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Comparative analysis of SNP data and hybrid taxa information by using a classificatory linear mixed model to study the genetic variation and heritability of initial height growth in selected poplar hybrids

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Abstract

Advances in genomics increase the possibility of using SNP markers in selecting new parental genotypes for poplar breeding or new poplar varieties for commercial deployment. Here, we use a classificatory linear mixed model and a quantitative genetic approach for the combined analysis of SNP and phenotypic data from poplar hybrids at the beginning of their testing program. The main objective was to compare genetic parameters estimated based on two classification criteria: original (hybrid) taxa and SNP clustering. Height growth measurements obtained in three consecutive years from a poplar trial planted in 2002 in the center of Chile were included. In 2016, DNA was extracted from leaves of the same hybrids and genotyped by sequencing. An increasing number of clusters based on the similarity of SNP information was obtained. Broad sense heritability values observed at all levels of genomic clustering were larger than the only estimate obtained by using the original taxa classification. Thus, the method can help to predict a higher genetic gain in the early selection of poplars, based on initial height growth. The method did not affect the accuracy of the heritability estimation. The systematic increment in the intra-clonal covariance with the clustering level also suggests a highly positive genotype-by-time interaction effect at high levels of SNP clustering, which can also be positive for selection purposes. We concluded that the use of SNP clustering allowed the expression of larger genetic differences among hybrids in initial height growth, regardless of the original hybrid taxa.

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Acknowledgments

The field experiment complies with the current Chilean laws regarding safety and environmental issues.

Data archiving statement

The SNP data were submitted to the Dryad Digital Repository and can be accessed by using the following link: https://doi.org/10.5061/dryad.95x69p8f5

Funding

The research was funded by the Chilean National Science and Technology Commission (CONICYT), grant FONDEF ID14I10242.

Author information

Authors and Affiliations

Authors

Contributions

DNA isolation and processing was performed by F. Guerra. The sequencing procedure was conducted by D. Fuentes. The bioinformatics processing was conducted by A. Gonzalez. The genomic clustering was performed by M. Yañez. Data bases were handled by M. Yañez and F. Zamudio. The genetic and statistical analysis was performed by F. Zamudio. The manuscript was mainly written by F. Zamudio and authors added some specific paragraphs and commented on previous versions of the manuscript.

Corresponding author

Correspondence to Francisco Zamudio.

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Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by C. Kulheim

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Appendices

Appendix 1. The variance of the model

The alternative matrix representation of the model (1) using the mixed model representation is

$$ y={1}_N\mu +{\mathbf{X}}_T{\boldsymbol{\upbeta}}_T+{\mathbf{Z}}_s{\boldsymbol{\upupsilon}}_S+{\mathbf{Z}}_w{\boldsymbol{\upupsilon}}_w+{\mathbf{Z}}_{TS}{\boldsymbol{\upupsilon}}_{TS}+\mathbf{e}, $$
(A1.1)

where μ represents a general means, y is a N × 1 vector of phenotypic data, βT is a t × 1 vector of unknown fixed time effect (in our study, t = 3), XT is a known N × t design matrix, υS, υw, andυTS are the unknown vectors of groups (nothospecies or genomic clusters), hybrids (within groups), and interaction random effects, with dimensions p × 1, m × 1, andtp × 1, respectively, and ZS, Zw, andZTS are the known design matrices, with dimensions N × p, N × m, andN × tp, respectively, and e is a vector of residuals or intra-hybrid effects. Here, we are interested in measuring differences among a set of nothospecies or set of SNP clusters and extending this measure of variation in a larger population of nothospecies or clusters; therefore, nothospecies and genomic cluster effects are considered to be random effects. In this appendix, we will use the word nothospecies or cross according to the context; however, the genetic meaning will be the same, a group of individuals of similar genetic origin. Following Searle et al. (1992), a generalization of (A1.1) is y = Xβ +  + e with  = 1μ+ XTβT and Zυ = ZSυS + Zwυw + ZTSυTS.

Considering that the number of hybrids within each genetic group (nothospecies or cluster) was unbalanced, expression (A1.1) can be more precisely represented in terms of elements (see Searle et al. 1992, page 445). This notation is based on the direct product of two matrices, which uses the symbol ⊗, and sometimes it is also referred to as the Kronecker product. This means

$$ {\displaystyle \begin{array}{c}\mathbf{y}={\left\{{{}_c\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\mu +{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\boldsymbol{v}}_S+{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\boldsymbol{v}}_w+{\left\{{{}_c\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\boldsymbol{\upbeta}}_T+{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\boldsymbol{v}}_{TS}+\mathbf{e},\\ {}\mathbf{y}={\mathbf{1}}_N\mu +{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\boldsymbol{v}}_S+{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\boldsymbol{v}}_w+{\left\{{{}_c\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\boldsymbol{\upbeta}}_T+{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\boldsymbol{v}}_{TS}+\mathbf{e},\end{array}} $$
(A1.2)

and simplifying, we obtain

$$ \mathbf{y}={\mathbf{1}}_N\mu +{\left\{{{}_d\mathbf{1}}_{n_i}\right\}}_i^p{\boldsymbol{v}}_S+{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\boldsymbol{v}}_w+{\left\{{{}_c\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\boldsymbol{\upbeta}}_T+{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\boldsymbol{v}}_{TS}+\mathbf{e}, $$
(A1.3)

where ai is the number of hybrids per group (nothospecies or SNP cluster), ni is the total number of observations for the ith genetic group. Here, ni = ai ⋅ t, and by extension, \( N={\sum}_{i=1}^p{n}_i \) is the total number of observations in the data. Also, \( {\mathbf{1}}_{a_i} \) and \( {\mathbf{1}}_{n_i} \) are vectors of ones with dimensions ai × 1 and ni × 1, respectively, and \( {\mathbf{I}}_{a_i} \) and It are identity matrices of dimensions ai × 1 and t × 1, respectively.

