deletions | additions       diff --git a/Method.tex b/Method.tex  index e1f6097..61491fc 100644  --- a/Method.tex  +++ b/Method.tex 
...
  Where: \(Y_u\) is the deflection, \(\epsilon\) is the strain, \(L\) is the cut length and \(R\) is the big end cross-section radius.   Narrow-sense heritability for all properties was calculated using equation \ref{H2}. The constant of 2.5 used was suggested by \citet{Griffin1988Genetic} due to the unknown proportions of selfing, full-siblings and half-siblings within the open-pollinated families (loosely refereed to as half-sibling families in this article).  \begin{equation}  \label{H2}  H^2 = \frac{2.5 \times Var(Family)}{Var(Family) + Var(Tree) + Var(Residual)}  \end{equation}  The analyses were conducted in R \citep{RCORE} and JAGS \citep{RJAGS}, utilizing a Bayesian approach to estimate the posterior distributions for the heritability of growth-strain and other wood properties.   We implemented a hierarchical model where $y_{ijkl}$ $y_{ijklm}$  follows a left-censored normal distribution $N(\omega_{ijkl}, $N(\omega_{ijklm},  \tau)$ with predicted value $\omega_{ijkl}$ $\omega_{ijklm}$  and a trial-dependent  precision $\tau$. $\tau[x_1]$.  The precision $\tau$ ($1/\sigma^2$) (reciprocal variance) $\tau[x_1]$ for each trial  was given a vague gamma prior $\Gamma(0.01, 0.01)$. The predicted value for the $i^{th}$ tree assessment  is modelled as a function of an overall intercept, the effect of the $j^{th}$ trial, $k^{th}$  coppicing level,$k^{th}$ site and  $l^{th}$ family: family and $m^{th}$ tree (to account for repeated assessment pre- and post-coppicing):  \begin{equation*}  \omega_{jkl|i} \omega_{jklm|i}  = \mu + \alpha [x_{1 j|i}] + \beta [x_{2 k|i}] + \gamma [x_{3 l|i}] + \delta[x_{4 m|i}]  \end{equation*}  where $x_1$, $x_2$ and $x_2$,  $x_3$ and $x_4$  represent indicator variables for the levels of the factors. The overall intercept ($\mu$), and individual-level coefficients for coppicing ($\alpha_j$) and trial site  ($\beta_k$) were given vague normal prior distributions: \begin{equation*}  \begin{split} 
...
\end{split}  \end{equation*}  The familyeffects  ($\gamma_l$) and tree ($\delta_m$) effects  were assumed to come froma  normal distribution distributions $N(0, \tau_f)$ and  $N(0, \tau_f)$, \tau_t)$,  with vague gamma prior priors  $\tau_f \sim \Gamma(0.01, 0.01)$. 0.01)$ and $\tau_t \sim \Gamma(0.01, 0.01)$ respectively. The statistical model is also presented graphically following \cite{kruschke2014doing} in Figure \ref{graph:model}.  Narrow-sense heritability at the trial level for all properties was calculated using equation \ref{H2}. The constant of 2.5 used was suggested by \citet{Griffin1988Genetic} due to the unknown proportions of selfing, full-siblings and half-siblings within the open-pollinated families (loosely refereed to as half-sibling families in this article).  \begin{equation}  \label{H2}  h^2 = \frac{2.5 \times \sigma_f^2}{\sigma_f^2 + \sigma_t^2 + \sigma_^2}  \end{equation}  where the $\sigma^2$ variances were obtained as the reciprocal of the precisions ($\tau$) mentioned in the model description.  Although all specimens were grown on the same site, they were grown during different time periods which are confounded with the effect of the two provenances and hence are included as the trial effect. The tree effect accounts for the measurement unit, as the same trees were assessed as both seedlings and coppice. In addition, residuals were modeled separately for each section. The R/JAGS code is available as supplementary material.