent, were not considered. Here the age-dependence of the SHIB is explored further using published data. If the index has validity, it would imply that BM may be better predicted from SH than from total height. Although this ignoring of leg length may seem counterintuitive, a separate argument for doing it is provided in the Discussion. This is based on the finding that LL has a negative influence on the leg’s mean cross-sectional area in adults (Burton et al., 2012). If valid, the SHIB would be in competition with the BMI as an index of body build and an indicator of adiposity. As already noted, the BMI has the disadvantages of being very dependent on age in children and influenced by relative leg length. Another defect of the BMI is that, for geometrically similar (isometric) individuals of different sizes, the larger ones have higher BMIs. It is hoped that the SHIB will prove to be an improvement on the BMI, one that is much needed in these times of widespread obesity.
Methods This study utilizes only published information, consisting principally of two data sets (Hansman, 1970; Waaler, 1983). These include means, but not individual values, of BM, height and SH for wide age ranges and for males and females separately. Means of SHIB, RI and BMI were estimated from these means and means of LL were calculated as the differences between those given for height and SH. The data of Hansman (1970), collected in Denver, CO from
Female
Waaler (1983) Male
Female
Number of individuals 2664 2493 2017 1948 Number of age groups 42 39 14 14 Individuals/age group 24–92 12–95 37–201 50–182 Age range, years 1–25 1–24 3.2–16 3.2–16 SH range, m 0.49–0.94 0.47–0.88 0.58–0.90 0.56–0.88 LL range, m 0.27–0.86 0.26–0.79 0.41–0.85 0.41–0.78 Mean SHIB ± SD 86.2 ± 3.0 84.1 ± 2.8 80.2 ± 3.0 80.6 ± 2.3 Range of SHIB 81.2–92.7 78.4–90.4 76.0–84.7 76.1–84.2 Range of RI 11.5–23.0 11.6–23.2 11.4–16.4 11.7–16.4 Range of BMI 13.5–23.5 15.3–21.9 15.5–20.9 15.3–20.3 CV of SHIB, % 3.5 3.3 3.7 2.9 CV of RI, % 19.8 19.6 11.7 11.3 CV of BMI, % 12.8 12.2 8.8 10.2 All of the following relate to published means at each age. SH, sitting height; LL, leg length; SHIB, (body mass)/SH3; RI, Rohrer Index (body mass)/height3; BMI, body mass index, (body mass)/ height2; SD, standard deviation; CV, coefficient of variation (100 SD/mean).
1933–1966, are ‘‘mainly for upper-class ‘Old Americans’ of northwest European descent’’ (in Forward, p. viii). From ages 1–17.5 years (males) and 1–15.5 (females) the data are for half-year age groups. For older subjects the means are for 1year groupings. The data of Waaler (1983) were collected in Bergen, Norway in 1971–1974. The mean ages, ranging from 3.2–16.9 years, are for age groups of 0.9–1.1 years. Table 1 shows relevant characteristics of the data sets. The numbers of individuals for each age group are for SH, which were sometimes fewer than for height and BM. In the case of the data of Hansman (1970), the means are less reliable for individuals older than 17 years inasmuch as the sample sizes then average only 32 for the males and 33 for the females, compared with 72 and 71, respectively, for the younger individuals. BM1/3 was regressed on SH and LL to produce equations of the form: BM1=3 ¼ a:SH þ b:LL þ c ð1Þ Statistical analyses were carried out using Excel 2007 (Microsoft Corporation, Redmond, WA). The units used are kilograms and metres.
Results Although common experience shows that SH and LL are not proportional to each other throughout childhood, their correlation coefficient is 0.987–0.996 for each of the four sets of mean values. Regression of BM1/3 on SH and LL gave the following equations. Hansman ð1970Þ: Males BM1=3 ¼ 4:27:SH þ 0:20:LL 0:02 Hansman ð1970Þ: Females BM1=3 ¼ 4:83:SH 0:33:LL 0:12 Waaler ð1983Þ: Males BM1=3 ¼ 3:78:SH þ 0:58:LL þ 0:03
ð2Þ
ð3Þ
ð4Þ
Sitting height as a predictor of body mass
DOI: 10.3109/03014460.2014.954615
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Why is SH cubed?
Figure 2. Age dependence of the mean ratio BM/SH3. (a) Based on data of Hansman (1970). (b) Based on data of Waaler (1983). Squares and circles (œ, *) are for females. Crosses (+, ) are for males.
