Study population

We included for analysis data from adult individuals 20 years of age and older who participated in the National Health and Nutrition Examination Survey (NHANES) 1999–2006. NHANES 1999–2004 data (n = 12,581) were used for model development and NHANES 2005–2006 data (n = 3,456) were used for model validation. Participants selection for the development and validation datasets is shown in Fig. 1. Characteristics of the participants studied are described in Table 1. Mean values of whole-body fat percentage measured by dual energy X-ray absorptiometry (DXA) in the development and validation datasets were 39.9% and 39.4% in women, and 28.0% and 27.8% in men, respectively. The frequencies of DXA multiply imputed data in the development and validation datasets are described in Supplementary Tables 1 and 2, respectively.

Figure 1 Flow diagram of participant selection for the development and validation datasets. DXA, dual energy X-ray absorptiometry. Full size image

Table 1 Characteristics of adult individuals (≥20 years old) included in the study*. Full size table

Model development, performance, and selection

Supplementary Table 3 shows correlation matrix among the commonly used anthropometrics including body weight, height, BMI, triceps and subscapular skinfolds, arm and leg lengths, and waist, calf, arm and thigh circumferences. Since arm and leg lengths showed poor correlation with body fat percentage, they were excluded from further analysis. In total, 365 anthropometric indices were empirically generated and tested for correlation with body fat percentage (see Supplementary Table 4 for a full list of all indices generated).

Equations were derived using linear regression. Our selected regression models included those based on the simplest indices with the highest correlation with body fat percentage among women and among men. Among the 365 generated indices, height3/(waist × weight) showed the highest correlation with whole-body fat percentage among women (r = −0.81; P < 0.001). (√Height)/waist equation showed the highest correlation with whole-body fat percentage among men (r = −0.85; P < 0.001). Height3/(waist × weight) showed slightly stronger correlation than the simple 1/BMI (r = −0.79; P < 0.001) among women. Among men, (√height)/waist showed slightly stronger correlation than the simpler index height/waist (r = −0.84; P < 0.001). Height2/(waist × √weight) showed high correlations both among women and men. Thus, we finally selected the five aforementioned indices to evaluate model performance.

Given height/waist is the reciprocal of the widely used waist-to-height ratio, we also examined the predicting ability of waist/height index. Height/waist better predicted whole-body fat percentage and showed lower root mean squared error (RMSE) than waist/height among men and women, and across ethnic groups (Supplementary Table 5) and age categories (Supplementary Table 6). Thus, we dropped waist/height from further analysis. Supplementary Fig. 1 shows improved linear relationship between whole-body fat percentage and height/waist by sex and ethnicity. All selected models showed lower prediction of body fat percentage in older individuals (Supplementary Table 6). We found a progressive decline in body weight, height and fat-free mass after 50 years of age, and a steeper decline in fat mass and waist circumference after 70 years of age among women and men (Supplementary Fig. 5), which coincided with the lower predicting ability of all models in older individuals.

For practical reasons, performance analyses of all selected models presented here were tested using their rounded and simplest expression (details are provided in the Supplementary material). Raw equations are shown in Supplementary Table 7. Concordance coefficients between DXA-measured whole-body fat percentage and final selected models are shown in Supplementary Table 8.

All selected models showed higher accuracy than BMI among women, whereas precision was improved only in models based on three anthropometrics and 1/BMI (Supplementary Table 9). Among men, height/waist equation showed the highest accuracy, and was also superior to BMI. Models based on three anthropometrics but not 1/BMI were also more accurate than BMI. All models but not 1/BMI were more precise than BMI among men (Supplementary Table 9).

Height/waist equation, named as the relative fat mass (RFM), was the final model selected because of its simplicity (it requires only two common anthropometrics), it was superior to BMI in predicting body fat percentage among men, had similar predicting ability relative to BMI among women and had overall better performance than BMI among women and men, independently.

