Carrillo-Colin et al.: Bayesian estimation of the age and growth of Rhinoptera steindachneri 13 
The second model was a modified version of the GM 
(Ricker, 1975), described as follows: 
—log( BW \(1-e78) 
DW, = DWe - 
(6) 
where g = the completion growth parameter (Smart and 
Grammer, 2021). 
Model fitting was done by using the software JAGS (vers. 
4.3.0; Plummer, 2003). An additive error was assumed, 
indicating that, for individual i at given age t, DW; follows 
a normal distribution expressed by a mean of zero, and 
its precision (t), which is the inverse of its variance, is 
expressed as t=1/o” (Doll and Jacquemin, 2018). 
It is necessary to specify a prior distribution for the param- 
eters to fit the growth models by using Bayesian methods. 
The priors for the parameters DW.., k, and g were chosen to 
be relatively uninformative, expressed as a uniform distribu- 
tion (UD) based on growth estimates and information about 
maximum size in waters of Mexico (Villavicencio-Garayzar, 
1996; Gamez-Moedano et al., 2006): UD(80,200) and 
UD(0,1), respectively. The prior for size at birth DW) was 
informative and normally distributed (ND): ND(41.5,1.0) 
based on the reported range of DW, (38-45 cm; Bizzarro 
et al., 2007; Burgos-Vazquez et al., 2018). The prior for o” 
was UD(0,100). The posterior distribution for the parame- 
ters of the 2 growth models (Equations 5 and 6) was com- 
puted by using Bayes formula (Smart and Grammer, 2021): 
P(d | ®)P() 
J, P(d | @)P@) 00 
P(®| d) = (7) 
Given the complexity of this multiparameter equation, the 
evaluation of the denominator requires numerical meth- 
ods. The posterior probability distribution is sum- 
marized by drawing random samples from the 
vector of all parameters with the Markov chain 
Monte Carlo algorithm (Robert and Casella, 
2004). A total of 1 million runs were completed 
after an initial burn-in of 20,000 runs. Every 
tenth iteration was saved to test convergence 
(Gerrodette and Eguchi, 2011). 
The growth models (VBGF and GM) were fit 
to data and then compared by using the Wata- 
nabe—Akaike information criterion (WAIC). The 
WAIC is considered a better version of the devi- 
ance information criterion, which is commonly 
used to assess Bayesian models because it uses 
the entire posterior distribution to make an 
inference (Coll et al., 2019). The best model was 
chosen by considering a difference of at least 2 
units in WAIC as is typically done for the Akaike 
information criterion (Burnham and Anderson, 
2002). Once the best model was selected, we 
compared 2 versions of the best model, with and 
without sex variation, and the WAIC was used 
to choose the best model. All statistical analyses 
were performed by using Microsoft Excel 2015 
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oy 
® 
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Lo 
(Microsoft Corp., Redmond, WA) and R, vers. 3.6.1. 
(R Core Team, 2019). 
Results 
Age estimation 
A total of 256 specimens of golden cownose rays (129 females 
and 127 males) were analyzed. Seven vertebrae were dis- 
carded because the band pairs were unreadable. The size 
range for females was 29.4—96.9 cm DW (mean: 49.20 cm DW 
[standard deviation (SD) 14.62]), and males ranged in size 
from 21.5 to 92.9 cm DW (mean: 49.69 cm DW [SD 15.71)). 
A higher proportion of organisms were observed in the 
10-cm-DW interval of 35-45 cm DW (Fig. 2). The difference 
in mean size between sexes was 0.49 cm DW, and the mean 
difference in DW between sexes was —0.15 cm (SD 1.93), 
with a 95% credible interval (95% CI) of —1.49-1.27 cm. This 
difference in mean DW is considered irrelevant because 
the probabilities of males being larger (P=0.53) or smaller 
(P=0.47) than females are almost equal (odds=1.13). 
We found strong evidence of a linear relationship 
between VD and DW, given that the 95% CI of 12.49-13.52 
for the marginal posterior of the slope did not include zero. 
The equation that describes the relationship is as fol- 
lows: DW=1.98+12.98(VD). The bias plot shows 13 growth 
bands identified by the 3 readers, indicating that variabil- 
ity increases with the number of growth bands. For the 
readings of the 3 readers, the overall JAPE was 12.11%, 
the CV was 15.73%, and the precision index was 67.47%. 
The precision of readings between each pair of readers is 
shown in Figure 3. 
Results of the MI analysis indicate that deposition of 
growth bands on examined vertebrae was close to 
25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 
Disc width (cm) 
Figure 2 
Length-frequency distributions for female (black bars) and male 
(white bars) golden cownose rays (Rhinoptera steindachneri) sam- 
pled during 2008-2014 in the southern Gulf of California in Mexico 
(number of rays=256). 
