40 
Fishery Bulletin 11 7(1-2) 
tion for fecundity analysis. Batch fecundity was deter¬ 
mined volumetrically for 6 subsamples per individual 
for fish histologically verified in the actively spawning 
subphase (Bagenal and Braum, 1971). 
Reproductive parameters 
Reproductive parameters were calculated from original 
data when available. When only summary data were 
available, the mean, standard error, and number of 
samples were taken or calculated from tables or graphs 
provided in the publication. Spawning seasonality was 
defined on the basis of the Gonadosomatic Index (GSI), 
with 
GSI = (gonad weight / gonad-free weight ) x 100. (1) 
Fish were considered reproductively active if their GSI 
was >1.0, and reproductively inactive if their GSI was 
<1.0. This GSI threshold was based on histological as¬ 
sessments of developing and spawning-capable fish. 
Spawning interval (estimated days between spawn¬ 
ings) was calculated in 2 ways using 2 separate types 
of spawning markers. First, the reciprocal of the to¬ 
tal number of actively spawning females (i.e., those 
undergoing oocyte maturation (OM), including those 
with hydrated oocytes) was divided by the number of 
spawning capable females. For the second method, we 
used the reciprocal of the total number of females with 
postovulatory follicles (POFs) <24 h that were observed 
in the ovary divided by the number of spawning ca¬ 
pable females. We assumed that both types of spawn¬ 
ing markers are equivalent in both detectability and 
duration (approximately 24 hours, but see Porch et al., 
2015). Because batch fecundity is positively correlated 
with fish size (Lowerre-Barbieri et al., 2015), relative 
batch fecundity (RBF, number of eggs/g ovary-free body 
weight) was used in all models to remove the influence 
of fish size and was calculated as batch fecundity di¬ 
vided by ovary-free body weight. 
Data analysis 
Our analyses involved the use of hierarchical Bayesian 
models, which are well suited to account for statisti¬ 
cal uncertainty at multiple levels and from multiple 
sources (Gelman et al., 2013). An essential component 
to Bayesian analysis is prior distributions, which can 
be thought of as formalized assumptions of our uncer¬ 
tainty about the parameters. In particular, we used 
weakly informative priors that place a low probabil¬ 
ity on extraordinarily unreasonable parameter values 
without excluding anything in a broad range of plausi¬ 
bility. A Bayesian analysis combines the prior distribu¬ 
tions with the likelihood of the data to create a poste¬ 
rior distribution that describes the uncertainty around 
the parameter estimates. Posteriors are often described 
by their median and their credible intervals (usually 
50% and 95%), which are the corresponding quantiles 
of the distribution. The posterior distribution is gen¬ 
erally estimated with an algorithm that explores and 
samples parameter values over a series of iterations. 
Once the posterior distribution has been estimated, 
it can be used to perform a posterior predictive check 
(i.e., simulate new data from the parameters to evalu¬ 
ate the suitability of the model). If these generated 
data are not similar to the real data, the model may 
need revision. For further details, an introduction to 
Bayesian methodology is provided by McElreath (2016). 
We examined the potential change in red snapper 
reproductive variables (spawning seasonality, spawn¬ 
ing interval, and fecundity) over a 27-year period. Our 
data came from multiple studies conducted by different 
researchers at varying times throughout the northern 
GOM, and all data sets shown in Table 1 were used 
in our analyses. Although we have full data sets for 
many of these studies, a substantial subset provided 
only means and standard errors. For each reproduc¬ 
tive parameter, we combined a Gaussian process time 
series model, which is a flexible method that estimates 
temporal trends where there is similarity among near¬ 
by years but not a linear increase or decrease in years, 
with a random effects meta-analysis, which accounts 
for variation among studies. Unless otherwise noted, 
all estimates and predictions produced by these analy¬ 
ses are posterior distributions. 
To estimate spawning seasonality, we used a Gauss¬ 
ian process time series to estimate the mean GSI for 
each month and year within the data range; both raw 
and summarized data were used in this analysis. Sepa¬ 
rate Gaussian processes estimated monthly, yearly, and 
monthly-by-yearly interactions. We calculated the prob¬ 
ability of spawning activity for each month and year 
from the proportion of the posterior distribution cor¬ 
responding to a mean GSI estimate that was >1. The 
sum of monthly spawning probabilities for each year is 
the expected spawning period for the year. 
Because spawning interval is the reciprocal of a 
proportion (i.e., the proportion of fish spawning at a 
given time), it can be analyzed with a modified logis¬ 
tic regression. When available, we used the number of 
fish caught and the number spawning to fit the model. 
When original data were unavailable, we used pub¬ 
lished estimates of spawning intervals and calculated 
standard errors from sample sizes. To examine poten¬ 
tial differences between the northeastern and north¬ 
western GOM, we compared models with a single time 
series with models where the 2 regions were modeled 
separately. We ran these models for both of the meth¬ 
ods to determine spawning interval (for individuals 
undergoing OM or POF). In addition to the time se¬ 
ries and meta-analysis, our model re-estimated spawn¬ 
ing interval for each study from the raw data. This 
approach improves spawning interval estimates from 
studies with small sample sizes and allows a credible 
interval calculation. 
We used only individual-level data for our analysis 
of RBF. As with the spawning interval analysis, we 
compared a model that estimated separate time series 
for the northeastern and northwestern GOM with a 
model that pooled the regions. 
