## Abstract

Although body mass index (BMI) has been adopted by WHO as an international measure of obesity, it lacks a theoretical basis, and empirical evidence suggests it is not valid for all populations. We determined standard weight‐for‐height using a model calibrated by multivariate analysis of observational data on body dimensions and health status in the USA (NHANES III). A multiple linear regression model based on a simple mathematical formulation accurately described the observed weight variations in this normal adult population. A standardized reference model using just two measurements (upper arm length and sitting height), readily applied in both clinical and research settings using lookup tables, improved explanatory power substantially compared to the best BMI formulation (r2 increased 16.3% for males, 21.1% for females). Physical dysfunction and self‐reported poor health showed strong trends with excess body weight. These findings need confirmation from larger population samples.

## Introduction

Rapid increases in the prevalence of obesity in the US,1 the associated disease burden2 and additional mortality risk3–5 have highlighted the importance of ‘weight for height’ assessment in clinical practice. Despite numerous attempts at improvement,6 the international standard measure7—the Body Mass Index (BMI)—is based on Quetelet's original 19th century empirical observation8 that weight tends to vary with the square of standing height.

Most subsequent scholars have assumed that body weight can and should be measured independently of stature, yet with little anatomical or physiological basis. Keys et al.9 recommended the Quetelet index over other mathematical combinations of height and weight as showing least correlation with height and greatest correlation with body fat (skinfold thickness). Benn10 and Lee, Kolonel and Hinds11 argued strongly for height‐independence as a primary criterion for weight‐height indices of obesity, and recommended Benn's p exponent as the ideal basis for measurement. However, Garrow12 in his 1983 review commented: ‘It is fairly obvious that an index which makes all short (or tall) people appear obese is not useful, but it is not obvious to this reviewer that an index totally unrelated to height is therefore an ideal indicator of obesity’.

Unfortunately, BMI is not universally successful in practice. Garn, Leonard and Hawthorne13 observed, following analysis of NHANES I data: ‘it is clear that the BMI is not quite independent of stature, especially at the younger ages. Moreover, the BMI is influenced by body proportions (relative leg length or relative sitting height) such that shorter‐legged individuals have BMI values higher by as much as 5 units. Finally, BMI (like weight) is influenced nearly to an equal degree by both the lean and the fat compartments of the human body. It is as much a measure of Lean Body Mass as it is a measure of fatness or obesity.’

General applicability of BMI is challenged by results from studies of individual populations, reporting a wide range of estimates for Benn's p parameter (e.g. Garn and Pesick14 show p between 1.18 and 1.50 for females, and 1.65–1.83 for males). Even greater variations are likely in non‐Western populations where body composition varies markedly between ethnic groups.15 Although BMI is a generally convenient measure, it lacks a theoretical foundation, and may be compromised by ethnic, cultural or lifestyle differences.

Nonetheless, BMI has been used as a generic risk factor in epidemiological research of many pathological conditions, has been adopted as the main or sole measure of obesity in numerous clinical trials, and subsequently has been incorporated into clinical guidelines concerning screening for and treatment of various common diseases.16 Thus the weaknesses of the BMI could have important implications for public health specialists,17,18 for researchers and for clinical practitioners in many fields.

In this paper, we develop and calibrate a simple theoretical model of body weight and stature as a possible alternative to the BMI, using observational data from NHANES III, and then compare the relative performance of the two measures.

## Methods

The NHANES III survey was conducted in the USA between 1988 and 1994, and assembled data for a total of 31 409 individuals of all ages from under 1 year to over 90 years. As part of the survey, participants were subject to a detailed examination by a medical examiner, including a range of anatomical measurements and assessments of current morbidity and lifestyle factors. A subset of the Examination data‐set of NHANES III was extracted for analysis, including the body measurements shown in Table 1, together with basic demographic details (age, sex, ethnic group), smoking habits, and items relating to a variety of morbidities, self‐reported health status and physical functional capacity.

To restrict the analysis to normal healthy adults, subjects were excluded from consideration if under the age of 16 years or over 70 years in the first instance. Trend lines were then fitted to univariate plots of major variables related to stature (standing height, sitting height, leg length and upper arm length) and body weight by age, in order to determine the age range for which there was evidence of general stability at a population level. This indicated that full stature is achieved across the whole population by the age of 30, and that there is no evidence of elderly degeneration before the age of 60. Body weight tends to drift higher with age within this band, particularly for women.

