A new family of mathematical models describing the human growth curve—Erratum: Direct calculation of peak height velocity, age at take-off and associated quantities

Abstract

A new family of mathematical functions to fit longitudinal growth data was described in 1978. The ability of researchers to directly use parameters as estimates of age at peak height velocity resulted in them overlooking the possibility of directly calculating these quantities after model estimation. This erratum has corrected three mistakes in the original manuscript in the direct calculation of peak height velocity and age at take-off and has implemented the solutions in a STATA program which directly calculates the estimates, standard errors and confidence intervals for age, height and velocity at peak height velocity.

Keywords::

• Growth model
• peak height velocity
• Preece Baines

Erratum

Preece and Baines, in their original 1978 article (Preece and Baines 1978), proposed three new mathematical models of human growth. Model 1, described as their simple and robust model, has become popular and widespread in its simulation of childhood growth, and their original paper has been cited in excess of 300 times.

Uniquely, the models were derived from a differential equation dh/dt = s(t)·(h ₁ − h), where h ₁ is adult size and s(t) is a function of time satisfying the logistic differential equation ds/dt = (s ₁ − s(t))·(s(t) − s ₀), where s ₀ and s ₁ are parameters.

The utility of these models was partly due to the correlation between the estimated parameters and suggested biologically meaningful time-points including age at peak height velocity (PHV) and age at ‘take off’ (TO, onset of rapid pubertal growth). Suggested interpretations of each parameter are given in the original paper (Preece and Baines 1978). Of the three models proposed, only two had analytical solutions to these meaningful time points, and the errata concerns these calculations.

The two models of h = height and t = time which we address are described as model 1 and model 2 in the original paper. Model 2, see below, describes height in six parameters, and Model 1, see below, is a special case of model 2 when γ = 1 (as γ is now constant it ceases to be a parameter and, therefore, model 1 describes height in five parameters). For Model 1, the form waswhile for Model 2,Model 2 was less successful in reaching convergence and, therefore, the use of the more parsimonious model 1 was preferred.

Whilst applying these models to the analysis of height data, we have found three errors in the original paper. The first error, which concerns the acceleration function, is applicable to both models 1 and 2.

The acceleration function, obtained by differentiating dh/dt, isThe original paper suggests that, after the substitution of dh/dt (the derivative of height with respect to time, t, i.e velocity),and ds/dt (derivative of s with respect to time, t)the acceleration function is equal toHowever, the correct expression for the acceleration is in factThis acceleration expression simplifies in the special case of model 1 (when γ = 1) to the expressiongiven in the original paper (note that this expression does not include the parameter γ).

The second error we have noted, which is applicable to model 1, occurs in the process of calculating age at peak height velocity/take-off.

In the case of model 1 the acceleration function (Equation 8) is set equal to zero and solved for s. Apart from the trivial solution when h = h ₁, it follows thatwhich the original paper suggests can be solved to giveHowever, after applying the general solution to a quadratic equation and gathering and simplifying, we find that the correct solution is:As Equation (8) does not generalize to model 2, Equation (11) is also not appropriate for model 2.

The third error stems from this lack of generalization. The acceleration function (equation 7) (which is generalizable to both model 2 and model 1 assuming γ = 1), can be further manipulated into the formwhich is easily solved. After setting the right hand side to zero and ignoring the trivial solution when h = h ₁, the general solution to a quadratic equation is applied and, after subsequent simplification and gathering procedures, we find thatSubstituting s from Equation (13) into Equation (14),and solving for t,allows the calculation of time (age) at PHV to be calculated.

Once the time of PHV/TO is known, t may be substituted into either model 1 or 2 (Equation 1 or 2) and, thus, height at PHV/TO may be calculated. Similarly once the height at PHV/TO is known and S from Equation (13) is known, these can be substituted into Equation (4) to calculate velocity at TO and PHV.

Thus, the growth traits previously mentioned can be calculated directly, rather than by using the parameters (and their standard errors) estimated from the model as proxies.

With the advances in statistical computation it is now possible to automatically calculate standard errors of these derived parameters by applying the delta method, instead of conducting a one-step Taylor series expansion manually. Therefore, intervals and limits are easily available for the analytic calculations of age at peak height velocity and age at take-off. The solutions to the previous expressions have now been implemented into STATA (STATA 2011) package, which utilizes nl and nlcom routines, which yield parameters, standard errors and confidence intervals.

These corrections and advances in statistical software should enhance the ability of researchers to derive biologically meaningful growth parameters from sequential height data using either of the first two original Preece-Baines models.

The STATA program pbreg is available from the authors and from the SSC archive.

Declaration of interest : The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.