Model results
Of the three model specifications fitted, the full spatio-temporal model
provided the best fit as determined by the model marginal
log-likelihoods (Appendix C) and hence also the posterior model
probabilities. The INLA posterior model summary for the spatio-temporal
model is presented in Table 1. Posterior density plots for all model
parameters are provided in Appendix D.
In the full spatial and temporal model (M 3), we
found a strong effect of the NTR on the presence of urchin barrens, with
the odds of barrens presence being doubled outside of the NTR compared
to within the NTR. For an NTR effect, the median coefficient estimate of
-3.107 (Table 1) on the logit scale is equivalent to a multiplicative
effect on the odds-ratio scale of 0.045 (i.e., exp(-3.107) = 0.045).
Therefore, if other model factors are held fixed, the percentage change
in the odds of an image being classified as a barren is (exp(-3.107) -
1)x100 = -96% when moving from a site outside of the NTR to within the
NTR.
Over the five-year time period, there was a substantial overall increase
in the presence of barrens across all the sites (positive year effect in
Table 1). The median coefficient for the year effect of 0.277 equates to
a change in the odds-ratio of barrens presence of 1.319. This means
that, when holding all other model factors fixed, the overall percentage
increase in the odds of the presence of barrens each year is increasing
by 31.9%. However, the rate of change inside the NTR was not found to
be substantially different to the reference sites (95% central CI for
the NTR:year effect spans zero in Table 1). Therefore, there was a rapid
and substantial rate of increase in barrens across the region of
coastline in this study, but the rate of change inside the NTR was not
statistically distinguishable from that of the reference sites.
Decisive effects were found for the environmental covariates of
depth-squared and rugosity (Table 1 and Fig. 3). The empirical
distribution of barrens across depth in the survey region showed barrens
ranged between 15 and 37 m (Appendix E). The strong effect of
depth-squared (Table 1) indicates a concave relationship between urchin
barrens presence and depth. For a hypothetical site within the NTR, in
the year 2016 with mean rugosity and no spatial random effects, this
relationship is plotted in terms of the predicted probability of barrens
presence for different value of depth in Fig. 3A. The peak probability
for urchin barrens presence is approximately 20 m, with low probability
beyond 40 m. Thus, there is a quadratic effect of depth, with an overall
lower presence of barrens in the shallower and deeper images collected
(Fig. 3A).
For a hypothetical site within the NTR in the year 2016, but now at a
constant mean depth with rugosity varying and no spatial random effects,
rugosity was found to have a strong positive effect on the probability
presence of barrens (Fig. 3B). Rugosity values in the raw data ranged
from near zero to 0.544, with a mean of 0.015 ± 0.030 inside the NTR and
a mean of 0.011 ± 0.018 at the reference sites. Rugosity values of
unsampled bathymetric cells in the region ranged up to 0.758.
Model comparisons showed that both spatial and temporal correlation were
important in explaining the presence of urchin barrens (Appendix B).
Results for the spatial random effects showed that spatial correlation
occurred over a mean range of approximately 17 m (Fig. 4A) with mean
spatial standard deviation of 3.413 (Fig. 4B). The posterior
distribution of the AR1 temporal correlation parameter was found to have
a mean of 0.734 ± 0.053, implying a strong temporal correlation effect
for the presence of barrens with barrens status likely to persist
throughout the time period.