Fig 1.
Classes of BMI validity issues.
(a) The basis for the BMI formula: How is this formula derived? (b) Validity of BMI as an objective height-adjusted measure in a population with differences in height: Is BMI a height-independent measure? (c) Validity of BMI cutoff values: How do we determine the cutoff value?
Fig 2.
Relationship between weight-for-height and BMI-for-height distributions.
(a) Example of weight-for-height distribution. The weight-for-height relationship (the median curve of the weight) can be empirically approximated by a power law of w ∝ h2, where ∝ denotes a proportional relation. (b) The BMI-for-height distribution converted from (a). The horizontal solid lines represent the cutoff values for thinness, underweight, overweight, and obesity from bottom to top. (c) Weight-for-height cutoff curves correspond to the BMI cutoffs shown in (b).
Fig 3.
Schematic illustration of qth quantile (centile) regression.
In the extended allometric analysis, the qth centile (quantile) curve of a weight-for-height distribution is fitted by a power-law function wq(h) = Cq = hα(q).
Fig 4.
Schematic illustration of uni- and multi-scaling.
(a) Uni-scaling in weight-for-height distribution. In a double-logarithmic plot, all centile curves are parallel straight lines. (b) Multi-scaling in weight-for-height distribution. In a double-logarithmic plot, slopes of the centile curves vary with position. (c) Uni-scaling centile curves in BMI-for-height distribution. When uni-scaling, two points with the same BMI are on the same 90th centile curve, A and B. (d) Multi-scaling centile curves in BMI-for-height distribution. When multi-scaling, two points with the same BMI are on different centile curves, A and B.
Fig 5.
Meaning of the uni-scaling criterion.
The upper figure schematically depicts a body weight distribution of people with 2nd, 10th, 50th, 90th, and 98th centile weights from the three different height groups (red, green, and blue represent the tall, medium, and short height groups, respectively). If the uni-scaling condition of the weight-for-height distribution is fulfilled, the BMI values place the weight centile points of each group in the same position (indicated by triangles), as depicted in the middle row. Thus, the comparison of BMI corresponds to a comparison of weight centile positions adjusted by heights. In contrast, if multi-scaling, the BMI cannot make the lower and higher centile positions comparable, although it makes the 50th centile (median) weight at the same position, as depicted in the bottom row.
Fig 6.
Flow chart of the exclusion process from initial to final study population.
See Table 1 for details on age composition.
Table 1.
Number of samples for each age and sex after removing samples with missing values.
Fig 7.
Smoothed bootstrap by adding noise.
(a) Sample plot of integer height and weight data. (b) Histogram of integer height data. (c) Sample plot of height and weight data after adding noise. (d) Histogram of height data after adding noise.
Fig 8.
Extended allometric analysis results.
(a) 17-year males. (b) 17-year females. (c) 8-year males. (d) 8-year females. Solid lines show the quantile regression results for 2nd, 10th, 50th, 90th, and 98th centiles in a double logarithmic scale. Estimated scaling exponents (slopes of the plots), α(2), α(50), and α(98), are shown in each panel.
Fig 9.
Age dependence of scaling exponents.
The estimated scaling exponents of the 2nd, 10th, 50th, 90th, and 98th centiles for the weight-for-height distribution are plotted from bottom to top. (a) Male. (b) Female. If the five lines have almost the same value, it indicates uni-scaling; if the values are different, it indicates multi-scaling.
Fig 10.
Weight and standardized BMI of 17-year-old females.
Example of uni-scaling. (a) Weight-for-height distribution (gray points) in the log-log scale. (b) Standardized BMI w/h1.85-for-height distribution (gray points) in the linear scale. In panels (a) and (b), the contour lines are drawn with solid black lines; the 2nd, 50th, and 98th quantile regression curves are described by lines; the 2nd, 50th, and 98th centile points of the weight distribution in each height stratum are plotted as circles, triangles, and diamonds, respectively. (c) The weight distribution of each height stratum. (d) The standardized BMI w/h1.85 distributions of each height stratum. In panels (c) and (d), the 2nd, 50th, and 98th centile points of the distribution in each height stratum are plotted as circles, triangles, and diamonds, respectively. Ten height strata are defined in the range of the median ±3 SD.
Fig 11.
Weight and standardized BMI of 8-year-old females.
Example of multi-scaling. (a) Weight-for-height distribution (gray points) in the log-log scale. (b) Standardized BMI w/h2.73-for-height distribution (gray points) in the linear scale. In panels (a) and (b), the contour lines are drawn with solid black lines; the 2nd, 50th, and 98th quantile regression curves are described by lines; the 2nd, 50th, and 98th centile points of the weight distribution in each height stratum are plotted as circles, triangles, and diamonds, respectively. (c) Weight distributions per height stratum. (d) Standardized BMI w/h1.85 distributions per height stratum. In panels (c) and (d), the 2nd, 50th, and 98th centile points of the distribution in each height stratum are plotted as circles, triangles, and diamonds, respectively. Ten height strata are defined in the range of the median ±3 SD.
Fig 12.
Limitation of uncorrelated criterion.
Schematic depictions of the relationship between BMI and height. There is no correlation between BMI and height for either (a) or (b). Although panel (a) indicates no height dependence of each centile of BMI, panel (b) indicates height dependence of centiles except the median.
Fig 13.
Comparison between BMI centiles and international cutoffs?.
The higher and lower centiles of our data are plotted with the international cutoffs established by the International Obesity Task Force (IOFT) [39, 40]. Age dependence of 80th to 99.9th centiles for male (a) and for female (b), and 0.1th to 20th centiles for male (c) and for female (d).
Fig 14.
The upper figure schematically depicts a body weight distribution of people in the 2nd, 10th, 50th, 90th, and 98th centile weights from the three different height groups (red, green, and blue represent the tall, medium, and short height groups, respectively). The middle figure depicts people shown in the upper figure sorted by their values of extended BMI (BMIext(2)), in which the 2nd centile weight positions of each group coincide (indicated by triangle). The bottom figure depicts people shown in the upper figure sorted by their values of extended BMI (BMIext(98)), in which the 98th centile weight positions of each group coincide.
Fig 15.
Weight and standardized BMI of 11-year-old males.
Example of multi-scaling. (a) Weight-for-height distribution (gray points) in the log-log scale. (b) Conventional BMI w/h2-for-height distribution (gray points) in the linear scale. In panels (a) and (b), the contour lines are described by solid black lines; 2nd and 50th quantile regression curves are described by lines; 2nd and 50th centile points of the weight distribution in each height stratum are plotted as circles and triangles, respectively. (c) Weight distributions of each height stratum. (d) Conventional BMI w/h2 distributions of each height stratum. In panels (c) and (d), the 2nd and 50th centile points of the distribution per height stratum are plotted as circles and triangles, respectively. Ten height strata are defined in the range of the median ±3 SD.
Fig 16.
Extended BMI distributions of 11-year-old males.
(a) BMIext(2)-for-height distribution in the linear scale. (b) BMIext(2) distributions of each height stratum. Figure formats of (a) and (b) are the same as Fig 15(b) and 15(d), respectively, although for the extended BMI instead of conventional BMI.
Table 2.
Estimated parameters, α(q) and Cq for 2nd, 10th, 90th, and 98th centile for males.
Table 3.
Estimated parameters, α(q) and Cq for 2nd, 10th, 90th, and 98th centile for females.
Fig 17.
Age dependence of scaling exponents.
The average ages at which the major developmental phenomena occur are indicated on the same plot as Fig 9. Values of the scaling exponents can be interpreted as the weight increase rate adjusted for height.