13 
National Marine 
Fisheries Service 
NOAA 
Fishery Bulletin 
fb- estabiished in 1881 
Spencer F. Baird 
First U.S. Commissioner 
of Fisheries and founder 
of Fishery Bulletin 
Flexible Baf esiaii analysis of the 
won Bertalanffy growth function with 
the use of a log'Skew^t distribution 
Freddy Omar Lopez Quintero' 
Javier E= Contreras-Reyes (contact author)''* 
Rodrigo Wiff ^ 
Reinaldo 1. Arel!ano-¥alle^ 
Email address for contact author: javier.contrerasigifop.c! 
Abstract— The von Bertalanffy 
growth function is the model most 
widely applied to describe growth 
in fish populations. Parameters de- 
scribing this function usually are 
estimated from observed lengths at 
different ages by using maximum 
likelihood and by assuming Gauss- 
ian distributed errors. In harvested 
populations, observed length at age 
usually involves a high level of 
skewness and extreme values be- 
cause of the size-selective sampling 
process. Some approaches, based on 
the maximum-likelihood method for 
making inferences, have been devel- 
oped to resolve such issues. We pro- 
pose a Bayesian framework for esti- 
mating growth parameters for non- 
linear regression models — a frame- 
work that is based on the family of 
log-skew-^ distributions and which 
provides an approach that is flexible 
enough for modeling the presence of 
asymmetries and heavy tails. This 
framework based on a method in 
which 1) the error accounts for both 
skewness and heavy-tailed distribu- 
tions of a log-skev/-^ model, and 2) 
the observed length at each age has 
a heteroscedastic error distribution. 
The proposed method was applied 
and compared with the methods of 
previous models by using observed 
length-at-age data for the southern 
blue whiting (Micromesistius austra- 
lis), an important fish species har- 
vested in the southeast Pacific. Com- 
parisons indicated that the proposed 
model is the best for describing data 
on southern blue whiting. 
Manuscript submitted 6 January 2016. 
Manuscript accepted 30 September 2016. 
Fish. Bull. 115:13-26 (2017). 
Online publication date: 27 October 2016. 
doi: 10.7755/FB.115.1.2 
The views and opinions expressed or 
implied in this article are those of the 
author (or authors) and do not necessarily 
reflect the position of the National 
Marine Fisheries Service, NOAA. 
’ Departamento de Matematica 
Universidad Tecnica Federico Santa Maria 
Avenida Esparia 1680 
Valparaiso 2390123, Chile 
2 Division de Investigadon Pesquera 
Institute de Fomento Pesquero 
Avenida Almirante Blanco Encalada 839 
Valparaiso 2361827, Chile 
Growth is one of the most impor- 
tant measurable life-history traits 
in individual organisms because it 
is fundamental in creating an un- 
derstanding of both population and 
ecosystem functions. Several models 
have been proposed to describe ani- 
mal growth. The most widely applied 
model, however, is the von Berta- 
lanffy growth function (VBGF; von 
Bertalanffy, 1938). This model has 
been used to describe length at age 
for a wide range of species across 
several taxa, such as mammals (Eng- 
lish et al., 2012), birds (Tj0rve and 
Tj0rve, 2010), and reptiles (Lehman 
and Woodward, 2008), although it is 
most extensively applied to fish spe- 
cies (Pardo et aL, 2013). The VBGF 
is based on principles underpinning 
the physiology of growth (von Ber- 
talanffy, 1938; Wiff and Roa-Ureta, 
2008), gives an adequate description 
of growth with the use of only 3 pa- 
rameters, and states that the rate 
of growth of an individual is deter- 
mined by the difference between the 
buildup of body mass and loss due 
to energy expenditures for mainte- 
3 Center of Applied Ecology and Sustainability 
Pontificia Universidad Catolica de Chile 
Avenida Libertador Bernardo O'Higgins 328 
Santiago 8331150, Chile 
^ Departamento de Estadfstica 
Pontificia Universidad Catolica de Chile 
Avenida Vicuna Mackenna 4860 
Santiago 7820436, Chile 
nance. In harvested fish populations, 
the usual data available to estimate 
these parameters are cross-sectional, 
and a single length and age measure- 
ment is taken from each sampled 
individual. The VBGF describes the 
expected growth rate for the popu- 
lation, on the basis of length-at-age 
data composed of individuals with 
variable growth rates. 
Maximum-likelihood techniques 
derived from Gaussian and log- 
Gaussian errors normally are used 
to estimate VBGF parameters (Mil- 
lar, 2002; Siegfried and Sanso, 2006). 
Yet, in fish populations, this assump- 
tion often fails because these distri- 
butions are typically skewed, present 
heavy tails or have extreme values. 
Skewed distributions usually result 
from the size-selective sampling pro- 
cess (Montenegro and Branco, 2016). 
In addition, in harvested fish popu- 
lations, an accumulative effect of 
fishing exploitation exists for size at 
age. Growth rates vary among indi- 
viduals (Sainsbury, 1980), and fish- 
ing selectivity removes faster grow- 
ing individuals from each particular 
