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The term in parentheses shows that the impact of both predictors varied with species, S, thus describing the hierarchical aspect (equivalent to S being a random effect in a mixed effects model).
The data were modeled as Cd~E+G+GxE+e, with E being a random effect (G=Genotype, E=Environment=Field x Year).
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Replicate was a random effect with repeated measures.
Because species was a random effect, the fixed effect is the mean of all species medians.
In these models, each individual subject is a random effect while the fixed effects represent average trends.
Since θ i is a random effect it is typical to assume θ i is iid Normal, with mean 0 and variance τ2 to be estimated.
Environments are deemed to be a random effect, as they were chosen to represent a roughly random sample of a population of environments.
Time (duration of clamp being tight), limb (left or right), and clamping order (first or second foot) were fixed effects, whereas horse was a random effect.
αtri is a random effect for technical replicate i within each biological replicate r, which is assumed to be uncorrelated with mean zero and variance σ2α.
There was a random effect for subject.
Case ID is a random effect in this model.
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