Deli'Apa et al.; Modeling the distribution of Squalus acanthias, by sex 
93 
Table 1 
Summary of variables as potential fixed effects that influence the distribution of spiny dogfish (Squalus 
acanthias) in the mid-Atlantic and included in Bayesian models. SST=sea-surface temperature; chl- 
a=chiorophyll-a concentration; practical salinity is a ratio and does not have physical units. 
Variable 
Description 
Units 
Bathymetry 
Mean fishing depth of haul 
m 
Slope 
Seabed slope at the sampling station 
% grade 
Distance to shore 
Distance from the coast at the sampling station 
m 
SST 
SST monthly value of haul 
°c 
Chl-a 
Chl-a monthly value of haul 
mg/m® 
Salinity 
Salinity of the water 
- 
Season 
Season when haul was sampled 
Spring, fall 
Time 
Time when haul was sampled 
Morning, afternoon, night 
random-effect term, reducing their influence on the es- 
timation of the habitat variables (Gelfand et al., 2006). 
Hierarchical Bayesian spatiotemporai models were 
used to predict abundance of spiny dogfish, by sex, with 
respect to explanatory variables, as well as to describe 
the main spatial distribution changes over time, by 
sex. These models are extremely applicable to studies 
characterized by data observed at continuous locations 
within a defined spatial area, as was the case for the 
data set used in this study. Values of CPUE from the 
NEAMAP surveys were considered appropriate proxies 
for levels of abundance of spiny dogfish. 
The spatial variation of the CPUE values for spiny 
dogfish, by sex, was modeled by using a hierarchical 
Bayesian spatiotemporai approach, specifically a Pois- 
son point process model with log-linear intensity. It 
was assumed that the number of spiny dogfish at each 
station sampled, has a Poisson distribution with 
rate tj where is the observation time at site i and 
A.i is proportional to relative species abundance at sta- 
tion i and measures the survey expectation for a unit 
observation time, according to this general formulation: 
logikOi - a + ZijP + Fj + Wi, 
where a 
P 
Wi 
the intercept; 
the matrix of covariates at the year j and the 
station I; 
the vector of the regression coefficients; 
the component of the temporal unstructured 
random effect at the year j; and 
the spatially structured random effect at the 
station i. 
In this model, independence between the sampling 
locations is assumed. However, some spatial autocor- 
relation may be present in the data set because the 
abundance of a species at nearby stations is influenced 
by similar environmental parameters. Consequently, 
adjacent, or nearby, stations would be expected to be 
similar in terms of abundance of spiny dogfish. The W, 
accounts for this influence. For each of the models for 
both sexes, 8 potential fixed-effects were considered: 
6 environmental variables and 2 temporal variables 
(Table 1). 
For all the parameters considered in the fixed-effects 
model, a vague zero-mean Gaussian prior distribution 
with a variance of 100 was assigned, and a zero-mean 
Gaussian prior distribution with a Matern covariance 
structure was assumed for the spatial effect (for more 
details about the spatial component, see Munoz et al., 
2013). Finally, for the temporal effect, a LogGamma 
prior distribution with the parameters of shape and 
scale equal to 1 and 5x10®, respectively, was assumed 
for the log-precision parameter Aj, and j represented 
the year. 
For each parameter, a posterior distribution was 
obtained. Unlike the mean and confidence interval 
produced by classical analyses, this type of distribu- 
tion enables explicit probability statements about the 
parameter. Therefore, the region bounded by the 0.025 
and 0.975 quantiles of the posterior distribution results 
in an intuitive interpretation: for a specific model, the 
unknown parameter is 95% likely to fail within this 
range of values (95% credibility interval [CrI]). 
All models obtained by combining environmental, 
spatial, and temporal variables and the possible inter- 
actions were fitted and compared by using the mea- 
sures of the deviance information criterion (DIG) (Spie- 
gelhalter et al., 2002) and the cross-validated logarith- 
mic score (LCPO) (Roos and Held, 2011). Specifically, 
smaller DIG and LCPO values indicate better fit and 
predictive quality. 
All the analyses were performed with the integrated 
nested lapiace approximation (INLA) method that is 
implemented in Rue et a!., 2009; Martins et al., 2013 
and with the R-INLA package (website) in R software. 
Model validation 
Two approaches were used to assess the predictive ac- 
curacy of the selected model. First, the predicted and 
observed values from the full data set were compared. 