As a result the definitive analyses on ‘normal healthy adults’ were focused on subjects in the age range 30–59, and were stratified by sex, current smoking status and ethnicity, with exclusions for individuals with evidence of amputations of any sort, and of pregnant women.

The main objective of the analysis was to develop statistical models of variation in body weight within the population in terms of the available anatomical measures. As a baseline for comparison, non‐linear regression was used to fit a power‐exponent model of weight against standing height (where BMI has an exponent of 2), in order to determine the maximum correlation achievable with the traditional formulation. In addition, the Benn p‐exponent was estimated (for minimum correlation with height) as a secondary comparator. Thereafter, a series of alternative formulations were tested to ascertain which offered significantly improved descriptive performance.

Since the prime objective was to investigate the hypothesis that stature can be effectively standardized when designing an index of relative weight, a principal‐components analysis was performed to expose the underlying structure of physical measurement variables within the dataset. This identified two subsets of variables, one related principally to skeletal dimensions, and the other to principal sites of adipose tissue. The strength of univariate relationships led to the identification of a shortlist of candidate anatomical measures which might feature in any new model of body weight.

### Theoretical basis for modelling body weight

$\mathrm{Mass\ of\ body}{=}2{\times}\mathrm{mass\ of\ arm}{+}2{\times}\mathrm{mass\ of\ leg}{+}\mathrm{mass\ of\ trunk}{+}\mathrm{mass\ of\ sundries}$

Each of the body cylinders can be considered to consist of a central core (bone, muscle, etc.) surrounded by a layer of fatty tissue and skin. In principle, the calculations for each component then take the same form. Thus in the case of an arm, if L=arm length, Rc=radius of the arm's core, Ro= outside arm radius, Dc=density of the arm's core, Df=density of fatty layer and C=circumference of arm, then:

$\mathrm{Mass\ of\ an\ arm}{=}{\pi}LR\mathrm{_{c}^{2}}D_{\mathrm{c}}{+}{\pi}L{(}R_{\mathrm{o}}^{2}{-}R\mathrm{_{c}^{2}{)}}D_{\mathrm{f}}{=}{\pi}L\{R_{\mathrm{c}}^{2}{(}D_{\mathrm{c}}{-}D_{\mathrm{f}}{)}{+}C^{\mathrm{2}}D_{\mathrm{f}}{/}{(}4{\pi}^{2}{)}\}$
If it is assumed that the densities are constant, and that the core radius is relatively invariant in comparison to L and C, then the mass of an arm may be approximated by an expression of the form
$\mathrm{Mass\ of\ arm}{\approx}aL{+}bLC^{\mathrm{2}}$
where a and b are expressions exhibiting little variation. Summing the estimated masses of the body components yields a function of the form
$\mathrm{Mass\ of\ body}{=}{\alpha}_{0}{+}{\alpha}_{1}{\cdot}\mathrm{Arm\ length}{+}{\alpha}_{2}{\cdot}\mathrm{Arm\ length}{\cdot}{(}\mathrm{Arm\ circumference}{)}^{2}{+}{\alpha}_{3}{\cdot}\mathrm{Leg\ length}{+}{\alpha}_{4}{\cdot}\mathrm{Leg\ length}{\cdot}{(}\mathrm{Leg\ circumference}{)}^{2}{+}{\alpha}_{5}{\cdot}\mathrm{Trunk\ length}{+}{\alpha}_{6}{\cdot}\mathrm{Trunk\ length}{\cdot}{(}\mathrm{Trunk\ circumference}{)}^{2}$
in which the αis can be estimated by multiple linear regression if the corresponding body dimensions are known from observational studies.