Waaler ð1983Þ: Females BM1=3 ¼ 5:13:SH 0:57:LL 0:23
ð5Þ
Of the regression parameters, only the values of a are significant (p50.003). The ratios b/a are, respectively, 0.05, 0.07, 0.15 and 0.11. Table 1 shows the mean values of the SHIB over the whole age ranges, their coefficients of variation and the corresponding coefficients of variation for RI and BMI. Figure 1 illustrates, for white males, the age dependence of RI, BMI, (SH/height)3 and BM/SH3. The latter two are multiplied by 86 and 0.1, respectively, in order to adjust their positioning on the graph. Graphs for the other data sets show the same general features. Figure 2 shows the age dependence of the ratio BM/SH3 for all four data sets.
Discussion The four data sets are in good agreement. Mean SHIB, calculated as BM/SH3 but also equal to RI/(SH/height)3, shows a much lower age-dependence over the full age range than either mean BMI or mean RI (Table 1, Figure 1). The low age-dependence is also illustrated in Figure 2, but made less immediately obvious because the vertical scale was chosen to accentuate the small, but consistent fluctuations that occur. They arise from the slight mismatches in the time courses of RI and SH/height (Figure 1) and need not correspond exactly in timing to significant growth episodes. In Figure 2 the erratic scatter above the age of 17 years that is evident in the data of Hansman (1970) may be due to the smaller sample sizes; it relates more to BM than to SH. The concept of the SHIB accords with the similarity in time courses of change in RI and SH/height. However, two aspects of its form need to be discussed: the cubing of SH and the disregard of LL and total height.
As Quetelet (1842) wrote in the English edition of one of his books, ‘‘If man increased equally in all his dimensions, his weight at different ages would be as the cube of his height’’. Cubing of SH in the SHIB and of height in the RI is justified by dimensional or unit analysis (Burton, 2008; McMahon & Bonner, 1983). Appropriate units for SHIB, as for RI, are kg/m3. However, with BM replaced with body volume, e.g. by dividing BM by body density (which generally varies by only a few per cent), the indices become dimensionless (unit-free) numbers. They are then fully compatible with dimensionless measures of relative fat content (e.g. percentage). Unlike the Benn Index and the BMI, which are based on population statistics, the SHIB and RI have physical interpretations. Thus, they are constant for any hypothetical isometric (geometrically similar) bodies of equal density so that, if density and the index in question remain constant over a given period, then growth is isometric. If the SHIB or RI change, then growth in regard to BM and either SH or height, respectively, is not isometric. That BM/SH3 varies so little over a wide range of body sizes implies that if the height exponent were estimated by non-linear regression of BM on SH it should be close to three. Actual estimates, based on the mean data of Hansman (1970) and Waaler (1983), are 3.03–3.20 for the two sexes and are thus in close agreement. The influence of SH/height on RI in adults In order to understand the influence of SH/height on RI one must consider the relative contributions to BM of upper body mass (UBM) and of the combined mass of the two legs. These masses correspond, respectively, to the two lengths SH and LL. In a study of adults (Burton et al., 2012) the ratio of leg mass to UBM averaged 0.39 for men and 0.43 for women, with the two masses tending to vary proportionately with each other (with r ¼ 0.88 for men and 0.89 for women). (It is relevant here that the girths of the legs and torso, where they meet, visibly tend to vary together.) The low ratio and high correlation would both reduce the importance of LL in the prediction of BM, but probably not enough to justify the complete disregard of LL. More crucial findings were that the mean cross-sectional area (CSA) of one leg minus the foot tended to vary almost in proportion to UBM and that, for a given UBM, the mean CSA showed a marked negative correlation with leg length. With the volume of a leg being equal exactly to its mean CSA multiplied by its length, this volume (approximately proportional to its mass) will vary less than proportionately with LL and approximately proportionately with UBM. In fact the UBM, not LL, will be the dominant influence on leg mass. Multiple regression equations given previously for log(mean CSA) on log(UBM) and log(LL) in adults (Burton et al., 2012) allow a more quantitative approach. Expressed in non-logarithmic form and converted to kg, m and m2, these equations are: Mean CSA ¼
3:02:UBM0:91 for men LL0:52
ð6Þ
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Ann Hum Biol, Early Online: 1–5
and Mean CSA ¼
3:16:UBM LL0:39
0:93
for women
ð7Þ
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Unlike Equations (2)–(5), these are calculated for individual data. According to Equations (6) and (7), with each leg volume taken as the product of mean CSA and LL and with the density taken as 1050 kg/m3 (Burton et al., 2012), BM approximates to [UBM + 0.63.UBM0.91.LL0.48] for men and [UBM + 0.66.UBM0.93.LL0.61] for women. To relate BM just to SH and LL, one may, for the sake of illustration, take the UBM as exactly proportional to SH3 (corresponding to geometrical similarity) and, thus, equal to KmSH3 in men and KfSH3 in women, where Km and Kf are constants. Then: 0:48 LL BM ¼ Km þ 0:63:Km0:91 SH3 for men ð8Þ SH0:27 and BM ¼
