Height and Weight Tables Are Designed to Be Used Fo
Abstract
Background: Weight for height in children is often assessed by comparing the child's weight-for-age centile with their height-for-age centile. However, this assessment has not been validated statistically, and it differs from the body mass index (BMI) centile.
Objective: To study indices of weight-for-height based on weight centile-for-age adjusted for height centile-for-age, and to see how they relate to the BMI centile-for-age.
Design: Cross-sectional survey of data for 40 536 boys and girls aged 0–18 y from the 1980 Nationwide Dutch Growth Survey, using the British 1990 and US CDC 2000 growth references.
Outcome measures: Two measures of weight for height: (a) the difference between weight centile and height centile, and (b) BMI centile, with the centiles analysed as SD scores (SDS).
Results: BMI centile is correlated strongly with weight centile (r=0.77) but weakly with height centile (r=0.1). By contrast the difference between weight centile and height centile is correlated only weakly with weight centile (r=0.3) and strongly negatively with height centile (r=−0.5). BMI centile is predicted to high accuracy by the multiple regression on weight centile and height centile (93–97% of variance explained, s.e.e. 0.2 units).
Conclusions: A child's BMI centile can be calculated to high accuracy from their weight and height centiles as read off the weight and height charts. This avoids the need to calculate BMI or to plot it on the BMI chart. A chart is provided to simplify this calculation, which works throughout the spectrum of nutritional status. It can also be used to monitor individuals' weight, height and BMI centiles simultaneously as they change over time. However the simpler procedure of comparing weight and height centiles (eg a difference of two or three channel widths) is a poor measure of weight-for-height and should not be used.
Sponsorship: Medical Research Council programme grant no. G9827821.
Introduction
Weight adjusted for height has been used to assess overweight and underweight in children for many years. In the past weight-for-height was used (Waterlow, 1972), a simple index that ignores the child's age, while more recently body mass index (BMI) adjusted for age has become popular for measuring child obesity (Cole et al, 1995; Dietz & Robinson, 1998).
Whichever of the various proposed indices is used, the process of deriving weight adjusted for height can be involved. Take for example the percentage ideal weight for height (%WFH), recommended for assessing the nutritional status of children with cystic fibrosis (CF; Ramsey et al, 1992). Dietitians have great difficulty calculating it, with the mean variability in %WFH between 42 CF dietitians asked to assess six patients reaching 28% (Poustie et al, 2000).
Similarly BMI, defined as weight/height2, has to be calculated and plotted on the BMI centile chart. It is hardly surprising that, instead, many clinicians rely on the readily available charts of weight-for-age and height-for-age to assess underweight and overweight. They compare the child's weight and height centiles informally, and if the two are sufficiently dissimilar—if say they differ by three centile channel widths or more—this is interpreted as over- or underweight for height (Hulse & Schilg, 1996).
In recent years it has become possible to express not only weight and height, but also BMI, as exact centiles for age and sex, using for example the British 1990 reference (Cole et al, 1995; Freeman et al, 1995) or the US CDC 2000 reference (Kuczmarski et al, 2000). This provides an opportunity to examine the relationship between weight centile, height centile and BMI centile in individual children. Weight, height and BMI are directly connected through the formula BMI=weight/height2, but the same is not true for their centiles. For example a boy aged 3.6 y, 100 cm tall and weighing 18.6 kg, has a BMI of 18.6 kg/m2 by definition. His corresponding centiles for age are 52, 91 and 97, respectively, where weight and BMI are two and three channel widths above the median and height is on the median. So this is the question: is there a general rule that predicts his BMI centile from his weight and height centiles?
Mulligan and Voss (1999) compared BMI, weight and height centiles in hypothetical fat and thin children aged 2–9 y. To their surprise they found that, in fat children, weight exceeded height by four to five channel widths if they were short, but by only one channel width if tall. Conversely in thin children, weight was about two channel widths below height in short children, but four channel widths below in tall children. The boy in the previous paragraph illustrates the problem—his BMI is three channel widths above the mean, yet his weight is only two channel widths greater than his height. These discrepancies led Mulligan and Voss (1999) to doubt the validity of the BMI.
The aims of this study are two-fold: (i) to repeat the analysis of Mulligan and Voss (1999) comparing the BMI centile with the difference between weight and height centiles, but using real rather than hypothetical data; and (ii) to quantify the relationship between centiles of BMI, weight and height.
