Reformulation of BMI and Percent Body Fat to Remove the Height Bias in 8-year-olds


  • Richard D. Telford,

    Corresponding author
    1. Public Health Research Unit, Commonwealth Institute (Australia), Manuka, Australian Capital Territory, Australia
    2. Faculty of Medicine, Australian National University, Canberra, Australian Capital Territory, Australia
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  • Ross B. Cunningham

    1. Fenner School of Environment and Society, Australian National University, Canberra, Australian Capital Territory, Australia
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BMI and percent body fat (%BF) are both related to height (Ht) in prepubertal children, so may misrepresent childhood adiposity, especially in tall or short children. We sought to construct replacement functions for BMI and %BF that are independent of Ht. Fat mass (FM) was measured using dual-energy X-ray absorptiometry, together with Ht and body mass (BM) in 746 healthy boys and girls aged 8 years (0.34 s.d.). Relationships between BM, FM, and Ht were measured and values of p and q derived such that the functions BM. Htp and FM.BMq were unrelated to Ht. BM was not directly proportional to Ht2, BMI being significantly related to Ht in both boys and girls (P < 0.001). BM was proportional to Ht3, BM. Ht−3 being independent of Ht. Similarly, FM was not directly proportional to BM and %BF was significantly related to Ht (P < 0.001). While FM was proportional to BM2, FM.BM−1.5 was the function found to be independent of Ht. Using the 85th and 95th percentiles as the cutoffs for overweight and obesity respectively, 6.4% of the boys and 6.8% of the girls were classified differently by BMI and the Ht independent measure BM. Ht−3. Similarly, 10.1% boys and 13.7% girls were classified differently by %BF and the Ht independent measure FM.BM−1.5. We propose that improved diagnostic accuracy of body composition in 8-year-olds is provided by the BM function (BMF, BM. Ht−3) and FM function (FMF, FM.BM−1.5) replacing BMI and %BF, which both overestimate the adiposity of taller children and underestimate it in shorter children.


Despite criticisms on various grounds (1), both BMI (body mass (BM).height (Ht)−2) and percent body fat (%BF, the ratio of fat mass (FM) to BM expressed as a percentage) remain commonly used measures of adiposity both clinically and in scientific studies.

Underlying the use of BMI is the assumption that BM is directly proportional to Ht2, deviations from this proportionality indicating higher or lower relative mass than expected. Such direct proportionality would preclude any relationship between BMI and Ht. In accordance with the work of Benn (2), Cole (3) explored the value of p for which BM. Htp was unrelated to Ht. In effect, he found the value of p which gave the “best” linear regression relationship between LnBM and p.LnHt. He concluded that p was ∼2 in preschool years, increasing to 3 by 10–11 years old, decreasing back to 2 after puberty. Cole added that BM. Ht−2 is appropriate for preschool children and adults, but in between it tends to assess tall children as being overweight, a bias that could be avoided by introducing a more precise power of Ht according to the child's age. Other authors have since supported this strategy, indicating that p increases steadily between the age of 3 and 7–9 years, varying around puberty (4); and cautioning against using BMI as an indicator of adiposity, proposing P > 2.0 before the age of 15 years (5); or proposing P = 3 for children aged 3–12 years (6).

There is also evidence that %BF is positively related to Ht in children (7,8). However, underlying the above-cited work with BMI is the assumption that Ht should be unrelated to any measure of adiposity. This position is taken by Wells and colleagues (9) as they investigated the use of FM in relation to Ht as a measure of adiposity in 8-year-olds. In the current study, based on the absence of any sound rationale to expect taller 8-year-old boys and girls to be generally fatter than their shorter counterparts or shorter children to be generally leaner, we assume that Ht is not related to adiposity in this age group. Our position then is that BMI and %BF will systematically misrepresent adiposity in 8-year-olds and so contribute to both clinical and experimental errors in body composition assessment.

The purpose of the current study was to find replacements for BMI and %BF, which indicate relative weight and fat, respectively, using the same primary variables but where the values of p and q are such that BM. Htp and FM.BMq are unrelated to Ht. To do this, we studied the underlying relationships between the three primary variables BM, Ht, and FM. Given that relationships between FM, BM, and Ht vary with age and maturation (10), we chose to investigate relationships between these variables in a group of children all close to 8-years-old, thereby minimizing any influence of age and maturation. Our work is distinct from, but complements that of Wells et al. (9), who investigated indicators of adiposity in terms of FM and Ht determining that the function FM.Ht−6 was unrelated to Ht in 8-year-olds.

