with \(y\left(s_{i},\ t\right)\ \) the categorisation of barren
presence for an image located at \(s_{i}\)in year \(t\), space and
time-varying covariate matrix \(X\left(s_{i},\ t\right)\) and
separable spatio-temporal random effects \(z\left(s,t\right)\). The
temporal covariance function is a stationary autoregressive process,
such that \(-1<\ \phi<1\), where\(z\left(s,\ t\right)=\ \phi z\left(s,\ t-1\right)+\ \omega(s,\ t)\)with \(z(s,\ 1)\) drawn from the stationary distribution.
The spatial random effects in year \(t\) have a stationary spatial
covariance function with spatial correlation function\(H(s-\ s^{{}^{\prime}},\rho)\) for a site \(s\) and \(s^{{}^{\prime}}\). The spatial
range is defined by Lindgren et al. (2011) as the distance at which the
spatial correlation drops to close to 0.1. The internal parameterisation
of the range \(\rho\) and spatial variance \(\sigma^{2}\) used by INLA
is given in Appendix A.
The model covariates for which β coefficients were estimated
included fixed effects for the NTR (binary, whether an image was located
inside the NTR or not), year, an interaction term between NTR and year,
depth, depth-squared (to capture expected non-linear effects in deep and
shallow images), and rugosity (see above). Depth was included as it has
been previously found to be an important predictor of barrens presence
(Johnson et al., 2005, Perkins et al., 2015).
Model M 2 is obtained fromM 3 by omitting the temporal dependence component
(i.e., setting the temporal dependence parameter to zero). ModelM 1 is then obtained from modelM 2 by also omitting the spatial dependence
component (i.e., setting the spatial variance parameter to zero). The
hypothesised models are considered equally likely a priori and compared
on the basis of the marginal likelihoods given the observed data. The
marginal likelihood (or evidence) of each model is proportional to the
posterior model probabilities given the a priori equal model weights.
We used the Integrated Nested Laplace Approximation approach (INLA; see
Rue et al., 2009) for Bayesian spatial and spatio-temporal modelling.
All statistical analyses were conducted within the R statistical
computing package (R Core Team, 2019).
As the values of rugosity were right-skewed (i.e. mostly small values,
with a few larger values), to avoid leverage issues a logit
transformation was applied to the raw rugosity values. All physical
model covariates (i.e. depth, depth-squared and logit-transformed
rugosity) were centred by their mean and scaled by their standard
deviation.
We examined both the percentage change in odds ratios and predicted
changes in the probability of urchin barren presence to interpret the
relationships with respect to model covariates. For the percentage
change in odds ratios given by an increase of one unit in thei th covariate, we used the formula\(\operatorname{(exp}\left(\beta_{i}\right)-1)*100\). We examined
the influence of covariate effects by plotting the expected probability
of barrens presence over the range of values for that covariate in our
sample space while holding all other covariates at their mean and
excluding spatial random effects. That is, we plot the marginal
relationships between the probability of barrens presence for both depth
and rugosity. This was accomplished by taking 5000 joint posterior draws
of the unknown β coefficients from the fitted model. We
chose to examine the influence of each covariate within the NTR in 2016
(i.e. the last year surveyed).