0:61 LL Kf þ 0:66:Kf0:93 SH3 for women SH0:21
ð9Þ
Because SH and LL are correlated in adults (Burton et al., 2013), the expression in square brackets for each sex should vary so little that, for any plausibly realistic value of Km or Kf, BM/SH3 would vary by only a few percent. Therefore BM would be minimally sensitive to variations in LL. In illustration, the expressions in square brackets, calculated from the data of Burton et al. (2013), are 0.86–1.00 in men and 0.93–1.22 in women. Accordingly, with Km and Kf suitably chosen as, say, 60 and 55, respectively, BM/SH3 equals 82.5–86.1 in men and 80.5–88.5 in women, values comparable to those in Table 1. What is significant about these calculated ranges is not their actual values, but their narrowness. The forms of Equations (8) and (9) cannot be expected to represent exact and algebraically appropriate functional relationships, but, although tentative and approximate, they do clearly illustrate a much greater influence of SH on BM in adults as compared with LL. Equations (6)–(9) are for adults. The fact that BM in children of widely varying age and height depends much more on SH than on LL suggests that the negative influence of LL on a leg’s mean CSA applies to children also, but this remains to be tested. The average ratio SH/height varies amongst populations (e.g. Bardeen, 1923; Burton et al., 2013), doing so with ethnicity and socioeconomic status. The ratio is notably low in Australian Aborigines (Norgan, 1994b) and high in Canadian Inuits (Charbonneau-Roberts et al., 2005). The mean ratio is lower in children of Maya immigrants to the US than in their counterparts in Guatemala (Smith et al., 2003). These variations influence relationships between adiposity and BMI, but not between adiposity and SHIB. Future research Obvious limitations of this study are its basis in mean values rather than individual measurements, its restriction to particular populations and the lack of data on relative fat content. Further studies are, therefore, needed to test the usefulness of the SHIB, involving individual data and
contrasting populations. The most important question is whether the SHIB is a more reliable predictor of adiposity and health than the BMI. Therefore, relationships between SHIB and percentage body fat need to be established and, in that context, it would be particularly useful to explore the agedependence of the ratio (fat-free mass)/SH3. Another point to investigate is whether that ratio and the SHIB might be influenced sufficiently by developmental changes in the proportions of the upper body, such as relative head size, to justify their routine measurement. The BMI differs fundamentally from the SHIB in that the linear measure of body size (i.e. height) is squared rather than cubed. The height exponent of two is the round number that minimizes the correlation between BMI and height in adults and in some 1-year age groups of young people. This improves the correlation with percentage body fat. It is this basis in statistics rather than in dimensional analysis—coupled with the low correlation between BM and height—that produces the lower height exponent of the BMI (Burton, 2007, 2008, 2014). At the risk of introducing unhelpful complications, the height exponent of the SHIB may perhaps be lowered also to produce a better predictor of adiposity for particular age groups.
Conclusions It is shown that the sitting height is a better predictor of total body mass than is total height and that the ratio (body mass)/ (sitting height)3, the SHIB, is a valid index of build with advantages over the BMI. Another such advantage is that influences of childhood nutrition on SHIB may be studied independently from those on leg length. The exclusion of leg length from the index is largely justified by the fact that the length of a leg tends to have a negative influence on its mean cross-sectional area and, therefore, has little influence on leg mass. As noted under Future research, further studies are needed, notably on individuals and on fat-free masses, in order to explore the usefulness of the index as a predictor of adiposity and ultimately as an indicator of health risks in both clinical practice and population studies.
Declaration of interest The author reports no conflict of interest. The author alone is responsible for the content and writing of the paper.
References Bardeen CR. 1920. The height-weight index of build in relation to linear and volumetric proportions and surface-area of the body during postnatal development. Contributions to Embryology No. 46. Carnegie Institution of Washington Bulletin 9:483–554. Bardeen CR. 1923. General relations of sitting height to stature and of sitting height and stature to weight. Am J Phys Anthropol 6:355–388. Bogin B, Beydoun N. 2007. The relationship of sitting height ratio to body mass index and fatness in the United States, 1988–1994. In: Singh SP, Gaur R, editors. Human body composition. Human Ecology Special Issue No. 15. Delhi, India: Kamal-Raj. 8 p. Broman B, Dahlberg G, Lichtenstein A. 1942. Height and weight during growth. Acta Paediatr 30:1–66. Brundtland GH, Liesto¨l K, Wallo¨e L. 1975. Height and weight of school children and adolescent girls and boys in Oslo 1970. Acta Paediatr Scand 64:565–573. Burton RF. 2007. Why is the body mass index calculated as mass/ height2, not as mass/height3? Ann Hum Biol 34:656–663. Burton RF. 2008. Biomedical calculations. Chichester, UK: Wiley.