Table 1

NHANES III anthropometric measurements

 Measurement Unit Measurement Unit Arm circumference cm Thigh circumference cm Biacromial breadth cm Thigh skinfold mm Bi‐iliac breadth cm Triceps skinfold mm Buttocks circumference cm Upper arm length cm Elbow breadth cm Upper leg length cm Leg length* cm Waist circumference cm Sitting height cm Waist to hip ratio — Standing height cm Weight kg Sub‐scapular skinfold mm Wrist breadth cm Supra‐iliac skinfold mm
 Measurement Unit Measurement Unit Arm circumference cm Thigh circumference cm Biacromial breadth cm Thigh skinfold mm Bi‐iliac breadth cm Triceps skinfold mm Buttocks circumference cm Upper arm length cm Elbow breadth cm Upper leg length cm Leg length* cm Waist circumference cm Sitting height cm Waist to hip ratio — Standing height cm Weight kg Sub‐scapular skinfold mm Wrist breadth cm Supra‐iliac skinfold mm

*Calculated as sub‐ischial height=standing height−sitting height.

## Results

### Selection of variables

The selection of NHANES III data fields to populate the model was driven by exploratory analysis of the correlation matrix, univariate correlations with body weight, and the presence of relevant measures in the dataset. Since arm length was not available, upper arm length was used as a convenient proxy. Similarly, sitting height was chosen to represent trunk length. Although leg length and upper leg length were both available, it was found that upper arm length consistently yielded better results than either, and therefore was used as a proxy for leg length as well. Arm and thigh circumferences were available; however, an appropriate trunk circumference measure was not immediately obvious. Factor analysis suggested that buttocks and waist circumferences make independent contributions to trunk mass, and therefore two terms were introduced into the model in place of the original trunk circumference.

Thus, the form of the final model was:

$\mathrm{Mass\ of\ body}{=}{\alpha}_{0}{+}{\alpha}_{1}{\cdot}\mathrm{Upper\ arm\ length}{+}{\alpha}_{2}{\cdot}\mathrm{Upper\ arm\ length}{\cdot}{(}\mathrm{Arm\ circumference}{)}^{2}{+}{\alpha}_{3}{\cdot}\mathrm{Upper\ arm\ length}{\cdot}{(}\mathrm{Thigh\ circumference}{)}^{2}{+}{\alpha}_{4}{\cdot}\mathrm{Sitting\ height}{+}{\alpha}_{5}{\cdot}\mathrm{Sitting\ height}{\cdot}{(}\mathrm{Buttocks\ circumference}{)}^{2}{+}{\alpha}_{6}{\cdot}\mathrm{Sitting\ height}{\cdot}{(}\mathrm{Waist\ circumference}{)}^{2}$

### Primary model

All persons aged 30–59 were included in the regression analysis, with the exception of pregnant women and anyone with a reference to any form of amputation (3403 males and 3944 females). Multiple regression analysis was carried out using SPSS version 9.0 for Windows, and after listwise deletion of missing data, the models were calibrated with 3221 male and 3707 female records. The resulting models achieved adjusted R2 of 0.966 (male) and 0.963 (female), indicating remarkably effective model performance. The model coefficients and diagnostics are shown in Table 2, and include standardization variables for ethnicity and smoking status.

Table 2

Principal model coefficients for weight (in kg) of males and females aged 30–59 excluding amputees and pregnant women

