Lopez Quintero et al.: Bayesian analysis of the von Bertalanffy growth function 
19 
Table 3 
Estimates from fitted log-normal and log-skew-r models, with standard deviations (SDs) and 
95% highest posterior density (HPD) intervals. The parameters are the asymptotic length (L„), 
growth rate coefficient (if), theoretical age in years when the length is zero i-to), dispersion 
(o^), heteroscedasticity (r), skewness (A), and degrees of freedom (v). 
Model 
Parameter 
Estimate 
SD 
95% HPD 
Log-normal (type I) 
Constant 
59.249 
0.091 
(59.064, 59.439) 
K 
0.167 
0.001 
(0.165,0.170) 
-^0 
2.323 
0.035 
(2.250, 2.396) 
0.004 
3x10-5 
(0.0042, 0.0044) 
Log-normal (type II) 
Constant 
59.249 
0.090 
(59.060, 59.425) 
K 
0.167 
0.001 
(0.165,0.170) 
-h 
2.323 
0.034 
(2.248,2.391) 
0.004 
0.000 
(0.0042, 0.0044) 
Log-skew-t 
Constant 
59.212 
0.086 
(59.055, 59.386) 
K 
0.166 
0.001 
(0.164,0.169) 
“^0 
2.382 
0.034 
(2.322, 2.454) 
0.005 
1x10-4 
(0.0047, 0.0053) 
X 
-1.012 
0.051 
(-1.105,-0.916) 
V 
11.020 
0.643 
(9.853, 12.210) 
Log-skew-t 
Exponential 
59.666 
0.085 
(59.527, 59.815) 
K 
0.161 
0.001 
(0.159,0.163) 
-h 
2.488 
0.036 
(2.428, 2.549) 
p 
-0.039 
0.002 
(-0.043, -0.035) 
0-2 
0.008 
2x10-4 
(0.007, 0.008) 
X 
-1.080 
0.050 
(-1.181,-0.977) 
V 
13.351 
0.884 
(11.757, 15.144) 
Log-skew-t 
Power 
59.573 
0.090 
(59.386, 59.755) 
K 
0.162 
0.001 
(0.159,0.165) 
-tQ 
2.454 
0.042 
(2.367, 2.541) 
P 
-0.180 
0.009 
(-0.197,-0.161) 
0-2 
0.011 
0.001 
(0.010,0.013) 
X 
-1.096 
0.053 
(-1.200, -0.997) 
V 
14.322 
1.047 
(12.457, 16.586) 
symmetric length-at-age distribution. In addition, log- 
normal models yielded very similar posterior estimates 
for VBGF and Interestingly, standard errors for all 
parameters were very precise, in a similar way to that 
of previous studies of frequentist inference (Contreras- 
Reyes and Arellano-Valle, 2013; Contreras-Reyes et al., 
2014). This level of precision probably is a result of a 
strong underlying structure of the data. 
Using the DIG and WAIC criteria of Equations 12 
and 14, respectively, we found the log-skew-t model 
with power heteroscedastic function to be the best 
model (Table 4). As Table4 indicates, the log-normal 
model is the least useful among the selected models. 
The fitted curve of the power-variance log-skew-^ 
model to the observed length-at-age data is presented 
in Figure lA. The model is adequate for younger ages 
(1-8 years), but for older ages (>15 years) the observed 
length tends to converge to = 59.52 (Table 3). The 
log-skew-t model provides more precise 95% HPD in- 
tervals for older ages (>13 years; Fig. IB) and less pre- 
cise for young ages (0-5 years) in comparison with the 
log-normal model. Intervals of 95% HPD of log-skew-t 
model fit indicate that the observations were affected 
by the negative heteroscedastic parameter p (Fig. 1C). 
In addition, constant variance was assumed for the 
log-normal model and, therefore, the model underesti- 
mated the real variance in the age at length containing 
extreme values. The posterior densities of VBGF and 
variance parameters corresponding with the homosce- 
dastic log-normal and power-variance log-skew-^ mod- 
els are compared in Figure 2. The asymmetry and dis- 
persion of the posterior densities of VBGF were similar 
for the different error distributions. However, for the 
variance parameter, the posterior density was leptokur- 
tic when log-normal error distribution was used. 
Considering the boxplots of residuals by age from 