Subjects and methods
Subjects
Data were obtained from the Third Nationwide Dutch Growth Survey, a large nationally representative sample of 40 536 Dutch children aged 0–18 y measured in 1980 (Roede & Van Wieringen, 1985). BMI was calculated as weight/height2 (kg/m2). The weights, heights and BMIs were converted to standard deviation scores (SDS) using the revised British 1990 reference (Cole et al, 1998) and the US CDC reference (Kuczmarski et al, 2000). The British reference starts at birth, and the US reference at 2 y of age. Centiles were calculated using both references to check that the relationships were the same.
The British reference used the same reference sample for all three measurements past birth, so it should be internally consistent, with children of median weight and height on the median for BMI as well. The US reference omitted NHANES III children aged over 6 y from the weight and BMI charts, but retained them for height. However, mean height was similar in all three NHANES surveys, so the older NHANES III children, who contributed height but not weight or BMI, should not materially have biased the height centiles up or down. So again, the reference ought to be internally consistent.
Centiles and standard deviation scores
For mathematical reasons the comparison of weight and height should be based on SDS rather than centiles. The centile scale is nonlinear, being stretched in the tails of the distribution and bounded between 0 and 100. The SDS indicates how many standard deviations a measurement is above or below the median of the distribution, and it is these deviations for weight and height that are to be compared. For the reference population the mean SDS is 0 and the standard deviation 1. The LMS method (Cole & Green, 1992) was used to fit the British and US references and takes into account skewness in the distributions.
The size of the gap between centiles on the chart, here called the channel width, gives an indication of the underlying SDS scale. The US charts have seven centiles, the 3rd, 10th, 25th, 50th, 75th, 90th and 97th, where the channel widths vary between 0.60 and 0.68 SDS units. On the British charts there are nine centiles defined to be 0.67 (two-thirds) of an SDS apart, giving (approximately) the 0.4th, 2nd, 9th, 25th, 50th, 75th, 91st, 98th and 99.6th centiles. The channel width is a useful unit of measurement for comparing chart centiles.
Statistical methods
There are two standard forms of weight adjusted for height:
and
where the coefficient b is positive (Cole, 1991). A third form, that adjusts for age as well, uses weight SDS and height SDS instead of weight and height (Cole, 1994):
Several authors (eg Hulse & Schilg, 1996; Mulligan & Voss, 1999) have proposed using the difference between weight and height centiles as an index. Numerically this gives:
which is here called the W-H index—'W minus H'. It is of the same form as equation (1) except that b is 1.
BMI centile is an index of weight adjusted for height and age, which is the same as weight centile adjusted for height centile. For this reason it should fit equation (1). To test this, the coefficient b can be estimated from the multiple regression of BMI SDS on weight SDS and height SDS:
The coefficient a is included because BMI SDS has unit standard deviation, and equation (1) needs scaling appropriately. The intercept c is the mean BMI SDS for a child of median weight and height, and is likely to be close to 0. Note that b has a negative sign, as in equation (1). The coefficient of determination (r 2) from the multiple regression indicates how much of the variance in BMI SDS is explained by weight SDS and height SDS. The standard error of the estimate (s.e.e.) is the standard deviation of predicted BMI SDS. It is much less than the BMI SDS standard deviation of 1, reflecting the reduction in variance due to the regression.
Equation (3) is here fitted to the Dutch data for the two sexes separately and combined, using the British 1990 and US CDC references.
BMI centile chart
The results of equation (3) can be presented as a chart of weight centile plotted against height centile, which simultaneously shows the BMI centile on transverse axes. This arises because equation (3) can be rearranged as:
When BMI SDS=0 and c=0, this is a straight line of slope b/a on a plot of weight SDS vs height SDS. The line joins points where weight SDS and height SDS correspond to a BMI SDS of 0. Lines parallel to this represent other BMI centiles, the lines being shifted up or down by 1/a units per unit of BMI SDS.
Results
Table 1 summarises the validation dataset of Dutch boys and girls relative to the British and US references. Dutch children in 1980 were on average taller and thinner than British children in 1990 aged 0–18 y, but with SDS standard deviations close to 1 (as they should be). The same was true relative to the US CDC reference (for age 2–18 y), although the standard deviations were slightly smaller. This indicates greater variability in the US than the British reference.
Full size table
Tables 2 and 3 give the correlations by sex between weight SDS, height SDS, BMI SDS and the W-H index, based on the British 1990 and US CDC references respectively. Correlations for males are below the diagonal and females above. Weight and height are correlated at about 0.7, while the BMI–weight and BMI–height correlations are near 0.8 and 0.1, respectively. By contrast W-H has a lower correlation with weight (0.3) than with height (−0.5). So in broad terms BMI is strongly correlated with weight and uncorrelated with height, whereas W-H is only weakly correlated with weight and strongly negatively correlated with height.