The linear mixed model theory requires a description of the variance of the model for the maximum likelihood principle to be applied. The correct formulation of variance V in the model (1) is

$$ {\displaystyle \begin{array}{c} Var\left(\mathbf{y}\right)= Var\left(\mathbf{X}\boldsymbol{\upbeta } +{\mathbf{Z}}_S{\boldsymbol{\upupsilon}}_S+{\mathbf{Z}}_w{\boldsymbol{\upupsilon}}_w+{\mathbf{Z}}_{TS}{\boldsymbol{\upupsilon}}_{TS}+\mathbf{e}\right)\\ {}= Var\left(\mathbf{X}\boldsymbol{\upbeta } +\mathbf{Z}\boldsymbol{\upupsilon } +\mathbf{e}\right)=\mathbf{Z} Var\left(\boldsymbol{\upupsilon} \right){\mathbf{Z}}^{\prime }+ Var\left(\mathbf{e}\right)\end{array}} $$
(A1.4)

We also assume E(y) =  and Var() = 0, and no covariance between the vector of residual and random effects. The variance-covariance matrix of the vector of random effects is usually called as G and has the following form in our study

$$ \mathbf{G}= Var\left(\boldsymbol{\upupsilon} \right)=\left[\begin{array}{ccc} Var\left({\boldsymbol{\upupsilon}}_S\right)& \mathbf{0}& \mathbf{0}\\ {}\mathbf{0}& Var\left({\boldsymbol{\upupsilon}}_w\right)& \mathbf{0}\\ {}\mathbf{0}& \mathbf{0}& Var\left({\boldsymbol{\upupsilon}}_{TS}\right)\end{array}\right] $$
(A1.5)

The vector of random group effects (either nothospecies or genomic clusters, depending on the context) is represented by \( {\boldsymbol{v}}_S^{\prime }=\left[{S}_1\kern0.5em {S}_2\kern0.5em \cdots \kern0.5em {S}_g\right]={\left\{{{}_cS}_i\right\}}_{i=1}^p \). If we assume that E(υS) = 0, the variance-covariance matrix of group effects is

$$ Var\left({\boldsymbol{v}}_S\right)=E\left({\boldsymbol{v}}_S{\boldsymbol{v}}_S^{\prime}\right)=E\left({\left\{{{}_cS}_i\right\}}_{i=1}^p\cdot {\left\{{{}_rS}_i\right\}}_{i=1}^p\right)={{\left\{{}_mE\left({S}_i\cdot {S}_{i^{\prime }}\right)\right\}}_{i=1}^p}_{i^{\prime }=1}^p={\mathbf{I}}_p{\sigma}_S^2, $$
(A1.6)

where Ip is an identity matrix of dimensions p × p and \( E\left({S}_i^2\right)={\sigma}_S^2 \) is the variance among groups. We also assume \( E\left({S}_i\cdot {S}_{i^{\prime }}\right)=0, \) or any pair of sets is not statistically correlated. More details of the principles of matrix algebra used here are found in Johnson and Wichern (1992).

The vector of random hybrid effects can be expressed as \( {\boldsymbol{v}}_w^{\prime }=\left[{\mathbf{w}}_1^{\prime}\kern0.5em {\mathbf{w}}_2^{\prime}\kern0.5em \cdots \kern0.5em {\mathbf{w}}_S^{\prime}\right]={\left\{{{}_c\mathbf{w}}_i\right\}}_{i=1}^p \), where wi is the sub-vector of hybrids within the ith group. In our case, each vector wi has an unequal number of hybrids, which depicts the unbalanced nature of our data. Therefore, wi can also be expressed as \( {\boldsymbol{w}}_i^{\prime }=\left[{w}_{i1}\kern0.5em {w}_{i2}\kern0.5em \cdots \kern0.5em {w}_{ia_i}\right]={\left\{{{}_cw}_{ij}\right\}}_{j=1}^{a_i} \) where ai is the number of hybrids “nested” within the ith group.

We expect that \( E\left({w}_{ij}^2\right)={\sigma}_w^2 \) be the variance of hybrid effects within groups, the result of combining hybrids from several types of genetic groups (nothospecies or genomic clusters) in the same trial. We also assume that any pair of hybrids from the same or different nothospecies is not statistically correlated, or \( E\left({w}_{ij},{w}_{i{j}^{\prime }}\right)=E\left({w}_{ij},{w}_{i^{\prime }{j}^{\prime }}\right)=0 \). If we also assume E(wi) = 0, the variance of the sub-vector of hybrid effects is

$$ {\displaystyle \begin{array}{c} Var\left({\mathbf{w}}_i\right)=E\left({\mathbf{w}}_i-E\left({\mathbf{w}}_i\right)\right){\left({\mathbf{w}}_i-E\left({\mathbf{w}}_i\right)\right)}^{\prime }=E\left({\mathbf{w}}_i{\mathbf{w}}_i^{\prime}\right)=E\left({\left\{{{}_cw}_{ij}\right\}}_{j=1}^{a_i}\cdot {\left\{{{}_rw}_{ij}\right\}}_{j=1}^{a_i}\right)\\ {}={{\left\{{}_mE\left({w}_{ij}\cdot {w}_{i{j}^{\prime }}\right)\right\}}_{j=1}^{a_i}}_{j^{\prime }=1}^{a_i}={\mathbf{I}}_{a_i}{\sigma}_w^2,\end{array}} $$
(A1.7)

where each identity matrix has a dimension of ai, the number of hybrids in the ith cross. With this information, and also assuming E(υw) = 0, we derive the variance-covariance matrix of the vector of (combined) unknown hybrid random effects as

\( Var\left({\boldsymbol{v}}_w\right)=E\left({\boldsymbol{v}}_w-E\left({\boldsymbol{v}}_w\right)\right){\left({\boldsymbol{v}}_w-\mathrm{E}\left({\boldsymbol{v}}_w\right)\right)}^{\prime }=E\left({\boldsymbol{v}}_w{\boldsymbol{v}}_w^{\prime}\right)=E\left({\left\{{{}_c\mathbf{w}}_i\right\}}_{i=1}^p\cdot {\left\{{{}_r{\mathbf{w}}^{\prime}}_i\right\}}_{i=1}^p\right)={{\left\{{}_mE\left({\mathbf{w}}_i\cdot {\mathbf{w}}_{i^{\prime}}^{\prime}\right)\right\}}_{i=1}^p}_{i^{\prime }=1}^p, \)or

$$ Var\left({\boldsymbol{v}}_w\right)=\left[\begin{array}{cccc}E\left({\mathbf{w}}_1{\mathbf{w}}_1^{\prime}\right)& E\left({\mathbf{w}}_1{\mathbf{w}}_2^{\prime}\right)& \cdots & E\left({\mathbf{w}}_1{\mathbf{w}}_p^{\prime}\right)\\ {}E\left({\mathbf{w}}_2{\mathbf{w}}_1^{\prime}\right)& E\left({\mathbf{w}}_2{\mathbf{w}}_2^{\prime}\right)& \cdots & E\left({\mathbf{w}}_1{\mathbf{w}}_p^{\prime}\right)\\ {}\vdots & \vdots & \ddots & \vdots \\ {}E\left({\mathbf{w}}_p{\mathbf{w}}_1^{\prime}\right)& E\left({\mathbf{w}}_p{\mathbf{w}}_2^{\prime}\right)& \cdot & E\left({\mathbf{w}}_p{\mathbf{w}}_p^{\prime}\right)\end{array}\right]=\left[\begin{array}{cccc}{\sigma}_w^2{\mathbf{I}}_{a_1}& \cdot & \cdot & \cdot \\ {}\cdot & {\sigma}_w^2{\mathbf{I}}_{a_2}& \cdot & \cdot \\ {}\cdot & \cdot & \ddots & \cdot \\ {}\cdot & \cdot & \cdot & {\sigma}_w^2{\mathbf{I}}_{a_p}\end{array}\right]={\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_{i=1}^p{\sigma}_w^2. $$
(A1.8)

Each dot in (A1.8) represents a null matrix.

