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):