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DOI: 10.3109/03014460.2014.954615
Burton RF. 2014. Adult fat content: reinterpreting and modelling the Benn Index and related sex differences. Ann Hum Biol. DOI: 10.3109/ 03014460.2014.903997. Burton RF, Nevill AM, Stewart AD, Daniell N, Olds T. 2012. A negative relationship between leg length and leg cross-sectional areas in adults. Am J Hum Biol 24:562–564. Burton RF, Nevill AM, Stewart AD, Daniell N, Olds T. 2013. Statistical approaches to relationships between sitting height and leg length in adults. Ann Hum Biol 40:64–69. Charbonneau-Roberts G, Saudny-Unterberger H, Kuhnlein HV, Egeland GM. 2005. Body mass index may overestimate the prevalence of overweight and obesity among the Inuit. Int J Circumpolar Health 64:163–169. Cole TJ. 1986. Weight/heightp compared to weight/height2 for assessing adiposity in childhood: influence of age and bone age on p during puberty. Ann Hum Biol 13:433–451. Cole TJ. 1991. Weight-stature indices to measure underweight, overweight, and obesity. Chapter 5. In: Himes JH, editor. Anthropometric assessment of nutritional status. New York: Wiley-Liss. p 83–111. Fredriks AM, van Buuren S, van Heel WJM, Dijkman-Neerincx RHM, Verloove-Vanhorick SP, Wit JM. 2005. Nationwide age references for sitting height, leg length, and sitting height/height ratio, and their diagnostic value for disproportionate growth disorders. Arch Dis Child 90:807–812. Gasser T, Ziegler P, Seifert B, Prader A, Molinari L, Largo R. 1994. Measures of body mass and of obesity from infancy to adulthood and their appropriate transformation. Ann Hum Biol 21:111–125. Gleiss A, Lassi M, Blu¨mel P, Borkenstein M, Kapelari K, Mayer M, Schemper M, Ha¨usler G. 2013. Austrian height and
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body proportion references for children aged 4 to under 19 years. Ann Hum Biol 40:324–332. Hansman C. 1970. Anthropometry and related data. In: McCammon RW, editor. Human growth and development. Springfield, IL: Charles C Thomas. p 103–154. Khosla T, Lowe CB. 1967. Indices of obesity derived from body weight and height. Br J Prev Soc Med 21:122–128. Lebiedowska MK, Alter KE, Stanhope SJ. 2008. Human body shape index based on an experimentally derived model of human growth. J Pediatr 152:45–49. McMahon TA, Bonner JT. 1983. On size and life. New York: Scientific American Library. Norgan NG. 1994a. Relative sitting height and the interpretation of the body mass index. Ann Hum Biol 21:79–82. Norgan NG. 1994b. Interpretation of low body mass indices: Australian Aborigines. Amer J Phys Anthropol 94:229–237. Quetelet A. 1835. Sur l’homme et le de´veloppement de ses faculte´s, ou essai de physique sociale. Volume 2. Paris, France: Bachelier. Quetelet LAJ. 1842. A treatise on man and the development of his faculties. Facsimile reproduction (1969) of the English translation of 1842. Gainsville, FL: Scholars’ facsimiles & reprints. Rohrer F. 1908. Eine neue Formel zur Bestimmung der Ko¨rperfu¨lle. Korrespondenze-Blatt der Deutschen Gesellschaft fu¨r Anthropologie, Ethnologie und Urgeschichte 39:5–7. Smith PK, Bogin B, Varela-Silva MI, Loucky J. 2003. Economic and anthropological assessments of the health of children in Maya immigrant families in the US. Econ Hum Biol 1:145–160. Waaler PE. 1983. Anthropometric studies in Norwegian children. Acta Paed Scand 308:1–41.
RESEARCH PAPER
Sitting height as a better predictor of body mass than total height and (body mass)/(sitting height)3 as an index of build Richard F. Burton
Ann Hum Biol Downloaded from informahealthcare.com by University of Ottawa on 11/24/14 For personal use only.
