where \(Y_{u}\) is the deflection, \(\epsilon\) is the strain, \(L\) is
the cut length and \(R\) is the big end cross-section radius.
All specimens were grown on the same site; however, they were grown
during different time periods which are confounded with the effect of
the two provenances and hence are included as a trial effect. The tree
effect accounts for the measurement unit, as the same trees were
assessed as both seedlings and coppice.
The analyses used a Bayesian approach to estimate the posterior
distributions for the heritability of growth-strain and other wood
properties. We implemented a hierarchical model where
\(y_{\text{ijklm}}\) follows a left-censored normal distribution
\(N(\omega_{\text{ijklm}},\tau_{j|i})\) with predicted value
\(\omega_{\text{ijklm}}\) and a trial-dependent precision
\(\tau_{j|i}\). The precision (reciprocal variance)
\(\tau[x_{1}]\) for each trial was given a vague gamma
prior \(\Gamma(0.01,0.01)\).
The predicted value for the \(i^{\text{th}}\) assessment is modelled as
a function of an overall intercept, the effect of the \(j^{\text{th}}\)
trial, \(k^{\text{th}}\) coppicing level, \(l^{\text{th}}\) family and
\(m^{\text{th}}\) tree (to account for repeated assessment pre- and
post-coppicing):