 Variable Coefficient SE t p Males Constant −6.310 1.727 −3.65 0.000 Sitting height 22.499 2.065 10.90 0.000 Upper arm length 6.058 3.397 1.78 0.075 Upper arm length*(arm circumference)2 275.074 11.537 23.84 0.000 Upper arm length*(thigh circumference)2 128.207 5.837 21.97 0.000 Sitting height*(buttocks circumference)2 17.648 0.878 20.11 0.000 Sitting height*(waist circumference)2 28.646 0.584 49.07 0.000 Age (years) −0.005 0.007 −0.72 0.474 White smoker −0.071 0.183 −0.39 0.697 Black non‐smoker 2.079 0.206 10.08 0.000 Black smoker 2.273 0.197 11.53 0.000 Mexican‐American non‐smoker 0.470 0.179 2.63 0.009 Mexican‐American smoker 0.565 0.215 2.63 0.008 Other non‐smoker −0.873 0.377 −2.31 0.021 Other smoker −0.197 0.416 −0.47 0.636 Females Constant −4.751 1.707 −2.78 0.005 Sitting height 20.975 2.092 10.03 0.000 Upper arm length 10.353 3.691 2.81 0.005 Upper arm length*(arm circumference)2 256.907 9.975 25.76 0.000 Upper arm length*(thigh circumference)2 119.852 4.647 25.79 0.000 Sitting height*(buttocks circumference)2 17.086 0.656 26.03 0.000 Sitting height*(waist circumference)2 25.452 0.523 48.64 0.000 Age (years) −0.008 0.007 −1.09 0.277 White smoker −0.407 0.209 −1.94 0.052 Black non‐smoker 1.247 0.189 6.61 0.000 Black smoker 1.368 0.220 6.22 0.000 Mexican‐American non‐smoker 0.296 0.182 1.63 0.104 Mexican‐American smoker 0.787 0.286 2.75 0.006 Other non‐smoker 0.270 0.311 0.87 0.385 Other smoker −0.014 0.658 −0.02 0.983
 Variable Coefficient SE t p Males Constant −6.310 1.727 −3.65 0.000 Sitting height 22.499 2.065 10.90 0.000 Upper arm length 6.058 3.397 1.78 0.075 Upper arm length*(arm circumference)2 275.074 11.537 23.84 0.000 Upper arm length*(thigh circumference)2 128.207 5.837 21.97 0.000 Sitting height*(buttocks circumference)2 17.648 0.878 20.11 0.000 Sitting height*(waist circumference)2 28.646 0.584 49.07 0.000 Age (years) −0.005 0.007 −0.72 0.474 White smoker −0.071 0.183 −0.39 0.697 Black non‐smoker 2.079 0.206 10.08 0.000 Black smoker 2.273 0.197 11.53 0.000 Mexican‐American non‐smoker 0.470 0.179 2.63 0.009 Mexican‐American smoker 0.565 0.215 2.63 0.008 Other non‐smoker −0.873 0.377 −2.31 0.021 Other smoker −0.197 0.416 −0.47 0.636 Females Constant −4.751 1.707 −2.78 0.005 Sitting height 20.975 2.092 10.03 0.000 Upper arm length 10.353 3.691 2.81 0.005 Upper arm length*(arm circumference)2 256.907 9.975 25.76 0.000 Upper arm length*(thigh circumference)2 119.852 4.647 25.79 0.000 Sitting height*(buttocks circumference)2 17.086 0.656 26.03 0.000 Sitting height*(waist circumference)2 25.452 0.523 48.64 0.000 Age (years) −0.008 0.007 −1.09 0.277 White smoker −0.407 0.209 −1.94 0.052 Black non‐smoker 1.247 0.189 6.61 0.000 Black smoker 1.368 0.220 6.22 0.000 Mexican‐American non‐smoker 0.296 0.182 1.63 0.104 Mexican‐American smoker 0.787 0.286 2.75 0.006 Other non‐smoker 0.270 0.311 0.87 0.385 Other smoker −0.014 0.658 −0.02 0.983

### Reference model

Standardization of body weight for stature requires the estimation of the component of overall weight which is attributable to frame dimensions (skeletal structure and essential soft tissue and organs) rather than to adipose tissue. To determine the ‘standard’ weight for a particular frame, it is necessary to specify desirable values for the four circumference factors in the principal model. However, these are not independent of skeletal dimensions, and moreover the prevalence of obesity may vary within the population by stature. Three possible criteria were derivable from the NHANES III data to delineate the reference sub‐population of ‘standard’ weight for normal fit adults: freedom from morbidity, freedom from limitations to physical performance, and self‐reported general health status. Within the 30–59 age‐group, the general level of morbidity was sufficiently low to render this inadequate as a single discriminatory measure. Both the other factors involved self‐reported elements and could be considered to some degree subjective. Three criteria were therefore combined to select a subset of adults with no recorded morbidity, no reported serious limitations on physical performance (for walking, running, stooping or physical tasks) and self‐reported health status in either of the best two categories (‘Excellent’ or ‘Very good’). This yielded groups of 2333 males and 2323 females as the basis for defining ‘standard weight’ for normal healthy adults.

