Lopez Quintero et al.: Bayesian analysis of the von Bertalanffy growth function 
23 
Table S 
Summary of the log-skew-^ model fitted with power heteroscedastic variance function for the full and restricted data and its 
respective percentage of relative change (EC) for the probabilities (pj), of 0.70, 0.60, 0.55 and 0.51. The parameters are the 
asymptotic length (L^), growth rate coefficient (®, theoretical age in years when the length is zero {-^o), heteroscedastic- 
ity (p), dispersion (o®), skewness (1), degrees of freedom (v), sample size (re), and number of influential observations (Influ. 
observ.). 
Restricted data (S_i) 
Parameters 
Pi = 0.70 
EC (%) 
Pi = 0.60 
RC {%) 
Pi = 0.55 
RC (%) 
Pi = 0.51 
RC (%) 
L„ (cm) 
59.559 
0.024 
59.550 
0.039 
59.445 
0.215 
59.588 
0.025 
Kiy-^) 
0.163 
0.617 
0.163 
0.617 
0.164 
1.235 
0.161 
0.617 
-toiy) 
2.446 
0.326 
2.455 
0.041 
2.391 
2.567 
2.501 
1.915 
p 
-0.179 
0.556 
-0.180 
0.000 
-0.168 
6.667 
-0.141 
21.667 
(j2 
0.012 
9.091 
0.013 
18.182 
0.012 
9.091 
0.008 
27.273 
A 
-1.181 
7.755 
-1.231 
12.318 
-1.305 
19.069 
-1.801 
64.325 
V 
19.198 
34.046 
22.304 
55.732 
30.223 
111.025 
80.796 
464.139 
n 
24936 
24932 
24902 
21836 
Influ. observ. 
6 
10 
40 
3106 
will improve the estimation of ^o• In addition, consider- 
ing the influential analysis, the estimated parameters 
from the restricted data indicate significant differences 
with those obtained with the full data set, particularly 
for the degree of freedom parameter of the error dis- 
tribution. However, numerous subjects were not evalu- 
ated in this article — topics such as other sources of un- 
certainly or data-related problems that can lead to bias 
in an estimation (Ortiz and Palmer^). Particularly, we 
did not address the direct influence of the prior specifi- 
cation on the final estimates (Fig. 5C), a topic that will 
be of interest for future research. 
Siegfried and Sanso (2006) and Hamel (2015) con- 
sidered log-normal distributions to be appropriate for 
the asymmetry observed in the length-at-age data in 
harvested fish populations. However, in such data, we 
can usually find different degrees of skewness and 
heavy-tailed and extreme values in which log-normal 
distribution does not give a good description of obser- 
vations. The log-normal model may underestimate the 
real variance contained in the data (Slatkin, 2013). In 
such cases, log-skew-^ models (such as the one proposed 
here), could yield a fair description of the observed 
length-at-age data, as was the case for the southern 
blue whiting, in which the log-skew-^ model turned out 
to be the best among all competing models. In addition, 
the proposed model gives great flexibility in modelling 
heteroscedasticity by adding a function dependent on 
the scale of and a heteroscedastic parameter p. The 
assumption of asymmetry and heavy tails and the log- 
transformed nature of the log-skew-^ model reduces 
the standard errors of the estimated parameters of the 
VBGF (Contreras-Reyes et ai., 2014). 
1 Ortiz, M., and C. Palmer. 2008. Review and estimates of 
von Bertalanffy growth curves for king mackerel Atlantic 
and Gulf of Mexico stock units. SEDAR16-DW-12, 20 p. 
[Available at website.] 
Age (yr) 
Figure 3 
Boxplots of residuals versus ages of southern 
blue whiting (Micromesistius australis) from 
the log-skew-^ model with power heteroscedastic 
function. The dark black lines correspond to the 
observed median, the gray shaded boxes repre- 
sent the observed interval from the 25% residual 
quartile to the 75% residual quartiie, the error 
bars are the observed interval from minimum to 
maximum residual value, and the dots are atypi- 
cal residual values. 
The Bayesian analysis that we developed and de- 
scribe in this article provides a flexible framework that 
allows biologically meaningful estimates of the VBGF. 
This method also takes into account the uncertainty 
and kurtosis produced by extreme values common in 
