Torres-Paiados et al.: Age and growth parameters of Urotrygon aspidura 
171 
x100, (2) 
where = the i th reading for individual j; 
Xj = the average band-pair count for individual j; 
R - the number of readings (one for each reader); 
and 
n = the number of readings for the entire sample. 
Low values of these indices indicate greater precision 
between readers (Campana, 2001). 
Periodicity in band-pair formation 
The periodicity in the formation of vertebral band pairs 
in Panamic stingrays was evaluated through the analysis 
of edge type (qualitative approach) and of the marginal 
increment (MI) index (quantitative approach). For the 
edge type, the band at the edge of each vertebra (opaque 
or translucent and growing or ending band) was identi¬ 
fied, and the monthly frequency of each combination was 
calculated. The MI was estimated on the basis of the fol¬ 
lowing measures taken of each vertebral sagittal section 
with the Zen lite microscope software: radius (R, or dis¬ 
tance from the focus to the vertebral edge), and distance 
from the focus to the last (R n ) and next-to-last (R n _i) com¬ 
pletely formed band. The MI was estimated by using this 
equation: M/=CR-i? n )/(i? n -J? n . 1 ) (Hayashi, 1976), and the 
average value was calculated and plotted per month. Dif¬ 
ferences in Mis among months were evaluated by using a 
Kruskal-Wallis test. 
Growth models and parameters 
Two data sets of age were built; the first considers the 
months with peaks of reproduction, and the second does 
not include this reproductive information (Harry et al., 
2010; Mejia-Falla et al., 2014). In the first case (adjusted 
analysis), the time between birth and the first band for¬ 
mation was considered, on the basis of the reproductive 
cycle of the species obtaining a mean value for all data. 
Along the Pacific coast of Colombia, Panamic stingray 
present 2 birth peaks per year (in January and August; 
P. Mejia-Falla, unpubl. data), and the month of growth band 
formation occurs in October-November (see the “Results” 
section). Therefore, the times between birth and formation 
of the first growth band (9-10 or 2-3 months) have a mean 
of 6 months (0.5 years), a value that was subtracted from 
the total number of translucent bands counted. For the 
second data set, it was assumed that the first band pair 
is formed 1 year after birth, irrespective of reproductive 
seasonality; therefore, a value of 1 was subtracted from 
the total number of translucent bands counted. For exam¬ 
ple, an individual with 3 translucent bands counted had 
an age of 2.5 years for the first data set and an age of 
2.0 years for the second data set. 
A multi-model approach was used to describe growth. 
Eight models were fitted by using age and DW data of 
individuals: the von Bertalanffy growth model (VBGM; 
von Bertalanffy, 1938), Gompertz growth model (Ricker, 
1979), and logistic growth model (Ricker, 1979) were fit¬ 
ted with 2 and 3 parameters, and the two-phase growth 
model (TPGM; Soriano et al., 1992) was fitted with 4 and 
5 parameters (Suppl. Table 1) (online only). Models with 2 
and 4 parameters include a DW 0 parameter, for which the 
birth DW at 7.5 cm was used (P. Mejia-Falla, unpubl. data). 
The parameter BY/.. is the theoretical asymptotic width, 
representing the average disc width at age that individ¬ 
uals in a stock would attain if they grew indefinitely. The 
annual k is the relative growth rate at which a stingray 
reaches DW„ at age. The parameter t 0 is the theoretical 
age at zero, a point on the time axis when mean DW at age 
is zero. Models with 4 and 5 parameters included t h , which 
is the age at which the transition between the 2 phases 
occurs (inflection point), and h is the maximum difference 
in DW at age between the VBGM and the TPGM at the f h . 
Parameters of the models were estimated by maximum 
likelihood method. The most adequate model was chosen 
on the basis of biological and statistical fit. Biological fit 
was based on known maximum DW, DW at birth, and DW 
at maturity (26.5, 7.5, and 15 cm DW, respectively; P. Mejia- 
Falla, unpubl. data). Statistical relevance was evaluated 
by calculating Akaike information criterion (AIC,, Akaike, 
1973), as well as the difference in AICj between models (Aj) 
and Akaike weight (w^, for each i th model, as follows: 
A/C, 
= 21n(L) + 2 p, 
(3) 
A, 
= AJC i -A/C min , and 
(4) 
w i 
g (-°.5A i ) 
(5) 
y J-°- SA i) ’ 
where p = the number of parameters; 
L = maximum likelihood; 
A ; = the difference between each model’s AIC;; and 
A/C m in = the lowest AIC ; of all models. 
Models with A ; values between 0 and 2 have substantial 
statistical support, those with values between 4 and 7 
have moderate support, and those with values >10 have 
no support (Burnham and Anderson, 2002). With multi¬ 
model inference, if one model does not obtain an Akaike 
weight value over 0.9 or if several models have sub¬ 
stantial statistical support, it is advisable to estimate a 
weighted average of parameters on the basis of the mod¬ 
els with support, along with the associated unconditional 
standard error (SE) (Burnham and Anderson, 2002), as 
follows: 
0 : = Y" w, x 0 and 
1 i 1 
varfe^ + fe.-e ) 2 
where SE(Q) = the standard error of the mean 
0 = the parameter to be averaged; 
SE(d) = x 
X 
0 ; 
( 6 ) 
(7) 