Full size table
Full size table
The multiple regression analysis of BMI centile on weight centile and height centile (equation (3)), based on the British reference, shows a very strong relationship (Table 4). The equations for the two sexes account for around 97% of the variance in BMI SDS, with an s.e.e. of 0.17 SDS units or a quarter of a channel width on the BMI chart. So given a child's weight centile and height centile, the equation predicts their BMI centile to within 2 s.e.e.s or half a channel width with 95% confidence.
Full size table
Table 5 gives the corresponding results based on the US reference. The relationship is less strong but, even so, more than 93% of the variation in BMI SDS is explained by weight SDS and height SDS. The s.e.e. is 0.24, predicting a single measurement to within half an SDS (two-thirds of a channel width) with 95% confidence.
Full size table
Comparing the regression coefficients a and b in Table 4 and 5 shows them to be almost identical. In addition the intercepts c are effectively 0. The numerical relationship between BMI SDS, weight SDS and height SDS is essentially independent of the underlying growth reference.
Figure 1 represents the British reference regression equations of Table 4 graphically. The two lines through the origin are the equations by sex for BMI SDS 0 based on equation (4), and the upper and lower lines correspond to BMI SDS −2 and 2. The slopes of the lines and the spacings between them are similar for the two sexes. The graph demonstrates that plotting weight SDS against height SDS provides BMI SDS as well, to within the error of prediction.
BMI SDS predicted from weight SDS and height SDS using the regression equations of Table 4 for the British 1990 reference (boys solid, girls dashed). The lines join points where weight SDS and height SDS correspond to constant values of BMI SDS.
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Due to the large sample the small sex differences in Table 4 and 5 are highly significant. However, combining the sexes does not materially reduce the predictive power of the equations, as shown by the s.e.e.s in Table 4 and 5. So the sexes combined equations for the two references are averaged: a=1.434, b=0.794 and c=0 (ie forced through the origin). Substituting the coefficients into equation (4) gives centile lines with slope 0.55, spaced 0.70 standard deviations apart per standard deviation of BMI.
This leads to the charts in Figures 2 and 3, which relate weight centile and height centile to BMI centile and apply to both sexes. Figure 2 gives the nine British centiles (0.4th, 2nd, 9th, etc) and Figure 3 the seven US centiles (3rd, 10th etc). The points and dotted line in Figure 2 are discussed later. The point in Figure 3 is the boy on the 52nd height centile and 91st weight centile described in the Introduction. The figure predicts his BMI centile to be 97, which indeed it is.
A chart to convert weight and height centiles to the corresponding BMI centile, with an accuracy of conversion to within half a channel width on the British 1990 reference (95% confidence interval). Also shown is the line of equality (dashed), and four points representing children discussed by Mulligan and Voss (1999).
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A chart to convert weight and height centiles to the corresponding BMI centile, with an accuracy of conversion to within half an SDS on the US CDC reference (95% confidence interval). The point represents a boy discussed in the text.
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Discussion
The analysis here shows a remarkably close relationship between centiles of BMI, weight and height, with 93–97% of the variance in BMI SDS explained. Although surprising, this is to some extent inevitable as BMI is defined to be an index of weight adjusted for height (Cole, 1991).
This means that the BMI centile can for clinical purposes be derived graphically from weight and height centiles, avoiding the need to calculate BMI and plot it on a chart. The strength of the relationship is what makes this feasible. Figures 2 and 3 convert an individual child's weight and height centiles to a BMI centile with an accuracy (95% confidence interval) of around half a channel width. Although not exact this is a perfectly good approximation for most purposes.
Despite the good fit, it is worse with the US reference (93% explained) than with the British reference (97% explained). This may be because the British data were all based on the same children, whereas weight and BMI were omitted for US children over 6 y in NHANES III. The greater variability of the US reference, as shown by the smaller standard deviations for the Dutch data in Table 1, may also be relevant.
The charts in Figures 2 and 3 are suitable for both sexes and all ages from birth to 18 y. They are also independent of the underlying growth reference, as shown by the closely similar coefficients for the British and US references (Tables 4 and 5). External data from the Third Dutch Growth Survey were used, as opposed to British or US reference data, in order to provide extra validation of the chart. For these reasons it is likely that the chart applies to other countries of the developed world as well, which makes it of quite general applicability.
The W-H index is a simpler and clinically more intuitive approach to the assessment of over- or underweight, which compares the weight and height centiles directly. Mulligan and Voss (1999) criticised the BMI because it differed from the W-H index. Yet the W-H index has very undesirable properties—it is only weakly correlated with weight and strongly negatively correlated with height (Tables 2 and 3), exactly the opposite of what is required. A good weight-for-height index should be strongly correlated with weight and hardly at all with height (Cole, 1991), properties exemplified by the BMI in Tables 2 and 3.
