The explanatory variables are listed in the order of importance in and show a major additional influence of weight and percent body fat beyond age, gender and race. On the other hand, sexual maturity and height contribute little to the model once the other factors have been included.

| **Table 1**Importance of explanatory variables in describing spine BMD for North American children age 5–20 years |

Different combinations of explanatory variables resulted in different models (). The traditional model, model A, approximates the previous analysis [

6], but instead of using medians and a power transformation as the LMS method does, model A uses the more standard parametric technique employing means, no power transformation and a symmetric error distribution. We find that the previous results via the LMS model and model A are almost identical. We will use model A as an analog to the output of the LMS method for comparison purposes.

| **Table 2**Models considered in the analysis |

The full model, model B, is the “kitchen sink” model, in which everything is included, even parameters that may not add much information to the model.

Accurately evaluating sexual maturity in practice requires careful cross-training of the assessors and may be uncomfortable for both the subject and the assessor. Measuring the percent body fat means another DXA measurement, slightly increasing radiation dose to the subject. The other parameters can be gathered quickly and agreeably, resulting in the practical model, model C.

Because of computational difficulty with orthogonalizing the nominal variable of maturity, fully smoothed models for some parameter sets were not calculated. The results for the unsmoothed model provide us with a reasonable estimate of the effectiveness of the final model. As seen in and , dropping sexual maturity as a predictor does not significantly reduce the predictive power of the model.

Statistically, model F may be optimal in terms of creating the most powerful model while adding the fewest predicting parameters, but Tukey's confidence intervals indicate that, at the black/non-black level, race still matters for some age/gender combinations. Including both weight and percent body fat will intrinsically account for lean mass in the model [

9]. Model D may be optimal in terms of balancing the needs of the subjects with the most powerful model, and this model D will be investigated further.

When segmented by age, gender, and race, there are 64 simultaneous fits for each model. For the sparse model, 13 of these groups show a lack of normality in the underlying BMD values at the 0.05 level using a Shapiro-Wilk's test for normality. This lack of normality was the reason previous investigators chose to use the LMS model. For the smoothed model D, the residuals of only eight of these fits do not show normality due to outliers that make the distributions look leptokurtic. Since the outliers tend to appear in an uneven manner, the leptokurtosis leads to perceived skewness, and all eight of these groups have a moment of skewness twice the error of skewness, meaning the residuals show more than mild skewness [

18]; however, selectively removing one outlier from the extreme tail of the sample allows all but one of these groups to pass a test of normality. In samples of this size, this result reinforces the notion that the underlying distribution is actually less skewed and more normal.

By examining the smoothed coefficients of each parameter, we may glean some information about the parameter's effect and how this effect changes with age. Weight is positively related to BMD (), but the influence declines with increasing age. For most groups, greater percent body fat is associated with lower bone density (). Since the parameters are adjusted for weight, this negative association between percent body fat and BMD indicates that for two children of the same sex, race, age, height, and weight, the child with a higher percent body fat has a lower lean body mass and lower BMD, reflecting the known strong association of lean body mass and BMD [

9]. Also note that for late teen black females, the parameter reverses sign, but this is the area where data are most sparse.

Whereas weight usually has a strong positive contribution to BMD, height produces a much smaller negative correction after weight, age, and percent body fat are accounted for (). This can be interpreted as an effect of the given bone mass, defined largely by body weight for a child with given sex, race, age, and percent body fat being distributed over a larger projected area in a taller child and, thus, resulting in a lower measured BMD. For children at older ages, both black males and females show a reversal of this pattern, and height contributes positively to BMD.

The intercepts of each group represent a baseline before the other effects are layered in (). Again, a difference between the black and non-black groups is striking.

Building a more complete model, including anthropometric factors, should allow for narrower distributions than would be allowed by simply using age, gender, and race. We can indeed confirm that the expected distribution narrows as parameters are added; however, after age, gender, race, weight, height, and percent body fat are included in the model, sexual maturity does not contribute in narrowing the distribution further.

In addition to narrowing the estimated distributions, the inclusion of anthropometric variables may produce shifts in the expected BMD. Subject weight creates a major shift in the mean, whereas subject height has little influence (). The sparse model (model A) shows a wider distribution as it includes all variations in weight and height. Gender and race produce additional shifts in the model curves ().

When comparing the total root mean squared error (RMSE) of the sparse model, model A, to the total RMSE produced by a more complex model, model D, a lower total RMSE is observed for the more complex model (). Furthermore, plotting the race/gender subgroups' individual RMSEs by age () shows that model D also has a consistently lower RMSE at the group level. Thus, a model including relevant anthropometric data does produce narrower distributions. Also note that, since the RMSEs are expressed in the same scale as the dependent variable, they tend to increase as the subjects grow older just as the subjects' BMD levels increase with age.

Comparing model D, the full model, and model A, the reduced model, the *F* statistic returns a value above 14, where the degrees of freedom in the denominator and the numerator are both above 7000. The *p* value below 0.001 means that adding the anthropometric parameters makes numeric sense and helps to conclude that model D is superior to model A.

Not factoring in the effect of weight as a determinant of BMD leads to the possibility that children of normal weight, based on the 50% value of the CDC growth chart, will be judged as having low BMD when that is not the case (). The sparse model agrees better with individuals who are at the 75th percentile of weight and height by CDC standards () than it does with those at the 50th percentile of the CDC standards.

Two cases of independent longitudinal data from a study that follows human development in a normal population [

19] are depicted in . Differences in Z-scores are apparent between the sparse model (model A) and the model without sexual maturity (model D). Although neither of these selected cases approaches a critical value suggesting abnormality, they serve as examples clearly showing that inclusion of explanatory variables in the model can easily produce a difference of more than one Z-score when compared to the sparse model.