The combination of repeated measures from several hybrids and various nothospecies in a single-factor analysis allows the estimation of one common variance of hybrid effects that account for the whole variation among all hybrids, regardless of the hybrid taxa. This unique hybrid variance \( \left({\sigma}_w^2\right) \) will be the expression of the different genetic backgrounds of hybrids included in the single-factor experimental trial.

If we assume that each genetic group is measured at each corresponding time, then the vector of random interaction effects is \( {\boldsymbol{v}}_{TS}^{\prime }=\left[{TS}_{11}\kern0.5em {TS}_{12}\kern0.5em \cdots \kern0.5em {TS}_{1t}\kern0.5em \cdots \kern0.5em {TS}_{p1}\kern0.5em {TS}_{t2}\kern0.5em \cdots \kern0.5em {TS}_{pt}\right]={\left\{{{}_c\left\{{}_cT{S}_{ik}\Big\}\right\}}_{k=1}^t\right\}}_{i=1}^p \). If we also assume that E(υTS) = 0, the variance-covariance matrix of the vector of interaction effects is

$$ {\displaystyle \begin{array}{c} Var\left({\boldsymbol{v}}_{TS}\right)=\mathrm{E}\left({\boldsymbol{v}}_{TS}-E\left({\boldsymbol{v}}_{TS}\right)\right){\left({\boldsymbol{v}}_{TS}-E\left({\boldsymbol{v}}_{TS}\right)\right)}^{\prime }=E\left({\boldsymbol{v}}_{TS}{\boldsymbol{v}}_{TS}^{\prime}\right)=E\left({\left\{{{}_c\left\{{}_cT{S}_{ik}\Big\}\right\}}_{k=1}^t\right\}}_{i=1}^p\cdot {\left\{{{}_r\left\{{}_rT{S}_{ik}\Big\}\right\}}_{k=1}^t\right\}}_{i=1}^p\right)\\ {}={\left\{{}_mE\left({TS}_{ik}\cdot {TS}_{i^{\prime }{\mathrm{jk}}^{\prime }}\right)\right\}}_{i=1}^p{{{}_{i^{\prime }=1}^p}_{k=1}^t}_{k^{\prime }=1}^t=\left({\mathbf{I}}_p\otimes {\mathbf{I}}_t\right){\sigma}_{TS}^2\end{array}} $$
(A1.9)

As an example, we have a trial encompassing two genetic sets (crosses or genomic clusters) and genotypes within sets measured four times, then the vector of interaction effects is \( {\boldsymbol{\upupsilon}}_{TS}^{\prime }=\left[{TS}_{11}{TS}_{12}{TS}_{13}{TS}_{14}{TS}_{21}{TS}_{22}{TS}_{23}{TS}_{24}\right] \), and its variance is \( Var\left({\boldsymbol{\upupsilon}}_{TS}\right)=\left({\mathbf{I}}_2\otimes {\mathbf{I}}_4\right){\sigma}_{TS}^2 \). We can use the Kronecker product in (A1.9) because each set is measured in every time, or all combinations set-by-time are present in the data. However, if for example set 1 is not measured at time 4, then the combination TS14 is not present in the data, and we have \( {\boldsymbol{\upupsilon}}_{TS}^{\prime }=\left[{TS}_{11}{TS}_{12}{TS}_{13}{TS}_{21}{TS}_{22}{TS}_{23}{TS}_{24}\right] \). In this case, the variance of the vector is \( Var\left({\boldsymbol{v}}_{TS}\right)={\left\{{{}_d\mathbf{I}}_{t_i}\right\}}_i^p{\sigma}_{TS}^2 \), t1 = 3 and t2 = 4.

The vector of residual effects when we combine data from different sets is expressed as

$$ {\mathbf{e}}^{\prime }=\left[{\mathbf{e}}_{11}^{\prime}\kern0.5em {\mathbf{e}}_{12}^{\prime}\kern0.5em \cdots \kern0.5em {\mathbf{e}}_{1{a}_1}^{\prime}\kern0.5em {\mathbf{e}}_{21}^{\prime}\kern0.5em {\mathbf{e}}_{22}^{\prime}\kern0.5em \cdots \kern0.5em {\mathbf{e}}_{2{a}_2}^{\prime}\kern0.5em \cdots \right]={\left\{{{}_c\left\{{{}_c\mathbf{e}}_{ij}\right\}}_{j=1}^{a_i}\right\}}_{i=1}^p, $$

where each eij is at the same time the vector of residual effects for the jth clone nested within the ith genetic set, or \( {\mathbf{e}}_{ij}^{\prime }=\left[{e}_{ij1}\kern0.5em {e}_{ij2}\kern0.5em \cdots \kern0.5em {e}_{ij t}\right]={\left\{{{}_ce}_{ij k}\right\}}_{k=1}^t \). Here, k = 1,...,t, which is the number of times trees have been measured at the moment of analysis. In our research, t = 3; therefore, \( {\mathbf{e}}_{ij}^{\prime }=\left[{e}_{ej1}\kern0.5em {e}_{ej2}\kern0.5em {e}_{ej3}\right]={\left\{{{}_ce}_{ij k}\right\}}_{k=1}^3 \).