Final equations are as follows:

$${\rm{Equation}}\,{\rm{for}}\,{\rm{women}}:\,76-(20\times ({\rm{height}}/{\rm{waist}}))$$ (1)

$${\rm{Equation}}\,{\rm{for}}\,{\rm{men}}:\,64-(20\times ({\rm{height}}/{\rm{waist}}))$$ (2)

or

$${\rm{RFM}}:\,64-(20\times ({\rm{height}}/{\rm{waist}}))+(12\times {\rm{sex}})$$ (3)

In equations (1–3), height and waist (circumference) are expressed in meters. In (3), sex = 0 for male and 1 for female. The coefficients for equations (1) and (2) were rounded for practical purposes.

Supplementary Fig. 3 shows good agreement between RFM and DXA.

Although we found a significant interaction between age and RFM among women (P < 0.001), that was not case among men (P = 0.088). However, inclusion of age in the final model did not improve R2 among women (RFM model: R2 = 0.66; RFM and AGE model: R2 = 0.66) or among men (RFM model: R2 = 0.75; RFM and AGE model: R2 = 0.75). Likewise, inclusion of ethnicity in the final model did not substantially increased R2 among men (RFM and ethnicity model: R2 = 0.76). Among women, inclusion of ethnicity in the model did not improve body fat prediction (R2 = 0.66). Thus, age and ethnicity were not included in our final model selected.

Model validation and performance

In the validation dataset, compared with BMI, RFM had a more linear relationship with DXA whole-body fat percentage among women (adjusted coefficient of determination, R2: 0.69; 95% CI, 0.67–0.72; vs. 0.65; 95% CI, 0.63–0.67) and men (R2: 0.75; 95% CI, 0.72–0.77 vs. 0.61; 95% CI, 0.59–0.63) (Fig. 2 and Supplementary Table 10). RFM had less bias than BMI among women (0.9%; 95% CI, 0.6% to 1.1% vs. −10.9%; 95% CI, −11.2% to −10.5%) and a similar low bias among men (RFM: 0.5%; BMI: 0.7%) (Table 2). Among women, RFM showed higher accuracy than BMI (91.5% vs. 21.6%; P < 0.001). RFM was also more precise than BMI (4.9%; 95% CI, 4.6–5.2% vs. 5.8%; 95% CI, 5.5–6.2%). Among men, RFM showed higher accuracy than BMI (88.9% vs. 81.9%; P < 0.001) and better precision (RFM: 4.2%; 95% CI, 3.9–4.6% vs. BMI: 5.1%; 95% CI, 4.9–5.4%) (Table 2 and Supplementary Fig. 4). Among women, RFM was more accurate than BMI across ethnic groups (P < 0.001 for all comparisons). Among men, RFM was also more accurate among European-Americans (P < 0.001) and African-Americans (P < 0.001) (Table 2). RFM also showed better performance than BMI across age categories (Supplementary Fig. 5) and across body fat quintiles (Supplementary Fig. 6). Among men, RFM also showed better performance than CUN-BAE (Clinica Universidad de Navarra-body adiposity estimator), Gallagher, Deurenberg and Kagawa equations, including across ethnic groups. Among women, RFM was superior to Deurenberg and Kagawa equations (Table 2).

Figure 2 Prediction of whole-body fat percentage by RFM using linear regression in NHANES 2005–2006 (validation dataset). RFM, relative fat mass, which is based on height/waist. R2, coefficient of determination; RMSE, root mean squared error. Data plots correspond to DXA imputation 1. Full size image

Table 2 Comparison of performance between RFM and published equations based on BMI or waist-to-height ratio for prediction of body fat percentage among adult participants (n = 3,456) in the validation dataset (NHANES 2005–2006)*. Full size table

Internal validation with bootstrapping confirmed RFM was a better predictor of body fat percentage than BMI among women and men (Supplementary Table 11). RFM predicting ability decreased with age (Supplementary Table 12). RFM was more accurate and more precise than BMI (Supplementary Table 13) and had superior accuracy than BMI across age categories (Supplementary Fig. 7 and Supplementary Table 14) and body fat ranges; however, accuracy was lower in leaner individuals (Supplementary Fig. 8).