For these groups, each of the squared circumference variables was very well described by a linear or quadratic relationship to its corresponding length variable (upper arm length and sitting height), and adding higher polynomial powers did not significantly improve the fit. Substituting these relationships into the principal model yields the following equations to describe weight as a function of stature for White non‐smokers aged 30:

$\mathrm{{}Standard{\textquoteright}\ male\ weight\ {(}kg{)}}{=}{-}6.4704{-}3.1631{\ast}\mathrm{Upper\ arm\ length}{+}204.481{\ast}{(}\mathrm{Upper\ arm\ length}{)}^{2}{-}4.6114{\ast}\mathrm{Sitting\ height}{+}74.5385{\ast}{(}\mathrm{Sitting\ height}{)}^{2}$

$\mathrm{{}Standard{\textquoteright}\ female\ weight\ {(}kg{)}}{=}{-}4.7220{+}170.3938{\ast}\mathrm{Upper\ arm\ length}{-}848.4479{\ast}{(}\mathrm{Upper\ arm\ length}{)}^{2}{+}1585.0379{\ast}{(}\mathrm{Upper\ arm\ length}{)}^{3}{-}1.4922{\ast}\mathrm{Sitting\ height}{+}67.3340{\ast}{(}\mathrm{Sitting\ height}{)}^{2}$

where upper arm length, and sitting height are expressed in metres.

### Relationship of reference model to body mass index

The most basic test of the performance of the reference model in describing population variations in body weight is a comparison of correlation coefficients with recorded weight. Comparators were taken to be predicted weight based on the mean BMI for the study population, and predicted weight from fitted Benn p‐exponent model (where p was estimated by least squares to minimize correlation with height), and correlation coefficients were calculated for all individuals for whom all three measures could be calculated (3247 males and 3752 females). For men, Benn's p=1.987, very similar to the BMI exponent, so that both comparators exhibited a Pearson correlation with observed weight of 0.416 (r2=0.173). The reference model provided a correlation of 0.579 (r2=0.336). Benn's p for women took a value of 1.521, and correlations with weight were much weaker (BMI: r=0.257, r2=0.066; Benn's model: r=0.018, r2=0.0003). By contrast, the reference model yielded a correlation of 0.526 (r2=0.277).

The weight estimates differ very little between the two formulations in the main central portion of the height distribution (Figure 1). However, at extremes of standing height, discrepancies are evident between the weight estimates of as much as 5%. If the reference model is accepted as the more reliable, we conclude that the use of BMI tends to under‐estimate the prevalence of overweight and obesity among the tallest men and women, and over‐estimate it among the shortest.

Figure 1.

Reference model and BMI compared for expected weight by standing height.

Figure 1.

Reference model and BMI compared for expected weight by standing height.

### Defining ‘ideal’ weight compared to ‘standard’ weight

The ‘standard’ weight estimated by the reference model does not define the ‘ideal’ or target weight unequivocally. The choice of a combination of morbidity, physical performance and self‐reported health status as the selection criterion for a reference sub‐population is arbitrary and was driven by the available data: it reflects one of many possible bases on which to calibrate a body weight metric. A further issue of both theoretical and practical importance is the type of comparative measure most suited to describe variations from the ‘ideal’. BMI is a ratio measure, so differences in recorded BMI represent proportionate variations, whereas the derivation of the reference model by multiple linear regression implies that simple weight differences are more natural.

Figure 2.

Prevalence of physical dysfunction, poorer health status and any morbidity by difference from standard weight.

Figure 2.

Prevalence of physical dysfunction, poorer health status and any morbidity by difference from standard weight.

Figure 3.

Prevalence of obesity estimated by BMI and by standard weight.

Figure 3.

Prevalence of obesity estimated by BMI and by standard weight.

## Discussion

This paper demonstrates a formulation which relates height and weight, particularly at the extremes, better than the traditional BMI. This metric substitutes two linear measures (upper arm length and sitting height) in place of one (standing height): the additional measurement effort is unlikely to be prohibitive in a clinical setting, and a simple look‐up table can provide immediate interpretation in terms of a standard expected weight. By directly focusing on weight differences in common units rather than as a less intuitive ratio, it gives overweight patients a clear target weight loss to restore ideal weight.

The new metric is derived from a theoretical basis, in contrast to the purely empirical origins of BMI. Further refinement of the model is possible but probably not warranted given the quality of body measurements generally available. Not surprisingly, the two approaches display only minimal differences for the majority of people in the centre of the height distribution, but divergence becomes more apparent at the extremes, where it appears that BMI underestimates the weight appropriate for the shortest individuals and overestimates that of the tallest, resulting in a distortion of obesity prevalence patterns. Because of the small numbers of individuals of these types available for study, this finding needs to be further validated by analysis of a much larger population survey or selective studies of very tall and short people.