Methods and Procedures


The group consisted of 375 boys and 366 girls participating in the Commonwealth Institute LOOK (lifestyle of our kids) longitudinal study outlined in a methods paper (11). The children were recruited from 30 government-funded primary schools and the cohort represented 27% of all grade 2 pupils in these schools in the Australian Capital Territory. Using Australian Taxation Office statistics, the estimated average taxable family income for children in our study was close to the national average.


Ht was measured by a portable stadiometer to the nearest 0.001 m and BM by electronic scales to the nearest 0.05 kg. Body composition was measured using dual-energy X-ray absorptiometry (Hologic Discovery QDR Series; Hologic, Bedford, MA). Light clothing was worn and total body scans were analyzed using QDR Hologic Software Version 12.4:7 to generate lean tissue mass and FM and %BF was calculated. Spine and step phantoms provided by the manufacturer were scanned on a daily basis for quality control assessment; a whole-body phantom was also scanned on a weekly basis.

Statistical methods

Rather than undertake direct analysis of BMI as a response, we preferred to find a statistical model that was appropriate to study both the relationship between the means of BM and Ht as well as the deviations around the mean. Although analysis of BMI as the response variable would probably lead to similar conclusions, the variance properties of BMI, being a ratio of two random variables, may be complicated. The generalized linear model as described (12) allows us to separately address the relationships between the mean of BM and Ht and the variance of BM (itself conditional on Ht) and Ht. Our model is essentially a log-log model where the standard deviation depends on the mean, removing any need to deal with bias introduced by transformation of the response. This is achieved by specifying a log link between the linear predictor and the response. Hence we actually refer to Ln(predicted or estimated)BM which we abbreviate to LnE[BM].

Our model is formulated as:


and we assume BM has a gamma distribution, rather than a normal distribution, which deals with the nonconstant variance.

BMI assumes that the “best” allometric relationship between BM and Ht is one such that BM is proportional to Ht2. BMI for each individual is then equivalent to the simple residual from the log-log line with a slope of 2.

The appropriateness of BMI as a measure of adiposity is analyzed by quantifying the relationship between LnBM and LnHt, reformulating the model by including an offset variable as 2LnHt, and then assessing whether LnHt is a significant predictor. This allows us to assess the properties of BMI and to consider alternate measures of adiposity. Failure to include potential significant predictors in the regression equation will lead to an omission bias in the residuals.

Our model now becomes


and it follows that:


We also explored relationships using %BF within the same framework and the same model for the random variation. The ratio FM.BM−1 for each individual is then equivalent to the simple residual from the least squares line between LnFM and LnBM with a slope of 1.

In this case, LnBM is set as an offset variable and so


It follows that:


where BF is FM.BM−1 (or %BF/100).

Ethics approval

This study was approved by the Australian Capital Territory Health and Community Care Human Research Ethics Committee and the Ethics Committee of the Australian Institute of Sport. Participation by the children was entirely voluntary and informed consent was received from all the parents or guardians.


The characteristics of the group are summarized in Table 1. Recalling that our basic objective was to reformulate BMI and %BF to remove the Ht bias associated with these measurements in young children, we firstly used regression analysis to establish better allometric relationships between their component variables. We determined (i) the value of p such that HTp formed the best relationship with BM and (ii) the value n such that BMn formed the best relationship with FM. We then explored whether FM.BMn was unrelated to Ht, this being automatically the case for BM. Htp. As set out below, the allometrically optimal FM.BMn was related to Ht. Consequently, using statistical modeling to explore adjustments of n in BMn, we developed a new value q such that FM.BMq was unrelated to Ht.

Table 1.  Characteristics of the subjects
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The relationship between BM and Ht

The relationships between BM and Ht are shown in Table 2 and Figure 1a,b. The slope (β1) estimates of 2.94 and 3.22 for the boys and girls respectively indicate that a coefficient of 2 for the term LnHt is not supported by our data. On the other hand, a coefficient of 3 is well supported inferring that BM is closely related to Ht3 but not to Ht2. Therefore, the value of p for BM. Htp, where this function is unrelated to Ht, is 3. It is convenient here to refer to this as the BM function (BMF), so the BMF for 8-year-old children is BM. Ht−3.

Table 2.  The relationships between BM and Ht in 8-year-old boys and girls
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Figure 1.

The relationship of BM (kg) with Ht (m) in (a) boys and (b) girls. The dark curved line is the line of best fit, the gray line is BM vs. Ht3 (obscured or partly obscured by line of best fit), and the third line is BM vs. Ht2. Also shown are comparisons of classifications of overweight (85th–95th centile) and obesity (>95th centile) using BMI (BM. Ht−2) and BMF (BM. Ht−3). BM, body mass; BMF, BM function; Ht, height.

