Lopez Quintero et al.: Bayesian analysis of the von Bertalanffy growth function 
17 
then the rth observation is considered influential. Ad- 
ditionally, because the integral in Equation 16 cannot 
be written in closed form, it still can be approximated 
by sampling from the posterior distribution of 6 via the 
MCMC algorithm. In fact, if 6i,...,6b is a sample of size 
B from 7r(0 1 S), then the MCMC estimator of K(P,P_i) 
is computed as 
K{P,P_i) = 
logjiE 
S=1 
+ iIls=i^ogf{yl\x,0J, 
(18) 
with /'(y[|x,0g) given by Equation? and 9^ = 
(e.g., Cancho et al., 2011). It should 
be noted that we computed the KL-divergence be- 
tween P and P_i using the ith marginal sample density 
/’(y(|x,0s), but we did so without considering the pos- 
teriors 7r(0|S) and 7r(0|<S_j). 
In addition, given the new sample with removed 
observations, we quantified the change produced for 
each new estimate with respect to the full sample. In 
several cases, the estimates were notably different for 
these samples (Contreras-Reyes and Arellano-Valle, 
2013). We used restricted data in which a set of 
J observations was removed. Then, we computed the 
percentage of relative change (RC) of estimates by fol- 
lowing Contreras-Reyes and Arellano-Valle (2013). The 
RC was defined by 
RCi0^,e^j)^lOO 
(19) 
where and ^ are the posterior median estimates of 
^th component of 9 obtained from the posterior distri- 
butions Tr{9\S-j) and 7r(0 | A), respectively. Therefore, 
we computed the change (in percentage) of each para- 
meter of the VBGF. 
Application 
Data We evaluated the performance of the proposed 
model, using the available data for southern blue 
whiting. This dataset was based on 24,942 individuals 
collected from a region spanning latitudes from 46°S 
to 6°S over the period 1997-2010 by the Institute de 
Fomento Pesquero (Contreras-Reyes, unpubl. data). 
Random samples of fish were collected by onboard sci- 
entific observers during each catch haul of southern 
blue whiting were caught. All these fish were mea- 
sured to the nearest centimeter, and both otoliths of 
each fish were extracted onboard. Otoliths were then 
taken to the laboratory, where age was determined 
by reading annual growth increments in the sagit- 
tal otoliths. The southern blue whiting is assumed 
to recruit once a year; therefore, age is treated as a 
discrete variable with a 1-year interval. Otolith age 
assignment involved killing sampled fish; therefore, 
each data point represents 1 individual fish. Fish in 
the catch had observed ages between 1 and 25 years 
and a size range of 20-75 cm in total length (Ces- 
pedes et al., 2013). Contreras-Reyes et al. (2014) re- 
ported extreme values in young and old age classes 
and reported asymmetry caused by fishing selectivity. 
Both these issues justify the use of heavy-tailed and 
skewed distributions in VBGF errors. 
MCMC sampling For inference, 4 chains were se- 
lected from each applied Bayesian model. The length 
of the chains necessary to reach convergence dif- 
fered depending on the treated model: it was around 
20,000 iterations for all log-skew-^ models and 
around 100,000 iterations for the log-normal ho- 
moscedastic model. We considered a burn-in period 
to be 10,000 iterations for the first model and 20,000 
for the log-normal homoscedastic model. In addition, 
all models conformed with the traditional diagnostic 
convergence tests, such as Geweke and Heidelberger- 
Welch, when tests were applied to individual chains 
(Cowles and Carlin, 1996; Carlin and Louis, 2000). 
However, results of the Raftery-Lewis test, also ap- 
plied to individual chains, indicated that we should 
take the largest thinning of chains (Link and Eaton, 
2011) because values were highly correlated. In ad- 
dition, visual examination throughout trace and au- 
tocorrelation plots (not shown) indicated that conver- 
gence was reached for all parameters in all models. 
This situation was transferred to the effective sample 
size, which can be interpreted as the number of in- 
dependent samples necessary to yield the same pre- 
cision as the (serially dependent) MCMC samples. 
Effective sample size is especially important in re- 
sampling and should not be confused with the de- 
gree of over dispersionusually found in length-at-age 
compositions. 
For all models, the parameters with higher and low- 
er values of effective sample size were v and K, respec- 
tively. Finally, R Gelman’s indexes (Gelman and Rubin, 
1992) were all near 1, indicating that the specific pa- 
rameter had good convergence after the burn-in period 
was eliminated. This test was applied to 4 chains for 
each parameter and each model. 
Software Statistical methods used in this article were 
implemented in the software R, vers. 3.1.0 or higher 
(R Core Team, 2014). The MCMC was developed in 
C+-I- embedded in the R package RcppArmadillo, vers. 
0.4.300.0 or higher (Eddelbuettel and Sanderson, 2014). 
Diagnostic analysis was conducted with the coda pack- 
age, vers. 0.16-1 or higher, in R (Plummer et al., 2006). 
Von Bertalanffy growth curves were estimated in each 
realization by simulating estimated parameters several 
times (e.g., 10,000) with the models proposed previous- 
ly in the Comparisons and selection of models section. 
Such simulated observations are called fake data, ac- 
cording to Gelman and Hill (2007, Ch. 16). Afterward, 
the 95% highest posterior density (HPD) was computed 
across ages. The generation of the log-skew-^ values 
was conducted by using fake data in the R package sn, 
vers. 0.4-11 (Azzalini, 2008). 
