Directed Acyclic Graph showing how <i>Acropora</i> cover <i>y_ij</i> at a site <i>i</i> and time <i>j</i> is fitted in three stages.

<p>Here, the transformed data <i>y_ij</i> are modelled using a normal distribution with an expected value <i>µ_ij</i> and variance σ_ε. The expected value <i>µ_ij</i> is a function of four spatial scales (sub-region <i>f_s</i>, habitat <i>f_h</i>, reef <i>f_r</i>, and site <i>f_i</i>). Linear trends at sub-region and habitat scale are modelled using fixed effects (<i>β</i>, <i>γ</i>). At the site and reef scale, trend parameters (<i>θ</i>, <i>δ</i>) are considered as random. Trend parameters are denoted with the subscript 0 corresponding to the intercept and 1 corresponding to the slope. Random effects require specification of a variance component (<i>σ</i>), specified for each spatial scale of the model. Refer to the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0110968#s2" target="_blank">methods</a> section for equations.