In the alternate formulation of our model, an offset of 2lnHt (i.e., lnHt2) shows that BM. Ht−2 (i.e., BMI) is significantly related to Ht (P < 0.001, Table 2). The relationship between BMI and Ht is described this way to remove the direct involvement of ratios in this statistical model as outlined above.

Figure 2a,b are plots of the observed values of BMI and Ht illustrating their strong relationship (P < 0.001). On the other hand, Figure 3a,b illustrate the absence of a significant relationship between BMF and Ht in the 8-year-olds.

Figure 2.

The statistically significant (P < 0.001) relationship between BMI and height (m) in (a) boys and (b) girls.

Figure 3.

The absence of a significant relationship between BMF (BM. Ht−3) and Ht (m) in (a) boys and (b) girls. BM, body mass; BMF, BM function; Ht, height.

Table 3 shows a comparison of the classifications of the children as normal, overweight, and obese according to the 85th and 95th centile cutoff points using the BMI and BMF. Twenty-four (6.4%) boys and 25 (6.8%) of the girls are classified differently as normal, overweight, or obese by BMI and BMF. Figure 1a,b illustrate how BMI overestimates overweight compared with the Ht independent BMF in taller children and underestimates it in shorter children.

Table 3.  Comparison of the classifications of children by BMI and BMF (BM. Ht−3)
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The relationship between FM and BM

The relationships between the logarithms of FM and BM are shown in Table 4 and Figure 4a,b for boys and girls, respectively. The slope estimates of 1.97 and 1.89 for the boys and girls respectively indicate that a coefficient of 1 for LnBM is not supported by our data. A coefficient of 2 is well supported indicating that FM is more closely related to BM2 than to BM. Applying an offset of LnBM (Table 4) shows that FM/BM (and therefore %BF) is significantly related to Ht (P < 0.001). The relationship between %BF and Ht is depicted in Figure 5a,b.

Table 4.  Relationships between FM and BM in the 8-year-old boys and girls
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Figure 4.

The relationship of FM (kg) with BM (kg) in (a) boys and (b) girls. The dark curved line depicts the line of best fit, the gray line FM vs. BM1.5, and the broken line FM vs. BM. Also shown are comparisons of classifications of overweight (85th–95th centile) and obesity (>95th centile) by %BF and FMF (FM.BM−1.5). %BF, percent body fat; BM, body mass; FM, fat mass; FMF, FM function.

Figure 5.

The significant (P < 0.001) relationship between %BF and height (m) in (a) boys and (b) girls. %BF, percent body fat.

Our objective was to determine a value of q for which FM.BMq is independent of Ht. In parallel with BMF, we refer to FM.BMq where this function in unrelated to Ht as the FM function (FMF). Our analysis in 8-year-old boys and girls separately indicates that if q = 1.5, then this condition is satisfied. FM.BM−1.5 then is the FMF specific to 8-year-olds. The absence of any relationship between FMF and Ht is illustrated in Figure 6a,b.

Figure 6.

The absence of a significant relationship between FMF (BM.FM−1.5) and height (m) in (a) boys and (b) girls. BM, body mass; FM, fat mass; FMF, FM function.

Table 5 shows a comparison of the classifications of the children as normal, overweight, and obese according to the 85th and 95th centile cutoff points using the %BF and FMF indicators. Thirty-eight (10.1%) of the boys and 50 (13.7%) of the girls are classified differently by %BF and FMF. Figure 4a,b illustrate how %BF overestimates fatness, as categorized by the Ht independent FMF, in taller children and underestimates it in shorter children.

Table 5.  Comparison of classification of overfat and obesity by BMI and FMF (FM.BM−1.5)
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We also found that whereas %BF is not linearly related to BMI in the boys or the girls, FMF is directly proportional to BMF in the boys and is independent of Ht, a 1:1 relationship. Thus for a 1% change in BMF in the boys there is a 1% change in FMF. This relationship does not hold in the girls, however, where the slope coefficient is not 1 but 0.74 (calculations not shown).


Our data indicate significant positive associations between BMI and Ht in both 8-year-old boys and girls. The current findings on a large group of similarly aged children are consistent with previous work (3,6,13). We also found %BF to be related to Ht in both boys and girls, these highly significant relationships in a large group of 8-year-olds adding solid support to previous evidence of relationships across various age groups (7,8).

