More on the growth equations

The code, available in GitHub, is in the public domain. An R implementation of the double Box-Cox model, function unitran(), is included in the resde R package, which deals with estimation methods.

The following table shows simplified growth curve formulas and common names for the various special cases. Where possible, axes have been scaled and translated to have an origin at t=0, y=0, and an asymptote at y=1. The exceptions are the cases a ≤ 0, b=0, where t ranges between minus and plus infinity, and the case a > 0, b > 0, where the range of t is finite, here scaled to [0, 1]. To simplify, expressions are given in terms of positive shape parameters p = |1/a| and q = |1/b|, and t is re-scaled for the simplest form. In practice, scale and possibly location parameters would be included through substitutions t → k t + m, t → k(t - m), or t → (t - m) / k. Similarly for y, except, of course, when modelling probability distributions.

b > 0, b ≤ 1	1 - 1 / (1+t^q)^p	1 - exp(- t^q) (Weibull)	[1 - (1-t)^q]^pand 1 - (1-t^q)^p
b=0	1 / (1+e^-t)^p (Richards), and 1 - 1 / (1+e^t)^p	exp(-e^-t) (Gompertz), and 1 - exp(-e^t)	(1 - e^-t)^p (Bertalanffy)
b < 0	[t^q / (1+t^q )]^p (Levacovic I)	exp(-1/t^q ) (Korf)	[1 - 1 / (t+1)^q]^p
	a < 0	a=0	a > 0, a ≤ 1

I suggest reserving the name Richards for Richards' extension to negative a, and calling Bertalanffy or Bertalanffy-Richards the case a > 0 originally proposed by von Bertalanffy.

As noted in Garcia (2005), there is some confusion in the literature about the differential forms of growth models. Many authors give "the" growth equation dy/dt = f(t, y) corresponding to each growth curve. In fact, there are infinitely many differential equations of this type that generate exactly the same curve. Of these, the most interesting is the autonomous or time-invariant differential equation without t on the right-hand-side, graphed at the top-left in the applet. Or, for distributions, the probability density function that depends only on t. For our model, the general form of the first one, in terms of the Box-Cox transformation, is dy/dt = y^1-a B(y, a)^1-b. Expanding this into a table like the one above, and working out the densities, are left as an exercise to the reader ☺

Some common names for the probability distributions, and variable ranges:

b > 0, b ≤ 1	Burr Type XII 0 ≤ t < ∞	Weibull 0 ≤ t < ∞	Proctor 0 ≤ t ≤ 1
b=0	Burr Type II or VII, Generalized Logistic -∞ < t < ∞	Extreme Value, Gumbel -∞ < t < ∞	? 0 ≤ t < ∞
b < 0	Burr Type III 0 ≤ t < ∞	Extreme Value Type 2, Fréchet 0 ≤ t < ∞	? 0 ≤ t < ∞
	a < 0	a=0	a > 0, a ≤ 1

Chakraborty, B., Bhowmick, A. R., Chattopadhyay, J. and Bhattacharya, S. (2019). "A novel unification method to characterize a broad class of growth curve models using relative growth rate". Bulletin of Mathematical Biology 81, 2529-2552 (DOI).
García, O. (2005). "Unifying sigmoid univariate growth equations". Forest Biometry, Modelling and Information Sciences 1, 63-68 (Environmental Modelling and Software 23 (12), 1474-1475 (DOI).

Back to the applet page...