The variance-covariance matrix of combined intra-hybrid (residual) effects is usually called the R matrix. By definition, R = Var(e) = E(e − E(e))(e − E(e)). Assuming that E(eij) = 0, for every residual vector, then \( E\left({\mathbf{e}}^{\prime}\right)=\left({\mathbf{e}}_{11}^{\prime }{\mathbf{e}}_{12}^{\prime}\cdots {\mathbf{e}}_{1{a}_1}^{\prime }{\mathbf{e}}_{21}^{\prime}\cdots {\mathbf{e}}_{pa_p}^{\prime}\right)=\mathbf{0} \). Therefore, the variance-covariance can be expressed as

\( Var\left(\mathbf{e}\right)=E\left(\mathbf{e}{\mathbf{e}}^{\prime}\right)=E\left({\left\{{{}_c\left\{{{}_c\mathbf{e}}_{ij}\right\}}_{j=1}^{a_i}\right\}}_{i=1}^p\cdot {\left\{{{}_r\left\{{{}_r{\mathbf{e}}^{\prime}}_{ij}\right\}}_{j=1}^{a_i}\right\}}_{i=1}^p\right)={\left\{{}_mE\left({{}_c\mathbf{e}}_{ij}{\mathbf{e}}_{i^{\prime }{j}^{\prime}}^{\prime}\right)\right\}}_{i=1}^p{{{}_{i^{\prime }=1}^p}_{j=1}^{a_i}}_{j^{\prime }=1}^{a_i} \)

$$ Var(e)=\left[\begin{array}{ccccccccc}E\left({e}_{11}{e}_{11}^{\prime}\right)& E\left({e}_{11}{e}_{12}^{\prime}\right)& \cdots & E\left({e}_{11}{e}_{1{a}_1}^{\prime}\right)& E\left({e}_{11}{e}_{21}^{\prime}\right)& E\left({e}_{11}{e}_{22}^{\prime}\right)& \cdots & E\left({e}_{11}{e}_{1a2}^{\prime}\right)& \cdots \\ {}E\left({e}_{12}{e}_{11}^{\prime}\right)& E\left({e}_{12}{e}_{12}^{\prime}\right)& \cdots & E\left({e}_{12}{e}_{1{a}_1}^{\prime}\right)& E\left({e}_{12}{e}_{21}^{\prime}\right)& E\left({e}_{12}{e}_{22}^{\prime}\right)& \cdots & E\left({e}_{12}{e}_{1{a}_2}^{\prime}\right)& \cdots \\ {}\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & & \vdots & \\ {}E\left({e}_{1{a}_1}{e}_{11}^{\prime}\right)& E\left({e}_{1{a}_1}{e}_{12}^{\prime}\right)& \cdots & E\left({e}_{1{a}_1}{e}_{1{a}_1}^{\prime}\right)& E\left({e}_{1{a}_1}{e}_{21}^{\prime}\right)& E\left({e}_{1{a}_1}{e}_{22}^{\prime}\right)& \cdots & E\left({e}_{1{a}_1}{e}_{1{a}_2}^{\prime}\right)& \cdots \\ {}E\left({e}_{21}{e}_{11}^{\prime}\right)& E\left({e}_{21}{e}_{12}^{\prime}\right)& \cdots & E\left({e}_{21}{e}_{1{a}_1}^{\prime}\right)& E\left({e}_{21}{e}_{21}^{\prime}\right)& E\left({e}_{21}{e}_{22}^{\prime}\right)& \cdots & E\left({e}_{21}{e}_{1{a}_2}^{\prime}\right)& \cdots \\ {}E\left({e}_{22}{e}_{11}^{\prime}\right)& E\left({e}_{22}{e}_{12}^{\prime}\right)& \cdots & E\left({e}_{22}{e}_{1{a}_1}^{\prime}\right)& E\left({e}_{22}{e}_{21}^{\prime}\right)& E\left({e}_{22}{e}_{22}^{\prime}\right)& \cdots & E\left({e}_{22}{e}_{1{a}_2}^{\prime}\right)& \cdots \\ {}\vdots & \vdots & & \vdots & \vdots & \vdots & \ddots & \vdots & \\ {}E\left({e}_{1{a}_2}{e}_{11}^{\prime}\right)& E\left({e}_{1{a}_2}{e}_{12}^{\prime}\right)& \cdots & E\left({e}_{1{a}_2}{e}_{1{a}_1}^{\prime}\right)& E\left({e}_{1{a}_2}{e}_{21}^{\prime}\right)& E\left({e}_{1{a}_2}{e}_{22}^{\prime}\right)& \cdots & E\left({e}_{1{a}_2}{e}_{1{a}_2}^{\prime}\right)& \cdots \\ {}\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \ddots \end{array}\right] $$
(A1.10a)

The elements on the diagonal of (A1.10a) are Var(eij) the variance-covariance matrices of each vector eij.. Here, \( Var\left({\mathbf{e}}_{ij}\right)=E\left({\mathbf{e}}_{ij}-E\left({\mathbf{e}}_{ij}\right)\right){\left({\mathbf{e}}_{ij}-E\left({\mathbf{e}}_{ij}\right)\right)}^{\prime }=E\left({\mathbf{e}}_{ij}{\mathbf{e}}_{ij}^{\prime}\right) \), with E(eij) = 0. If we consider t = 3, we have

$$ {\displaystyle \begin{array}{c}E\left({\mathbf{e}}_{ij}{\mathbf{e}}_{ij}^{\prime}\right)=E\left({\left\{{{}_ce}_{ij k}\right\}}_{k=1}^3\cdot {\left\{{{}_re}_{ij k}\right\}}_{k=1}^3\right)={{\left\{{}_mE\left({e}_{ij k}{e}_{ij{k}^{\prime }}\right)\right\}}_{k=1}^3}_{k^{\prime }=1}^3\\ {}=\left[\begin{array}{ccc}E\left({e}_{ij1}^2\right)& E\left({e}_{ij1}{e}_{ij2}\right)& E\left({e}_{ij1}{e}_{ij3}\right)\\ {}E\left({e}_{ij2}{e}_{ij1}\right)& E\left({e}_{ij2}^2\right)& E\left({e}_{ij2}{e}_{ij3}\right)\\ {}E\left({e}_{ij3}{e}_{ij1}\right)& E\left({e}_{ij3}{e}_{ij2}\right)& E\left({e}_{ij3}^2\right)\end{array}\right],\end{array}} $$
(A1.10b)

and

$$ E\left({\mathbf{e}}_{ij}\mathbf{e}{\prime}_{ij}\right)={\boldsymbol{\Sigma}}_e=\left[\begin{array}{ccc} Var\left({e}_{ij1}\right)& Cov\left({e}_{ij1},{e}_{ij2}\right)& Cov\left({e}_{ij1},{e}_{ij3}\right)\\ {} Cov\left({e}_{ij2},{e}_{ij1}\right)& Var\left({e}_{ij2}\right)& Cov\left({e}_{ij2},{e}_{ij3}\right)\\ {} Cov\left({e}_{ij3},{e}_{ij1}\right)& Cov\left({e}_{ij3},{e}_{ij2}\right)& Var\left({e}_{ij3}\right)\end{array}\right] $$
(A1.10c)

The element in the diagonal is \( E\left({e}_{ijk}^2\right)= Var\left({e}_{jk}\right) \), the variance of the residual effects at the tth moment of phenotypic evaluation (here, t = 1, 2, and3). The element out of the diagonal is \( E\left({e}_{ij k}{e}_{ij{k}^{\prime }}\right)= Cov\left({e}_{ij k},{e}_{ij{k}^{\prime }}\right) \), the covariance between residual effects of the ijth hybrid measured in two instances, times k and k′th. Therefore, phenotypic measurements of residual effects from the same hybrid taken at two instances are correlated. Both, Var(eijk) and \( Cov\left({e}_{ij k},{e}_{ij{k}^{\prime }}\right) \) are representations of variance and covariance linked to a generic jth hybrid from the ith genetic group (SNPs set or nothospecies), and they must be estimated by using the whole set of hybrids. The possible options for the structure of the Σe matrix are discussed by Verbeke and Molenberghs (2000) and Littell et al. (2006). Below, we briefly describe the structures tested here. The dimension of each matrix Σe is 3 × 3, the number of instances our hybrid poplars were measured.

