Lopez Quintero et al.: Bayesian analysis of the von Bertalanffy growth function 
21 
Age 
16 18 20 22 24 
Age 
0 . 010 - 
0.008 
0.006 
<b 
0.000 
Figure 1 
(A) Observations of length-at-age composition for southern blue whiting {Micromesistius australis) collected from a region 
spanning latitudes 46°S to 56°S over the period 1997-2010 (gray shaded areas), with von Bertalanffy growth function 
(VBGF) fits: the solid black line corresponds to the fit of the log-skew-model with a heteroscedastic power variance function 
(Table 3). Black points correspond to the log-normal model fit with constant (homoscedastic) variance function (Table 3). 
Gray dotted lines and points correspond to the 95% highest posterior density intervals for log-skew-^ and log-normal model 
fits, respectively. (B) The log-skew-i model fit includes the respective zoom subplot for fish at ages 14-26. (C) Heteroscedas- 
tic variance function (af) for log-skew-i (solid line) and log-normal (points) model fits. 
normal. Among the VBGF parameters, -^o showed the 
largest RC variation, although RC values of the VBGF 
parameters were small given the absence of influential 
observations. 
The relationship between the paucity of observa- 
tions for young (1-5 years) and old (16-24 years) age 
classes and heteroscedastic variance can be interpreted 
from Table 5 for the cases when p; = 0.55 and 0.51 
as follows. For p\ = 0.55, 67.5% and 17.5% of young 
and old individuals, respectively, were obtained for the 
sample from 40 influential observations. For = 0.51, 
35.4% and 23.0% of young and old individuals were 
obtained for the sample, respectively, from 3106 influ- 
ential observations. When the estimates for cF, v, and 
p, from the Log-skew-t von Bertalanffy growth model 
section, were considered, the heteroscedastic variance 
erf decreased mainly when young and old individuals 
(extreme values) were excluded from the sample. 
CorreSation analysis 
An important aspect in fisheries research related to 
VBGF analysis is the correlation between parameters 
(Pilling et ah, 2002; Siegfried and Sanso, 2006; Shelton 
and Mangel, 2012). High correlation among the 3VBGF 
parameters is common in fish populations (Ratkowsky, 
1986; Pardo et ah, 2013). Correlation between param- 
eters was analyzed by using the scatter plots in Figure 
5. The highest correlation was found between K and -^o 
(-0.94), followed by the correlation between and K 
(-0.89) (Xiao, 1994; Pilling et ah, 2002; Siegfried and 
Sanso, 2006) and by the correlation between and 
-to (0.71) (Ratkowsky, 1986; Pilling et ah, 2002). Rat- 
kowsky (1986) found that correlations between VBGF 
parameters may depend on the parameters that are 
used. Other choices of parameters should produce a low 
correlation between the VBGF parameters. However, in 
our model, the solution for K is affected by values of 
L„ and -to under the classical VBGF parametrization. 
The relationship between estimates of L„ and K 
are similar to the ones found by Siegfried and San- 
so (2006), but, in contrast to their results, we found 
a large correlation between {K, -to) and (L„, -to). This 
finding could have occurred for different reasons, such 
as the species studied and the specific Bayesian meth- 
od employed. However, the use of maximum-likelihood 
estimation (not shown) also verified high correlations 
between those parameters. The scatter plots did not 
show a clear correlation for error distribution parame- 
ters, except for the relationship between and p given 
by the heteroscedastic power function, where the cor- 
relation was -0.82. 
Discussion 
In this study, we embedded previous log-skew-^ distribu- 
tion analyses in a Bayesian framework. This approach, 
namely using log-skew-^ distribution, has several ad- 
vantages over previous frequentist inference. First, in 
a Bayesian framework, prior knowledge of the model 
parameters can be included in the modeling process in 
terms of a prior distribution, and our inferences were 
based on the posterior distribution, therefore, allowing 
