Modelling the Distribution of Anthropometrics: Gaussian Distributions, LMS Distributions, and Probability Plots

Abstract

In anthropologists' studies of growth and development, particularly in non-industrialized populations, sample sizes are often small and span a range of ages. These attributes can hamper analyses and hypothesis testing, and make comparisons to other populations difficult. $z$-scores are a commonly used statistic to mitigate these challenges. A $z$-score is the value of an individual's anthropometric ($y$) expressed in units of the standard deviation (SD) of the anthropometric for a suitable sex- and age-specific reference sample. That is, $z = (y - \mu)\big/\sigma$ where $\mu$ and $\sigma$ are the mean and SD of the reference sample, respectively. Depending on the research question, commonly used reference samples (e.g., WHO, CDC) are not necessarily suitable for all populations. Therefore, there are increasing efforts to construct population-specific growth references.

If a growth reference provides the mean and SD for each age/sex bin, it's easy to compute the $z$-score corresponding to any individual's measurement by assuming a Gaussian (normal) distribution. $z$-scores may be computed by either using the mean/SD for the individual's sex/age bin, or (for improved accuracy) interpolating tabulated means/SDs to the individual's age.

However, if the anthropometric has an asymmetric (skewed) distribution (as do weight, BMI, and many skinfolds), this approach results in systematically biased $z$-scores. Cole's LMS distribution can accurately represent the distributions of these and many other anthropometrics, avoiding this bias.

But sometimes only percentiles are provided for each age/sex bin. We describe how to: (a) determine the mean/SD of the Gaussian distribution, or the coefficients of the LMS distribution, that best fit the published percentiles; (b) visually assess the quality of such a fit; and (c) extrapolate the distribution beyond the range of the published percentiles and visually assess the quality of such an extrapolation.

We describe doing this fitting with common open-source software (Gnuplot, R, or SciPy (Python)), or with Microsoft Excel\texttrademark. The fitted coefficients can then be used to compute $z$-scores. We also describe how to extrapolate parameters (and thus compute $z$-scores) for an individual who is outside the tabulated age range, and we present a graphical assessment of any given extrapolation's quality.