RFM was a better predictor of trunk fat percentage than of whole-body fat percentage or whole-body fat mass (Supplementary Table 15).

Obesity misclassification

To compare the rates of obesity misclassification between BMI and our final model, we arbitrarily defined obesity as DXA-measured body fat percentage ≥33.9% for women and ≥22.8% for men, based on the corresponding cut-points between the first and second quintiles for each sex. These cut-points were calculated using combined datasets (NHANES 1999–2006). In the validation dataset, when using same DXA cut-points for obesity diagnosis (≥33.9% for women and ≥22.8% for men), RFM had higher sensitivity than BMI. Table 3 shows total positive and negative cases of obesity identified using either BMI or RFM. RFM resulted in fewer false negatives among women (5.0%; 95% CI, 3.1–6.8% vs. 72.0%; 95% CI, 67.3–76.6%; P < 0.001) and men (3.8%; 95% CI, 1.8–5.8% vs. 4.1%; 95% CI, 2.1–6.1%; P < 0.001). There were fewer false positives with RFM among men (32.3%; 95% CI, 25.8–38.8% vs. 49.7%; 95% CI, 44.2–55.3%; P < 0.001) but more false positives among women (41.0%; 95% CI, 32.2–49.9% vs. 0%; P < 0.001).

Table 3 Positive and negative cases of DXA-diagnosed obesity* identified using either BMI or RFM among adult participants (n = 3,456) in the validation dataset (NHANES 2005–2006). Full size table

Obesity total misclassification was also lower with RFM than with BMI among all women (12.7% vs. 56.5%; P < 0.001) and all men (9.4% vs. 13.0%; P < 0.001) (Fig. 3), and among all Mexican-Americans (8.2% vs. 35.4%; P < 0.001), all European-Americans (11.3% vs. 35.2%; P < 0.001) and all African-Americans (9.9% vs. 37.2%; P < 0.001).

Figure 3 Obesity total misclassification error in NHANES 2005–2006. Bars show comparison of total misclassification of obesity diagnosed by DXA-whole-body fat percentage (≥33.9% for women and ≥22.8% for men, based on the corresponding cut-points between the first and second quintiles for each sex) when using RFM and BMI at same DXA cut-points and a BMI of 30. Error bars are standard error. Full size image

In the internal validation dataset, compared with BMI, obesity total misclassification was lower with RFM among women (P < 0.001) and men (P < 0.001), among all Mexican-Americans, all European-Americans and all African-Americans (P < 0.001 for all three ethnic groups), and across age categories (P < 0.001 for all comparisons). Although we found a lower total misclassification rate with RFM among other ethnicities (Non-Hispanic Asians, Native Americans, and those who self-reported multiple ethnicity) (RFM: 12.9%, BMI: 41.9%; P < 0.001), these findings should be interpreted with caution as NHANES 1999–2006 did not oversample to get reliable estimates on these minority American ethnic groups.

Diagnostic accuracy for obesity and diabetes

In the validation dataset, compared with BMI, RFM showed better diagnostic accuracy for body fat-defined obesity among men (area under curve [AUC]: 0.94 vs. 0.91; P < 0.001) and similar diagnostic accuracy among women (AUC: 0.929 vs. 0.933; P = 0.52). RFM was also better than BMI in identifying diabetes cases among women (AUC: 0.79 vs 0.73; P = 0.002) and men (AUC: 0.80 vs. 0.76; P = 0.001).

Sensitivity analysis of the combined datasets showed RFM had a better diagnostic accuracy than BMI for high body-fat percentage among men (P < 0.001) regardless the DXA cut-point used to define obesity (Supplementary Fig. 9). RFM also showed a significant improvement over BMI and Gallagher, CUN-BAE and Deurenberg equations among men (Supplementary Table 16).

RFM was superior to DXA-measured trunk fat percentage in discriminating diabetes among women (P < 0.001) but not among men (P = 0.548) (Supplementary Fig. 10).