Two Bayesian methods for estimating parameters of the von Bertalanffy growth equation

Abstract

The von Bertalanffy growth equation (VBGE) is commonly used in ecology and fisheries management to model individual growth of an organism. Generally, a nonlinear regression is used with length-at-age data to recover key life history parameters: L _∞ (asymptotic size), k (the growth coefficient), and t ₀ (a time used to calculate size at age 0). However, age data are often unavailable for many species of interest, which makes the regression impossible. To confront this problem, we have developed a Bayesian model to find L _∞ using only length data. We use length-at-age data for female blue shark, Prionace glauca, to test our hypothesis. Preliminary comparisons of the model output and the results of a nonlinear regression using the VBGE show similar estimates of L _∞ . We also developed a full Bayesian model that fits the VBGE to the same data used in the classical regression and the length-based Bayesian model. Classical regression methods are highly sensitive to missing data points, and our analysis shows that fitting the VBGE in a Bayesian framework is more robust. We investigate the assumptions made with the traditional curve fitting methods, and argue that either the full Bayesian or the length-based Bayesian models are preferable to classical nonlinear regressions. These methods clarify and address assumptions␣made in classical regressions using von Bertalanffy growth and facilitate more detailed stock assessments of species for which data are sparse.

Appendix

The full Bayesian model

In our computations, we replaced t ₀ with q because it is easier to work with a strictly positive distribution. At the end of the model runs, we take the negative results for q and set them equal to t ₀.

Our joint posterior is the combination of the four priors—for k, q, L _∞, and σ²—and the likelihood:

$$\eqalign{p{\left( {l_{\infty } ,k,q,\sigma ^{2} \left| {l_{{ar}} } \right.} \right)} \propto & \frac{1}{{\sigma ^{2} }} q^{{13}} \exp {\left\{ { - 4q} \right\}} k^{{ - 85}}\cr & \exp {\left\{ { - k{\left( {100} \right)}} \right\}} \times {\prod\limits_{a = 0}^{15} {{\prod\limits_{r = 1}^R {\frac{1} {{{\sqrt {2\pi \sigma ^{2} } }}}}}}}\cr &\exp {\left\{ { - \frac{1} {{2\sigma ^{2} }}{\left( {l_{{ar}} - {\left( {l_{\infty } + \log {\left( {1 - \exp {\left\{ { - k{\left( {a + q} \right)}} \right\}}} \right)}} \right)}} \right)}} \right\}} } $$

Marginalizing, or multiplying out each of the parameters one at a time, give us either a formula to use in a Metropolis Hastings algorithm or a full conditional to use with a Gibbs sampler as detailed in the main text.