The elements out of the diagonal in (A1.10a) are the covariance matrices between vectors of residual effects from two different hybrids of the same nothospecies, eij and \( {\mathbf{e}}_{i{j}^{\prime }} \), or two hybrids from different nothospecies, eij and \( {\mathbf{e}}_{i^{\prime }{j}^{\prime }} \). Here.

\( Cov\left({\mathbf{e}}_{ij},{\mathbf{e}}_{i{j}^{\prime }}\right)=E\left({\mathbf{e}}_{ij}-E\left({\mathbf{e}}_{ij}\right)\right){\left({\mathbf{e}}_{ij}-E\left({\mathbf{e}}_{i{j}^{\prime }}\right)\right)}^{\prime }=E\left({\mathbf{e}}_{ij}{\mathbf{e}}_{i{j}^{\prime}}^{\prime}\right)=\mathbf{0} \), with \( E\left({\mathbf{e}}_{ij}\right)=E\left({\mathbf{e}}_{i{j}^{\prime }}\right)=\mathbf{0} \). This means that residuals from two ramets from two different hybrids of the same genetic set, regardless of the measurement time, are not statistically correlated, or

\( E\left({\mathrm{e}}_{ij k}{\mathrm{e}}_{ij{k}^{\prime }}\right)=E\left({\mathrm{e}}_{ij k}{\mathrm{e}}_{i{j}^{\prime }k}\right)=E\left({\mathrm{e}}_{ij k}{\mathrm{e}}_{i{j}^{\prime }{k}^{\prime }}\right)=0. \) Therefore, the matrix in (A1.10a) is newly

$$ \mathbf{R}= Var\left(\mathbf{e}\right)\left[\begin{array}{ccccccc}{\boldsymbol{\Sigma}}_e& .& .& .& .& .& .\\ {}.& {\boldsymbol{\Sigma}}_e& .& .& .& .& .\\ {}.& .& \ddots & .& .& .& .\\ {}.& .& .& {\boldsymbol{\Sigma}}_e& .& .& .\\ {}.& .& .& .& {\boldsymbol{\Sigma}}_e& .& .\\ {}.& .& .& .& .& \ddots & .\\ {}.& .& .& .& .& .& {\boldsymbol{\Sigma}}_e\end{array}\right]={\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\boldsymbol{\Sigma}}_e. $$
(A1.10d)

Each dot in (A1.10d) represents a null matrix. Similar to the case of the variance of hybrid effects, by combining hybrids from several nothospecies, the elements in Σe are estimated as a unique set of variances and covariances that are common to the whole set of hybrids and nothospecies. However, an alternative hypothesis is that each nothospecies has its matrix Σe, i.e., its own set of variances and covariances. This hypothesis will be addressed to our phenotypic data in another paper elsewhere. Now, the variance-covariance matrix of the whole vector of data in (A1.1) can be written as

$$ Var\left(\mathbf{y}\right)={\mathbf{Z}}_S Var\left({\boldsymbol{\upupsilon}}_S\right){\mathbf{Z}}_S^{\prime }+{\mathbf{Z}}_w Var\left({\boldsymbol{\upupsilon}}_w\right){\mathbf{Z}}_w^{\prime }+{\mathbf{Z}}_{TS} Var\left({\boldsymbol{\upupsilon}}_{TS}\right){\mathbf{Z}}_{TS}^{\prime }+ Var\left(\mathbf{e}\right) $$
(A1.11)

Each component in (A1.11) can be further expressed as

$$ {\displaystyle \begin{array}{c}{\mathbf{Z}}_S Var\left({\boldsymbol{v}}_S\right){\mathbf{Z}}_S^{\prime }=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\right]\cdot {\mathbf{I}}_S{\sigma}_s^2\cdot {\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\right]}^{\prime }=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\right]\cdot {\mathbf{I}}_S{\sigma}_S^2\cdot \left[{\left\{{{}_d{\mathbf{1}}^{\prime}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t^{\prime}\right]\\ {}=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\sigma}_S^2\right]\cdot \left[{\left\{{{}_d{\mathbf{1}}^{\prime}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t^{\prime}\right]=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\cdot {\left\{{{}_d{\mathbf{1}}^{\prime}}_{a_i}\right\}}_i^p\right]\otimes {\mathbf{1}}_t{\mathbf{1}}_t^{\prime }{\sigma}_S^2\\ {}={\left\{{{}_d\mathbf{1}}_{a_i}{\mathbf{1}}_{a_i}^{\prime}\right\}}_i^p\otimes {\mathbf{J}}_t{\sigma}_S^2={\left\{{{}_d\mathbf{J}}_{a_i}\right\}}_i^p\otimes {\mathbf{J}}_t{\sigma}_S^2,\end{array}} $$
(A1.12a)
$$ {\displaystyle \begin{array}{c}{\mathbf{Z}}_w Var\left({\boldsymbol{v}}_w\right){\mathbf{Z}}_w^{\prime }=\left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\right]\cdot {\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p{\sigma}_w^2\cdot {\left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\right]}^{\prime}\\ {}=\left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t\right]\cdot {\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p{\sigma}_w^2\cdot \left[{\left\{{{}_d{\mathbf{I}}^{\prime}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t^{\prime}\right]=\left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t{\sigma}_w^2\right]\cdot \left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{1}}_t^{\prime}\right]\\ {}=\left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\cdot {\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\right]\otimes {\mathbf{1}}_t{\mathbf{1}}_t^{\prime }{\sigma}_w^2={\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{J}}_t{\sigma}_w^2,\end{array}} $$
(A1.12b)
$$ {\displaystyle \begin{array}{c}{\mathbf{Z}}_{TS} Var\left({\boldsymbol{v}}_{TS}\right){\mathbf{Z}}_{TS}^{\prime }=\left[{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t\right]\cdot \left({\mathbf{I}}_S\otimes {\mathbf{I}}_t\right){\sigma}_{TS}^2\cdot {\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t\right]}^{\prime}\\ {}=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t\right]\cdot \left({\mathbf{I}}_S\otimes {\mathbf{I}}_t\right){\sigma}_{TS}^2\cdot \left[{\left\{{{}_d{\mathbf{1}}^{\prime}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t\right]\\ {}\begin{array}{c}=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\sigma}_{TS}^2\right]\cdot \left[{\left\{{{}_d{\mathbf{1}}^{\prime}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t\right]=\left[{\left\{{{}_d\mathbf{1}}_{a_i}\right\}}_i^p\cdot {\left\{{{}_d{\mathbf{1}}^{\prime}}_{a_i}\right\}}_i^p\right]\otimes {\mathbf{I}}_t{\sigma}_{TS}^2\\ {}={\left\{{{}_d\mathbf{1}}_{a_i}{\mathbf{1}}_{a_i}^{\prime}\right\}}_i^p\otimes {\mathbf{I}}_t{\sigma}_{TS}^2={\left\{{{}_d\mathbf{J}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\sigma}_{TS}^2.\end{array}\end{array}} $$
(A1.12c)