School of Life Sciences, College of Medical, Veterinary and Life Sciences, University of Glasgow, Glasgow, UK
Abstract
Keywords
Background: The Rohrer Index and the ratio of sitting height (SH) to height fall similarly with growth in early childhood, then level off and rise slightly towards adulthood. In adults the BMI correlates with SH/height. The mean cross-sectional areas of the legs of adults are correlated positively with upper body masses and negatively with leg lengths. Aim: To find an index of body build that is less dependent on relative leg length and age in children and adults than are the BMI and the Rohrer Index. Subjects and methods: Published data are analysed to establish the relative importance of SH and leg length as predictors of body mass and to investigate the age dependence of the ratio (body mass)/SH3. Results: SH is a much better predictor of body mass than height, with leg length being barely relevant. Average values of (body mass)/SH3 vary very little over the age range of 1–25 years, despite small non-random fluctuations. Conclusion: The ratio (body mass)/SH3 is proposed as a useful ‘‘sitting-height index of build’’ that is superior to the Rohrer Index and could prove better than the BMI as a predictor of adiposity. Further studies are needed, notably using individual data and fat-free masses.
BMI, children, Cormic Index, Rohrer Index, sitting-height index of build
Introduction Characterizing the dependence of body mass (BM) on age and height in children and adolescents, although long studied (e.g. Quetelet, 1835, 1842) and important to the assessment of adiposity, remains to some extent problematic, with the main complication being the progressive change in the proportions of the body during growth (e.g. Cole, 1991). Relationships between BM and height form the basis of indices such as the BMI, i.e. BM/height2, and the Rohrer Index (RI), BM/height3 (Rohrer, 1908), that are indicators of overweight, underweight and fat content. The BMI’s height exponent of two is the round number that generally minimizes the correlation between BMI and height in adults (Khosla & Lowe, 1967; Cole, 1986). In 1-year age groups of children and adolescents the corresponding round number may be 2 or 3 (Cole, 1986). The height exponent of three of the RI is justified instead by dimensional analysis (see Discussion). Inconveniently, both indices are very dependent on age in young people (see Figure 1). Moreover, at least in adults, the BMI and presumably also the RI tend to increase with the ratio of sitting height (SH) to height (Bogin & Beydoun, 2007; Norgan, 1994a), although the influence of this ratio is not
Correspondence: Dr. Richard F. Burton, West Medical Building, University of Glasgow, Glasgow G12 8QQ, UK. Fax: +44 (0) 330 5481. E-mail: [email protected]
History Received 9 April 2014 Revised 3 July 2014 Accepted 1 August 2014 Published online 17 September 2014
generally taken into account in the interpretation of BMIs in terms of fatness and health risks. This paper explores the possibility that a different index, namely BM/SH3, is much less dependent on both age and leg length (LL). It is called here the ‘‘Sitting-Height Index of Build’’ (SHIB), in keeping with ‘‘height–weight index of build’’, the name used by Bardeen (1920) for the RI. Evidence for the appropriateness of the SHIB comes partly from a comparison of the parallel changes during growth in RI and the ratio SH/height (the Cormic Index). The RI typically falls from the age of 1 year, doing so progressively less steeply towards the age of 7–9 years when a fairly constant ‘‘plateau’’ is reached. Subsequently, mean values of the RI tend to rise a little from 12–16 years. This time course is illustrated graphically by Broman et al. (1942), Burton (2008), Cole (1991) and Gasser et al. (1994), with all these studies being on European males and females. Similar graphs may be derived from other data (Bardeen, 1920, 1923; Brundtland et al., 1975; Lebiedowska et al., 2008; Quetelet, 1835, 1842). During the period of growth when the RI remains fairly constant, the BMI rises with increasing body size, as is mathematically inevitable. This is because BMI ¼ RI height. The ratio SH/height changes with age in a similar manner to the RI, although imperfectly matched to it in timing (Fredriks et al., 2005; Hansman, 1970; Gleiss et al., 2013; Waaler, 1983). These four studies show a gradual decline in the ratio from infancy to a minimum reached at 12–15 years
2
R. F. Burton
Ann Hum Biol, Early Online: 1–5
Table 1. Characteristics of the four population samples. Hansman (1970) Male
Ann Hum Biol Downloaded from informahealthcare.com by University of Ottawa on 11/24/14 For personal use only.