A key feature of the reference model is that when only fit healthy people are used for model calibration there appears to be no justification for varying weight thresholds by height—a kilogram is a kilogram for everyone. Is it therefore reasonable to expect a similar effect on morbidity from an extra 10 kg on an already obese person as for 5 kg weight gain in a slightly underweight person? There is no simple answer, as each type of disease or disability may have its own peculiar metric, but there are certainly a number of common conditions (e.g. load‐bearing articular disorders) where a linear relationship with body mass is more likely than one based on a ratio.

Preliminary analysis suggests that there is probably a range of body weight (± about 10 kg) in which little variation in morbidity or physical limitation is likely. However, a threshold can be defined above which prevalence will increase sharply, and the identification of the threshold appropriate for each disease/condition may be more important than estimating a single ‘ideal’ weight. There are indications in the analyses of possible increases for very low weights, but there are too few such observations in the survey to draw firm conclusions.

Address correspondence to A. Bagust, YHEC, University of York, Heslington, York YO10 5DD. e‐mail: ab13@york.ac.uk

## References

1
Mokdad AH, Serdula MK, Dietz WH, Bowman BA, Marks JS, Koplan JP. The spread of the obesity epidemic in the United States, 1991–1998.
JAMA

1999
;
282(16)
:
1519
–22.
2
Must A, Spadano J, Coakley EH, Field AE, Colditz G, Dietz WH. The disease burden associated with overweight and obesity.
JAMA

1999
;
282(16)
:
1523
–9.
3
Allison DB, Fontaine KR, Manson JE, Stevens J, VanItallie TB. Annual deaths attributable to obesity in the United States.
JAMA

1999
;
282(16)
:
1530
–8.
4
Calle EE, Thun MJ, Petrelli JM, Rodriguez C, Heath CW. Body‐mass index and mortality in a prospective cohort of U.S. adults.
N Engl J Med

1999
;
341
:
1097
–105.
5
Stevens J, Cai J, Pamuk ER, Williamson DF, Thun MJ, Wood JL. The effect of age on the association between body‐mass index and mortality.
N Engl J Med

1998
;
338
:
1
–7.
6
Rosenbaum S. 100 years of heights and weights.
J Roy Statist Soc A

1988
;
151(2)
:
276
–309.
7
WHO.
Physical status: the use and interpretation of anthropometry.
WHO Technical Report Series 854. WHO, Geneva,
1995
.
8
Quetelet LAJ.
Physique sociale
, vol. 2. Brussels, C. Muquardt, 1869:92.
9
Keys A, Fidanza F, Karvonen MJ, Kimura N, Taylor HL. Indices of relative weight and obesity.
J Chron Dis

1972
;
25
:
329
–43.
10
Benn RT. Some mathematical properties of weight‐for‐height indices used as measures of adiposity.
Br J Prev Soc Med

1971
;
25
:
42
–50.
11
Lee J, Kolonel LN, Hinds MW. The use of an inappropriate weight‐height derived index of obesity can produce misleading results.
Int J Obes

1982
;
6
:
233
–9.
12
Nutr Abstr Rev

1983
;
53(8)
:
697
–708.
13
Garn SM, Leonard WR, Hawthorne WM. Three limitations of the body mass index.
Am J Clin Nutr

1986
;
44
:
996
–7.
14
Garn SM, Pesick SD. Comparison of the Benn index and other body mass indices in nutritional assessment.
Am J Clin Nutr

1982
;
32
:
573
–5.
15
Norgan NG. Population differences in body composition in relation to the body mass index.
Eur J Clin Nutr

1994
;
48 (Suppl. 3)
:
S10
–27.
16
Willett WC, Dietz WH, Colditz GA. Guidelines for healthy weight.
N Engl J Med

1999
;
341
:
427
–34.
17
Koplan JP, Dietz WH. Caloric imbalance and public health policy.
JAMA

1999
;
282(16)
:
1579
–81.
18
Williamson DF. The prevention of obesity.
N Engl J Med

1999
;
341(15)
:
1140
–1.