Therefore, the correct formulation of variance in (A1.11) is

$$ Var\left(\mathbf{y}\right)={\left\{{{}_d\mathbf{J}}_{a_i}\right\}}_i^p\otimes {\mathbf{J}}_t{\sigma}_S^2+{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\mathbf{J}}_t{\sigma}_w^2+{\left\{{{}_d\mathbf{J}}_{a_i}\right\}}_i^p\otimes {\mathbf{I}}_t{\sigma}_{TS}^2+{\left\{{{}_d\mathbf{I}}_{a_i}\right\}}_i^p\otimes {\boldsymbol{\Sigma}}_e. $$
(A1.13)

Combining phenotypic data from several genetic sets (nothospecies or SNPs clusters) and measured in several instances requires the estimation of three variances and a matrix of variance-covariance of statistically correlated residuals.

The structures of Σe that were tested here are represented in Table 1. The tested structures were as follows:

  • CS structure: Variances are homogeneous; therefore, residual (intra-hybrid) effects are homoscedastic respect to time, and they are invariant over time. There is only one covariance, but it is assumed to be constant, regardless of how far apart the measures are over time. The simplest covariance structure that includes within-subject correlated errors is compound symmetry (CS). Here we see correlated errors between time points within-subjects and note that these correlations are presumed to be the same for each set of times, regardless of how distant in time the repeated measures are made.

  • UN structure: It specifies a completely general (unstructured) covariance matrix parameterized directly in terms of variances and covariances. The variances are constrained to be non-negative, and the covariances are unconstrained. This structure is not constrained to be non-negative definite in order to avoid non-linear constraints (SAS Institute Inc. 2011). Therefore, within-hybrid or residual effects are variable over time (heterogeneity of variance), and covariances between residuals for each pair of time points are unique, which makes the estimation of the matrix to be more complex.

  • HF structure: Specifies the Huynh-Feldt covariance structure. This structure has the same number of parameters and heterogeneity along the main diagonal than the UN structure. However, it constructs the off-diagonal elements by taking the arithmetic means of diagonal variances.

Table 1 Variance-covariance structures for Σe (adapted from Wolfinger 1996) tested in the research. For simplification purposes, we assume there are only three measurement times, therefore t = 3. The first two structures can be considered as homogeneous, whereas the last one is a heterogeneous structure

Appendix 2. Genetic parameters

Estimation of the genetic and environmental variance

The linear model in (1) let the estimation of what is called the observational components of variation. In our research, they are \( {\sigma}_w^2,{\sigma}_S^2,{\sigma}_{TS}^2,\boldsymbol{and}{\boldsymbol{\Sigma}}_e \). These components were estimated directly from the phenotypic values. However, the genetic parameter reported here also required the simultaneous estimation of the causal components of variation (Falconer and MacKay 1996), which are the components of the phenotypic variance attributable to genetic and non-genetic causes. These components cannot be directly estimated from the phenotypic values, but we must establish a statistical link between both types of components. This is described here for our data.

Following Falconer and MacKay (1996), and based on Kempthorne (1957, page 225), the scalar linear model that describes the causal components of variation in our data is

$$ {y}_{ij k}={\mu}_{Ec}+{g}_{ij}+{Es}_{ij k}, $$
(A2.1)

where yijk is the phenotypic value, gij is the genotypic value of the ijth hybrid within the ith group (either nothospecies or genomic cluster),μEc is a fixed parameter due to the general environmental effect that is common to all genotypes in the single-factor trial, and Esijk, this is the within-individual effect at time kth arising from temporary and localized circumstances (specific environmental effect) for the ijth hybrid (See Falconer 1989, page 140). Now, let us consider that at k = 1, Esij1 involved a 1-year-old root system; that at k = 2, Esij2 involved a 2-year-old root system; and at k = 3, Esij3 involved a 3-year-old root system. As the root system grows, the competition for soil (water and nutrients) begins. A 2-year-old root system is larger than at age one and new and more interactions with the specific micro-environment can be present, and that can be variable from hybrid to hybrid. A 3-year-old root system in poplars is more exposed to inter-hybrid competition. Therefore, we are assuming here that the specific and localized environmental effect of ijth hybrid is variable from time k to k′. However, we do not know the exact trend of changes from hybrid to hybrid. For some hybrids, Esij1 and Esij2 can be positively correlated. For others, the correlation can be negative. And, for a third group, it can be null. Therefore, we are assuming that \( Cov\left({Es}_{ij k}{Es}_{ij{k}^{\prime }}\right) \) tends to zero, to cancel out, and to be negligible.

Assuming E(gijk) = E(Esijk) = 0, E(yijk) = μEc, and there is no covariation among genetic and non-genetic effects, or \( E\left({g}_{ij}\cdot {Es}_{ij{k}^{\prime }}\right)=0 \), we can estimate the covariance between measurements from two different stems of the same hybrid measured at two different times, kth and k′th as

$$ {\displaystyle \begin{array}{c} Cov\left({y}_{ij k},{y}_{ij{k}^{\prime }}\right)=E\left({g}_{ij}+{Es}_{ij k}\right)\left({g}_{ij}+{Es}_{ij{k}^{\prime }}\right)\\ {}=E\left({g}_{ij}^2\right)+E\left({g}_{ij}{Es}_{ij{k}^{\prime }}\right)+E\left({g}_{ij}{Es}_{ij k}\right)+E\left({Es}_{ij k}{Es}_{ij{k}^{\prime }}\right)\\ {}= VG,\end{array}} $$
(A2.2)

where VG is the genetic variance.