Figure 1. Age dependence of the RI (+), 86(SH/height)3 (), (BM/SH3)/10 (œ) and the BMI (*) in males according to the mean data of Hansman (1970). BM/SH3 is the SHIB.
that is followed by a much smaller rise and a levelling off towards adulthood. This suggests that some function of the ratio SH/height and of RI might vary very little with age and provide a useful index of body build. Dimensional analysis (see Discussion) suggests that an appropriate function might be RI/(SH/height)3. This is numerically equal to BM/SH3. Bardeen (1923) found mean values of 100.BM/SH3 to be nearly constant (8.19–8.50 g/cm3) and uncorrelated with age in girls over the age range 6.5–17.5 years (so that, in terms of the units kg and m, mean BM/SH3 varied from 81.9–85.0 kg/m3). He did not suggest any rationale or significance for this index and it has attracted little subsequent attention, possibly because it ignores the obvious contribution of the legs to BM and because younger children, in which SH/height is most age-dependent, were not considered. Here the age-dependence of the SHIB is explored further using published data. If the index has validity, it would imply that BM may be better predicted from SH than from total height. Although this ignoring of leg length may seem counterintuitive, a separate argument for doing it is provided in the Discussion. This is based on the finding that LL has a negative influence on the leg’s mean cross-sectional area in adults (Burton et al., 2012). If valid, the SHIB would be in competition with the BMI as an index of body build and an indicator of adiposity. As already noted, the BMI has the disadvantages of being very dependent on age in children and influenced by relative leg length. Another defect of the BMI is that, for geometrically similar (isometric) individuals of different sizes, the larger ones have higher BMIs. It is hoped that the SHIB will prove to be an improvement on the BMI, one that is much needed in these times of widespread obesity.
Methods This study utilizes only published information, consisting principally of two data sets (Hansman, 1970; Waaler, 1983). These include means, but not individual values, of BM, height and SH for wide age ranges and for males and females separately. Means of SHIB, RI and BMI were estimated from these means and means of LL were calculated as the differences between those given for height and SH. The data of Hansman (1970), collected in Denver, CO from
Female
Waaler (1983) Male
Female
Number of individuals 2664 2493 2017 1948 Number of age groups 42 39 14 14 Individuals/age group 24–92 12–95 37–201 50–182 Age range, years 1–25 1–24 3.2–16 3.2–16 SH range, m 0.49–0.94 0.47–0.88 0.58–0.90 0.56–0.88 LL range, m 0.27–0.86 0.26–0.79 0.41–0.85 0.41–0.78 Mean SHIB ± SD 86.2 ± 3.0 84.1 ± 2.8 80.2 ± 3.0 80.6 ± 2.3 Range of SHIB 81.2–92.7 78.4–90.4 76.0–84.7 76.1–84.2 Range of RI 11.5–23.0 11.6–23.2 11.4–16.4 11.7–16.4 Range of BMI 13.5–23.5 15.3–21.9 15.5–20.9 15.3–20.3 CV of SHIB, % 3.5 3.3 3.7 2.9 CV of RI, % 19.8 19.6 11.7 11.3 CV of BMI, % 12.8 12.2 8.8 10.2 All of the following relate to published means at each age. SH, sitting height; LL, leg length; SHIB, (body mass)/SH3; RI, Rohrer Index (body mass)/height3; BMI, body mass index, (body mass)/ height2; SD, standard deviation; CV, coefficient of variation (100 SD/mean).
1933–1966, are ‘‘mainly for upper-class ‘Old Americans’ of northwest European descent’’ (in Forward, p. viii). From ages 1–17.5 years (males) and 1–15.5 (females) the data are for half-year age groups. For older subjects the means are for 1year groupings. The data of Waaler (1983) were collected in Bergen, Norway in 1971–1974. The mean ages, ranging from 3.2–16.9 years, are for age groups of 0.9–1.1 years. Table 1 shows relevant characteristics of the data sets. The numbers of individuals for each age group are for SH, which were sometimes fewer than for height and BM. In the case of the data of Hansman (1970), the means are less reliable for individuals older than 17 years inasmuch as the sample sizes then average only 32 for the males and 33 for the females, compared with 72 and 71, respectively, for the younger individuals. BM1/3 was regressed on SH and LL to produce equations of the form: BM1=3 ¼ a:SH þ b:LL þ c ð1Þ Statistical analyses were carried out using Excel 2007 (Microsoft Corporation, Redmond, WA). The units used are kilograms and metres.
Results Although common experience shows that SH and LL are not proportional to each other throughout childhood, their correlation coefficient is 0.987–0.996 for each of the four sets of mean values. Regression of BM1/3 on SH and LL gave the following equations. Hansman ð1970Þ: Males BM1=3 ¼ 4:27:SH þ 0:20:LL 0:02 Hansman ð1970Þ: Females BM1=3 ¼ 4:83:SH 0:33:LL 0:12 Waaler ð1983Þ: Males BM1=3 ¼ 3:78:SH þ 0:58:LL þ 0:03
ð2Þ
ð3Þ
ð4Þ
Sitting height as a predictor of body mass
DOI: 10.3109/03014460.2014.954615
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Why is SH cubed?