Freedman and coauthors (7), in finding positive relationships between both BMI and %BF with Ht as did we, concluded that BMI is an appropriate indicator of adiposity in children. An alternative explanation, however, is that neither BMI nor %BF is an appropriate measure of adiposity. The latter position is the one we adopt in reference to prepubertal children as there does not seem to be any solid endocrine-related or other evidence to suggest that taller nonobese 8-year-olds are likely to be generally fatter (or shorter children leaner). If anything, the opposite might be expected if energy required for growth of lean tissue detracted from energy provided for an increase in adipose tissue.

As expressed in discussion by Freedman's group (7,13), there is evidence of a relationship between adiposity and Ht during pubertal development, and during this period it may be hypothesized that changes in hormonal secretion are influential. However, we believe any such effect is unlikely in our 8-year-olds (s.d. 0.3), none of whom were selected for overweight or obesity, for the following reasons. First, Freedman and colleagues found that relationships between measures of adiposity and Ht were stronger in 5- to 8-year-old subjects (99% of whom were in Tanner stage 1) than in older children and also that differences in sexual maturation could not explain associations between adiposity and Ht (7). Second, we found that the best description of the allometric relationship between FM and BM in our 8-year-olds emerged when FM was expressed as a function of BM2, and when we investigated the relationship of Ht with FM.BM−2 it was indeed negative rather than positive or nonexistent (data not shown). Third, although we did not measure sexual maturation in our subjects, it is unlikely that the number of 8-year-olds who may have surpassed Tanner stage 1 would have exerted sufficient impact to significantly influence the relationships displayed between BMI and %BF with Ht in the 376 boys as well as the 365 girls.

The practical significance of our work is that the adiposity of children aged 8 years and possibly of prepubertal children in general (4) may be misclassified by tables or charts based on BMI or %BF. Although charts based on BMI adopted by the International Obesity Task force (14) or by the Centre for Disease Control (15) make allowances for age and gender, our data suggest they should also take account of Ht. On the assumption that true adiposity is not related to Ht, our data indicate that misclassification of our 8-year-olds by BMI and %BF occurred in ∼6 and 12%, respectively, of the children. A Ht-biased misdiagnosis of adiposity has obvious clinical implications and misclassification may induce errors in the calculated incidences of overweight and obesity.

To further illustrate the clear practical significance of the relationships of BMI and %BF with Ht consider two 8-year-old boys of 10 cm difference in Ht, this being just under two standard deviations. Application of the above equations relating BMI and %BF with Ht using the means of BM, Ht, and %BF provides an idea of the impact of Ht in general terms. A 10 cm Ht difference corresponds with a 1.2 kg.m−2 increase in BMI, 2.1 kg increase in BM, and a 2.4 unit increase in %BF amounting to 0.7 kg of BF. The equivalent figures in the girls are more pronounced, 10 cm increase in Ht corresponding with 1.6 increase in BMI (2.8 kg extra BM) and 3.8 increase in %BF (1.1 kg of BF).

Of interest is that over recent decades the average Ht of children has increased an average of ∼1 cm per decade depending on the age, period, and ethnicity (16,17). The measurement used almost exclusively in reports of increased obesity levels over recent decades has been BMI. Should the relationships between BMI and Ht demonstrated in the current study apply to children measured longitudinally, then part of the increase in adiposity in 8-year-olds and possibly in prepubertal children in general over recent decades may be an artefact of the use of BMI.

On the assumption that in a typical nonselected group of Australian 8-year-olds taller children are generally no fatter than average and shorter children no leaner, we conclude that BMI and %BF are inappropriate measures of their body composition as both measures are related to Ht. However, BMI and %BF remain popular measures in clinical as well as scientific pediatric work. Therefore, with a view to maintaining the same variables in the numerator and denominator, we suggest that BMI be replaced by the BMF, BM. Ht−3 and %BF be replaced by the FMF, FM.BM−1.5, so removing the Ht bias and improving body composition classification. Further work is required to develop BMF and FMF specific to other age groups and ethnicities.


Drs Reynolds and Lafferty, Department of Pediatrics and Child Health and Dr Javaid of Nuclear Medicine at The Canberra Hospital for their comments on the manuscript; Rohan M. Telford of the Commonwealth Institute (Australia) and Department of Psychology, Australian National University for his assistance with the preparation of the manuscript and general administration of the project and Karen J. Gravenmaker of the Department of Geriatrics, Canberra Hospital, for performing the DXA scanning. We express our thanks to the Commonwealth Institute (Australia) and the Commonwealth Education Trust (UK) for funding this work.


The authors declared no conflict of interest.