Now, considering the scalar mixed model (1) and also assuming that E(yijk) = μ + Tk, then the same covariance can be estimated by

$$ {\displaystyle \begin{array}{l} Cov\left({y}_{ij k},{y}_{ij{k}^{\prime }}\right)=E\left(\Big({y}_{ij k}-E\left({y}_{ij k}\right)\right)\left(\Big({y}_{ij{k}^{\prime }}-E\left({y}_{ij{k}^{\prime }}\right)\right)=\\ {}\kern7.75em =E\left({S}_i+{w}_{ij}+T{S}_{ik}+{e}_{ij k}\right)\left({S}_i+{w}_{ij}+T{S}_{ik\prime }+{e}_{ij{k}^{\prime }}\right)\\ {}\begin{array}{c}\kern2.25em =E\left({S}_i^2+{w}_{ij}^2+{e}_{ij k}{e}_{ij{k}^{\prime }}+ crossproducts\right)\\ {}\begin{array}{l}\kern7.5em =E\left({S}_i^2\right)+E\left({w}_{ij}^2\right)+E\left({e}_{ij k}{e}_{ij{k}^{\prime }}\right)+E(crossproducts)\\ {}\kern7.5em ={\sigma}_S^2+{\sigma}_w^2+{\sigma}_{e{e}^{\prime }}\kern0.5em ,\end{array}\end{array}\end{array}} $$
(A2.3)

where \( {\sigma}_w^2 \) is the variance due to hybrid effects, which is estimated by combining data from either several nothospecies or genomic sets, and \( {\sigma}_{e{e}^{\prime }} \) is the covariance between intra-hybrid effects of the ijth hybrid measured at time k and k′th. Expression (A2.3) is also assuming that there is no statistical covariation among hybrid, interaction, and intra-hybrid effects. However, two residual (intra-hybrid) effects from the same hybrid are originated from measurements taken on two stems (1-year-old shoots) from the same ortet (genotype) and, henceforth, they are correlated. Now, \( {\sigma}_{e{e}^{\prime }} \) is estimated from Σe. However, the challenge is to determine the value of \( {\sigma}_{e{e}^{\prime }} \) if the likelihood ratio test (LRT) indicates that the most probable structure of the Σe matrix is the UN or HF. Combining (A2.2) and (A2.3), we obtain an estimate of the genetic variance

$$ VG={\sigma}_S^2+{\sigma}_w^2+{\sigma}_{e{e}^{\prime }} $$
(A2.4)

Assuming that genotypes and environments are associated in random, which means Cov(G, E) = 0, we can represent the phenotypic variance based on its causal components or VP = VG + VE + V(G × E) (Falconer and Mackay 1996), where VE is the variance of environmental effects and V(G × E) is the variance due to the genotype-by-environment interaction. We can also assume that VE = VEc + VEs, the sum of the environmental variance contributing to the permanent, non-localized circumstances (Falconer 1989), and that is common to a set of genotypes (VEc), plus the special environmental variance (VEs), which Falconer (1989, page 140) refers as the within-individual variance arising from temporary or localized circumstances. In the context of a one-single factor trial, VEc is not estimable because the permanent and common environmental effects are fixed and represented by the parameter μEc. We are in conditions to estimate VEc if the tested hybrids are cloned and ramets are planted in several blocks. Then, we can assess the variance that arises from permanent and common environmental factors that characterize each block. By extension, V(G × E) cannot be isolated and estimated in a single factor experimental trial. However, VEs can be estimated by equating the variance of model (A2.1) to its representation based on the observed variance component linked to the model (1). This is

$$ VG+ VEs= VP={\sigma}_{\boldsymbol{phen}}^2={\sigma}_S^2+{\sigma}_w^2+{\sigma}_{TS}^2+{\sigma}_e^2 $$
(A2.5)

If we replace (A2.4) in (A2.5), we obtain

$$ \left({\sigma}_S^2+{\sigma}_w^2+{\sigma}_{e{e}^{\prime }}\right)+ VEs={\sigma}_S^2+{\sigma}_w^2+{\sigma}_{TS}^2+{\sigma}_e^2, $$

which implies that the special environment variance can be estimated as

$$ VEs={\sigma}_{TS}^2+{\sigma}_e^2-{\sigma}_{e{e}^{\prime }} $$
(A2.6)

The genetic and phenotypic variances are combined to provide for an estimate of the broad-sense heritability:

$$ {H}^2=\frac{VG}{VP}=\frac{\sigma_S^2+{\sigma}_w^2+{\sigma}_{e{e}^{\prime }}}{\sigma_{\boldsymbol{phen}}^2}=\frac{\sigma_S^2+{\sigma}_w^2+{\sigma}_{e{e}^{\prime }}}{\sigma_S^2+{\sigma}_w^2+{\sigma}_{TS}^2+{\sigma}_e^2} $$
(A2.7)

The estimation of the genetic variance requires knowing the covariance between intra-hybrid effects. We can assess this covariance by modeling the variance-covariance matrix Σe. However, some decision criteria must be adopted if the selected structure of this matrix includes several estimates of \( {\sigma}_{e{e}^{\prime }} \)

Restrictions to the use of variance-covariance components in Σ e

If the LRT and the analysis of information criteria tell us that the CS structure is the most probable for Σe, then we obtain the unique needed variance from the diagonal of the estimated R matrix and use it as an estimate of \( {\sigma}_e^2 \). Conveniently, the CS structure gives us only one estimate of the off-diagonal covariance \( {\sigma}_{e{e}^{\prime }} \). However, if the LRT shows that the selected structure is either UN or HF, we have more than one option for both the variance and covariance involving residuals. Therefore, we must use an extra criterion for selecting only one variance and covariance to be used in expressions (A2.4) and (A2.7). In the analyses reported here, all selected structures were the UN and the extra criterion was using the simple average variance and covariance of residual effects (Zamudio et al. 2008). Considering that each genotype was measured three times in our research, thus

$$ {\hat{\sigma}}_{e{e}^{\prime }}={\overline{\sigma}}_{e{e}^{\prime }}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left({\sigma}_{e_1{e}_2}+{\sigma}_{e_1{e}_3}+{\sigma}_{e_2{e}_3}\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\Sigma}_{p<q}{\sigma}_{e_p{e}_q}; $$
(A2.8a)

and

$$ {\hat{\sigma}}_e^2={\overline{\sigma}}_e^2=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left({\sigma}_{e_1}^2+{\sigma}_{e_2}^2+{\sigma}_{e_3}^2\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\Sigma}_p{\sigma}_{e_p}^2, $$
(A2.8b)

where \( {\sigma}_{e_p}^2 \) is the variance of the residual (intra-hybrid) effects at the pth measurement time and \( {\sigma}_{e_p{e}_q} \) is the covariance of residual effects between times pth and qth.