Figure 2 provides a graphical comparison between BMI and the W-H index. The dashed line of equality marks points where the weight and height centiles are equal, ie where the W-H index is 0. Other values of the W-H index lie on lines parallel to the dashed line. The hypothetical children of Mulligan and Voss are also shown, fat and thin, short and tall. The tall fat child (top right) is much closer to the line of equality (ie nearer to W-H=0) than the short fat child (top left), yet they are on the same BMI centile. The reverse is true for the thin children. A given BMI centile does not correspond to a constant difference between weight and height centiles.
Why are the two indices different? It all comes down to the importance of height relative to weight in the assessment. The W-H index assumes that weight and height are of equal importance, so that a given number of height channel widths (relative to the median) can be 'traded off' against the same number of weight channel widths. By contrast the BMI, along with most other weight-for-height indices, expresses weight adjusted for height. This is a regression adjustment, where height is given sufficient importance for the index to be broadly uncorrelated with height. Numerically the regression line has a slope of 0.55 rather than 1, so the importance of height is only 55% that of weight. This height 'shrinkage' is an inevitable consequence of regression to the mean.
A further problem with the W-H index is that its centile bands are too wide. For children of average height, like the boy in Figure 3, the 90th weight centile and the 97th BMI centile coincide. His weight centile is two centile bands above his height centile, yet his BMI is 3 centile bands above the median. The BMI centile bands are relatively narrower, and they pick up cases that the W-H index misses.
Despite its appeal to clinicians on the grounds of simplicity, the W-H index really should not be used as it is grossly misleading. Figures 2 and 3 provide a simple and valid alternative which requires no calculation. A small version of the chart could be added to conventional weight and height for age charts. Taken together they would provide a child's weight and height centiles directly, which could then be plotted on the new chart to give the child's BMI centile.
The regression equations underlying the chart are based on data from early life to 18 y. The BMI–weight–height relationship depends on age to some extent, so that the equation's already impressive fit can be improved by allowing the coefficients to vary with age. The fit can be improved further by including terms in the equation for weight2, height2 and interactions between them. However, the improvement in fit is modest, and does not justify any greater complexity than Figures 2 and 3.
One clinical use for the chart is to follow concurrent changes in weight, height and BMI in individual children over time. Figure 4 gives three examples, children selected from the French Growth Study (Sempé et al, 1979) to show dramatic concurrent changes in weight, height and BMI centile. Case 1 remained on broadly the same length centile from 1 to 18 months, but became increasingly obese and then slimmed down again. Case 2's height centile increased from 1 month to 3 y while his BMI centile stayed constant. He then remained on the 98th height centile but gained weight to reach the 91st BMI centile. Case 3 had three distinct growth phases: getting relatively shorter while staying near the 2nd BMI centile, gaining weight to the 91st BMI centile, and then staying there but getting relatively shorter.
Three subjects from the French Growth Study (Sempé et al, 1979) followed from 1 month of age selected to show substantial changes in weight, height and BMI centile through childhood.
Full size image
The chart highlights how changes in weight and height affect nutritional status. For the BMI centile to remain constant, the height centile has to change about twice as fast as the weight centile in the same direction. There appear to be two broad patterns of change: (i) the height centile stays constant while weight and BMI change; and (ii) the weight and height centiles change in step, with height changing faster than weight, so that the BMI centile is conserved. The chart used in this way for longitudinal measurements can be hard to interpret, but these extreme examples do suggest underlying patterns in height and weight change.
The chart has been developed in the context of obesity indices, but it applies equally to the assessment of underweight. It displays the Waterlow classification (Waterlow, 1972) directly, showing how weight is partitioned into wasting (BMI) and stunting (height). This may be useful for example in tracking recovery from severe protein-energy malnutrition (Walker & Golden, 1988).
In conclusion, BMI centile, weight centile and height centile in children are seen to be closely related, and their relationship can be summarised in the form of a chart. The difference between weight and height centiles is not a suitable measure of weight-for-height, as it is poorly correlated with weight and negatively correlated with height.
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Acknowledgements
I am grateful to Mike Preece, Angie Wade and Jonathan Wells for their comments on the paper, and to Machteld Roede for providing the Third Nationwide Dutch Growth Survey data.
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Cole, T. A chart to link child centiles of body mass index, weight and height. Eur J Clin Nutr 56, 1194–1199 (2002). https://doi.org/10.1038/sj.ejcn.1601473
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DOI : https://doi.org/10.1038/sj.ejcn.1601473
Keywords
- child
- weight
- height
- BMI
- malnutrition
- obesity
- SDS
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Height and Weight Tables Are Designed to Be Used Fo
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