Figure 2. Age dependence of the mean ratio BM/SH3. (a) Based on data of Hansman (1970). (b) Based on data of Waaler (1983). Squares and circles (œ, *) are for females. Crosses (+, ) are for males.
Waaler ð1983Þ: Females BM1=3 ¼ 5:13:SH 0:57:LL 0:23
ð5Þ
Of the regression parameters, only the values of a are significant (p50.003). The ratios b/a are, respectively, 0.05, 0.07, 0.15 and 0.11. Table 1 shows the mean values of the SHIB over the whole age ranges, their coefficients of variation and the corresponding coefficients of variation for RI and BMI. Figure 1 illustrates, for white males, the age dependence of RI, BMI, (SH/height)3 and BM/SH3. The latter two are multiplied by 86 and 0.1, respectively, in order to adjust their positioning on the graph. Graphs for the other data sets show the same general features. Figure 2 shows the age dependence of the ratio BM/SH3 for all four data sets.
Discussion The four data sets are in good agreement. Mean SHIB, calculated as BM/SH3 but also equal to RI/(SH/height)3, shows a much lower age-dependence over the full age range than either mean BMI or mean RI (Table 1, Figure 1). The low age-dependence is also illustrated in Figure 2, but made less immediately obvious because the vertical scale was chosen to accentuate the small, but consistent fluctuations that occur. They arise from the slight mismatches in the time courses of RI and SH/height (Figure 1) and need not correspond exactly in timing to significant growth episodes. In Figure 2 the erratic scatter above the age of 17 years that is evident in the data of Hansman (1970) may be due to the smaller sample sizes; it relates more to BM than to SH. The concept of the SHIB accords with the similarity in time courses of change in RI and SH/height. However, two aspects of its form need to be discussed: the cubing of SH and the disregard of LL and total height.
As Quetelet (1842) wrote in the English edition of one of his books, ‘‘If man increased equally in all his dimensions, his weight at different ages would be as the cube of his height’’. Cubing of SH in the SHIB and of height in the RI is justified by dimensional or unit analysis (Burton, 2008; McMahon & Bonner, 1983). Appropriate units for SHIB, as for RI, are kg/m3. However, with BM replaced with body volume, e.g. by dividing BM by body density (which generally varies by only a few per cent), the indices become dimensionless (unit-free) numbers. They are then fully compatible with dimensionless measures of relative fat content (e.g. percentage). Unlike the Benn Index and the BMI, which are based on population statistics, the SHIB and RI have physical interpretations. Thus, they are constant for any hypothetical isometric (geometrically similar) bodies of equal density so that, if density and the index in question remain constant over a given period, then growth is isometric. If the SHIB or RI change, then growth in regard to BM and either SH or height, respectively, is not isometric. That BM/SH3 varies so little over a wide range of body sizes implies that if the height exponent were estimated by non-linear regression of BM on SH it should be close to three. Actual estimates, based on the mean data of Hansman (1970) and Waaler (1983), are 3.03–3.20 for the two sexes and are thus in close agreement. The influence of SH/height on RI in adults In order to understand the influence of SH/height on RI one must consider the relative contributions to BM of upper body mass (UBM) and of the combined mass of the two legs. These masses correspond, respectively, to the two lengths SH and LL. In a study of adults (Burton et al., 2012) the ratio of leg mass to UBM averaged 0.39 for men and 0.43 for women, with the two masses tending to vary proportionately with each other (with r ¼ 0.88 for men and 0.89 for women). (It is relevant here that the girths of the legs and torso, where they meet, visibly tend to vary together.) The low ratio and high correlation would both reduce the importance of LL in the prediction of BM, but probably not enough to justify the complete disregard of LL. More crucial findings were that the mean cross-sectional area (CSA) of one leg minus the foot tended to vary almost in proportion to UBM and that, for a given UBM, the mean CSA showed a marked negative correlation with leg length. With the volume of a leg being equal exactly to its mean CSA multiplied by its length, this volume (approximately proportional to its mass) will vary less than proportionately with LL and approximately proportionately with UBM. In fact the UBM, not LL, will be the dominant influence on leg mass. Multiple regression equations given previously for log(mean CSA) on log(UBM) and log(LL) in adults (Burton et al., 2012) allow a more quantitative approach. Expressed in non-logarithmic form and converted to kg, m and m2, these equations are: Mean CSA ¼
3:02:UBM0:91 for men LL0:52
ð6Þ
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R. F. Burton
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and Mean CSA ¼
3:16:UBM LL0:39
0:93
for women
ð7Þ
