#### 2.4. Statistical Analysis

Data on wood properties (Wood density, MoE and MoR) and growth (DBH, height and volume) were tested for normality and homogeneity with Kolmogorov-Smirnov D and normal probability plot tests in SAS software version 9.1.3 [

19]. In a few marginal cases, a graphic residual analysis was also performed; however, no data transformation was deemed necessary. Variance components and covariances were estimated using Mixed procedure and PROC VARCOMP in SAS. Data on wood properties (density, MoE and MoR) were subjected to the following linear mixed model:

where: Y

_{ijkl} is the observation in the

ijkl^{th} tree;

µ is the overall mean; H

_{i} and R

_{j} are fixed effects of stem height and stem radial position, respectively; F

_{k} is the random effect of family;

T_{l}(

F_{k}) is the tree-within-family effect; (

HR)

_{ij} is the interaction between stem height and stem radial position effect; (

RF)

_{jk} is the interaction between stem radial position and family effect;

HT(

F_{k})

_{il} is the interaction between stem height and tree-within-family effect;

RT(

F_{k})

_{jl} is the interaction between stem radial position and tree-within-family effect; (

HRF)

_{ijk} is the interaction between stem height, stem radial position and family effect; and e

_{ijl(k)} is the random error term.

A

X^{2} test was performed on the difference in the −2 residual log likelihood of the model to select the best model. All the interactions between parameters were removed from the analysis because their contribution to the total variance was negligible. Furthermore, the variance component for these terms could not be estimated or was otherwise insignificant. Equation (2) was therefore reduced to:

Growth data were subjected to a linear mixed model with replicate as a fixed factor and family as a random effect factor.

Family mean heritability and individual heritability were calculated using Equations (4) and (5), respectively [

20]:

where

${h}_{a}^{2}$ and

${h}_{w}^{2}\text{}$are heritability among and within families, respectively;

${\sigma}_{f}^{2}$ is family variance;

${\sigma}_{e}^{2}$ is residual error variance;

b is number of positions in the stem height where specimens were collected per tree for specimen-level analysis, while for tree-level analysis, it is the number of replicates; and

n is the number of positions in the radial direction where specimens were collected per log for specimen-level analysis, while for tree-level analysis, it is the number of trees per replicate. The within-family heritability estimate was one third due to the mixed mating that is expected in

Pinus species [

13], rather than one quarter as is appropriate for true half-sibs [

20]. Standard errors of heritabilities were estimated by the Delta method and PROC IML in SAS.

Genetic correlations (

r_{A}) between traits were estimated using the following formula [

20]:

where the numerator is the additive genetic covariance between traits

X and

Y;

${\sigma}_{ax}^{2}$ and

$\text{}{\sigma}_{ay}^{2}$ are the additive variance components for traits

X and

Y, respectively. Standard errors associated with genetic correlation were calculated using the following equation [

20]:

where

${r}_{A}$ is the genetic correlation between the traits;

${h}_{x}^{2}\text{}\mathrm{and}\text{}{h}_{y}^{2}$ are heritability for traits

X and

Y, respectively;

$SE\left({h}_{x}^{2}\right)\text{}\mathrm{and}\text{}SE\left({h}_{y}^{2}\right)$ are standard errors for heritability traits

X and

Y, respectively. Residual correlations (

${r}_{E}$) between traits were calculated using the same formula as for genetic correlation. However, residual variance components were used. Phenotypic correlations were calculated as Pearson product moment correlations (

${r}_{P}$) using the CORR procedure in SAS.

In most tree breeding programmes, stem wood quality (stiffness and strength) and volume are the most important breeding objective traits [

21]. Therefore, genetic gains and correlated response were calculated to improve these three traits (MoE, MoR and Volume) by direct selection and indirect selection using correlated traits. Genetic gains (

G_{x}) and correlated response (

CR_{x}), expressed in percentage, were calculated using the following formulas [

20]:

where

i is the selection intensity;

${h}_{x}^{2}\text{}$is the heritability of the objective trait;

${r}_{A}\text{}$is the genetic correlation between the traits;

${h}_{x}\text{}\mathrm{and}\text{}{h}_{y}$ are square roots of the heritability for the objective and selected traits, respectively;

$\text{}C{V}_{px}$ is the coefficient of phenotypic variance for the objective trait, expressed as:

where

${\sigma}_{px}$ is the phenotypic standard deviation for the objective trait;

$\mu \text{}$is the phenotypic mean value of the objective trait.