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Unlike Equations (2)–(5), these are calculated for individual data. According to Equations (6) and (7), with each leg volume taken as the product of mean CSA and LL and with the density taken as 1050 kg/m3 (Burton et al., 2012), BM approximates to [UBM + 0.63.UBM0.91.LL0.48] for men and [UBM + 0.66.UBM0.93.LL0.61] for women. To relate BM just to SH and LL, one may, for the sake of illustration, take the UBM as exactly proportional to SH3 (corresponding to geometrical similarity) and, thus, equal to KmSH3 in men and KfSH3 in women, where Km and Kf are constants. Then: 0:48 LL BM ¼ Km þ 0:63:Km0:91 SH3 for men ð8Þ SH0:27 and BM ¼
0:61 LL Kf þ 0:66:Kf0:93 SH3 for women SH0:21
ð9Þ
Because SH and LL are correlated in adults (Burton et al., 2013), the expression in square brackets for each sex should vary so little that, for any plausibly realistic value of Km or Kf, BM/SH3 would vary by only a few percent. Therefore BM would be minimally sensitive to variations in LL. In illustration, the expressions in square brackets, calculated from the data of Burton et al. (2013), are 0.86–1.00 in men and 0.93–1.22 in women. Accordingly, with Km and Kf suitably chosen as, say, 60 and 55, respectively, BM/SH3 equals 82.5–86.1 in men and 80.5–88.5 in women, values comparable to those in Table 1. What is significant about these calculated ranges is not their actual values, but their narrowness. The forms of Equations (8) and (9) cannot be expected to represent exact and algebraically appropriate functional relationships, but, although tentative and approximate, they do clearly illustrate a much greater influence of SH on BM in adults as compared with LL. Equations (6)–(9) are for adults. The fact that BM in children of widely varying age and height depends much more on SH than on LL suggests that the negative influence of LL on a leg’s mean CSA applies to children also, but this remains to be tested. The average ratio SH/height varies amongst populations (e.g. Bardeen, 1923; Burton et al., 2013), doing so with ethnicity and socioeconomic status. The ratio is notably low in Australian Aborigines (Norgan, 1994b) and high in Canadian Inuits (Charbonneau-Roberts et al., 2005). The mean ratio is lower in children of Maya immigrants to the US than in their counterparts in Guatemala (Smith et al., 2003). These variations influence relationships between adiposity and BMI, but not between adiposity and SHIB. Future research Obvious limitations of this study are its basis in mean values rather than individual measurements, its restriction to particular populations and the lack of data on relative fat content. Further studies are, therefore, needed to test the usefulness of the SHIB, involving individual data and
contrasting populations. The most important question is whether the SHIB is a more reliable predictor of adiposity and health than the BMI. Therefore, relationships between SHIB and percentage body fat need to be established and, in that context, it would be particularly useful to explore the agedependence of the ratio (fat-free mass)/SH3. Another point to investigate is whether that ratio and the SHIB might be influenced sufficiently by developmental changes in the proportions of the upper body, such as relative head size, to justify their routine measurement. The BMI differs fundamentally from the SHIB in that the linear measure of body size (i.e. height) is squared rather than cubed. The height exponent of two is the round number that minimizes the correlation between BMI and height in adults and in some 1-year age groups of young people. This improves the correlation with percentage body fat. It is this basis in statistics rather than in dimensional analysis—coupled with the low correlation between BM and height—that produces the lower height exponent of the BMI (Burton, 2007, 2008, 2014). At the risk of introducing unhelpful complications, the height exponent of the SHIB may perhaps be lowered also to produce a better predictor of adiposity for particular age groups.
Conclusions It is shown that the sitting height is a better predictor of total body mass than is total height and that the ratio (body mass)/ (sitting height)3, the SHIB, is a valid index of build with advantages over the BMI. Another such advantage is that influences of childhood nutrition on SHIB may be studied independently from those on leg length. The exclusion of leg length from the index is largely justified by the fact that the length of a leg tends to have a negative influence on its mean cross-sectional area and, therefore, has little influence on leg mass. As noted under Future research, further studies are needed, notably on individuals and on fat-free masses, in order to explore the usefulness of the index as a predictor of adiposity and ultimately as an indicator of health risks in both clinical practice and population studies.
Declaration of interest The author reports no conflict of interest. The author alone is responsible for the content and writing of the paper.
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