Sampling variances of variances, covariances, and heritability estimates

The heritability estimate used in our research is the ratio of two functions of variances and covariance components. For measuring the precision of our estimates, we used the Delta method, based on Taylor’s series expansion. This method gives the following general formula for the variance of a ratio (X/Y):

$$ Var\left(x/y\right)\approx {\left[E(x)/E(y)\right]}^2\cdot \left[ Var(x)/E{(x)}^2+ Var\left(\mathrm{y}\right)/E{\left(\mathrm{y}\right)}^2-2 Cov\Big(x,y\Big)/\left(E(x)\cdot E(y)\right)\right] $$
(A2.9)

where E(x) and E(y) are the mathematical expectation (or expected value) of the random variables x and y. Applying expression (A2.9) to our heritability estimate, we obtained:

$$ Var\left({H}^2\right)={\left(\frac{VG}{VP}\right)}^2\cdot \left[\frac{Var(VG)}{VG^2}+\frac{Var(VP)}{VP^2}-\frac{2 Cov\left( VG, VP\right)}{VG\cdot VP}\right] $$
(A2.10a)

With a further simplification of (A2.10a), we have

$$ Var\left({H}^2\right)=\left(1/{VP}^3\right)\cdot \left( VP\cdot Var(VG)+{VG}^2\cdot Var(VP)-2 VG\cdot Cov\Big( VG, VP\Big)\right) $$
(A2.10b)

Taking the squared root of the variance in (A2.10b) gave us the standard error, which is an expression of the precision in the estimation of the parameter.

Considering the estimates of VG and VP given by (A2.4) and (A2.5), respectively, we derived and used the following expressions in our research:

$$ {\displaystyle \begin{array}{l} Var(VG)= Var\left({\overset{\sim }{\sigma}}_S^2+{\overset{\sim }{\sigma}}_w^2+{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)\\ {}\kern4.5em = Var\left({\overset{\sim }{\sigma}}_S^2\right)+ Var\left({\overset{\sim }{\sigma}}_w^2\right)+ Var\left({\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)+\\ {}\kern4.5em +2\left[ Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_w^2\right)+ Cov\Big({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\left)+ Cov\right({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\Big)\right],\end{array}} $$
(A2.11)
$$ {\displaystyle \begin{array}{l} Var(VP)= Var\left({\overset{\sim }{\sigma}}_S^2+{\overset{\sim }{\sigma}}_w^2+{\overset{\sim }{\sigma}}_{ST}^2+{\overset{\sim }{\sigma}}_e^2\right)\\ {}\kern4.5em = Var\left({\overset{\sim }{\sigma}}_S^2\right)+ Var\left({\overset{\sim }{\sigma}}_w^2\right)+ Var\left({\overset{\sim }{\sigma}}_{ST}^2\right)+ Var\left({\overset{\sim }{\sigma}}_e^2\right)+\\ {}\kern4.5em \begin{array}{l}+2\left[ Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_w^2\right)+ Cov\Big({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_{ST}^2\left)+ Cov\right({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_e^2\Big)+\right.\\ {}\left.+ Cov\left({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_{ST}^2\right)+ Cov\Big({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_e^2\left)+ Cov\right({\overset{\sim }{\sigma}}_{ST}^2,{\overset{\sim }{\sigma}}_e^2\Big)\right]\kern0.5em ,\end{array}\end{array}} $$
(A2.12)

and

$$ {\displaystyle \begin{array}{l}\mathrm{Cov}\left( VG, VP\right)=\mathrm{Cov}\left[\left({\overset{\sim }{\sigma}}_S^2+{\overset{\sim }{\sigma}}_w^2+{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right),\left({\overset{\sim }{\sigma}}_S^2+{\overset{\sim }{\sigma}}_w^2+{\overset{\sim }{\sigma}}_{ST}^2+{\overset{\sim }{\sigma}}_e^2\right)\right]\\ {}\kern6.5em = Var\left({\overset{\sim }{\sigma}}_S^2\right)+ Var\left({\overset{\sim }{\sigma}}_w^2\right)+2 Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_w^2\right)+ Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_{ST}^2\right)+\\ {}\kern4.5em \begin{array}{l}\kern2em + Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_e^2\right)+ Cov\left({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_{ST}^2\right)+ Cov\left({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_e^2\right)+ Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)+\\ {}\kern2em + Cov\left({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)+ Cov\left({\overset{\sim }{\sigma}}_{ST}^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)+ Cov\left({\overset{\sim }{\sigma}}_e^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)\kern0.5em ,\end{array}\end{array}} $$
(A2.13)

Also, the sampling variance of the variance estimator of environmental effects was derived as:

$$ {\displaystyle \begin{array}{c} Var(VEs)= Var\left({\overset{\sim }{\sigma}}_{TS}^2\right)+ Var\left({\overset{\sim }{\sigma}}_e^2\right)+ Var\left({\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right)+\\ {}+2\left[ Cov\left({\overset{\sim }{\sigma}}_{TS}^2,{\overset{\sim }{\sigma}}_e^2\right)- Cov\Big({\overset{\sim }{\sigma}}_{TS}^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\left)- Cov\right({\overset{\sim }{\sigma}}_{TS}^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\Big)\right]\end{array}} $$
(A2.14)

Here, we adopted the terminology used by Searle et al. (1992) to denote the maximum likelihood estimators of variance and covariance components by \( {\overset{\sim }{\sigma}}_x^2 \) and \( {\overset{\sim }{\sigma}}_{xy} \), respectively.

As described above, all selected structures for Σe were UN in our research. Therefore, we derived the following expressions for measuring the variance of the variance and covariance estimates involving residual intraclonal effects and estimated by (A2.15a) and (A2.8b):

$$ {\displaystyle \begin{array}{l} Var\left({\overline{\sigma}}_e^2\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\left[ Var\left({\overset{\sim }{\sigma}}_{e_1}^2\right)+ Var\left({\overset{\sim }{\sigma}}_{e_2}^2\right)+ Var\Big(\right.{\overset{\sim }{\sigma}}_{e_3}^2\Big)+\\ {}\kern4.25em +\left.2\left( Cov\left({\overset{\sim }{\sigma}}_{e_1}^2,{\overset{\sim }{\sigma}}_{e_2}^2\right)+ Cov\Big({\overset{\sim }{\sigma}}_{e_1}^2,{\overset{\sim }{\sigma}}_{e_3}^2\left)+ Cov\right({\overset{\sim }{\sigma}}_{e_2}^2,{\overset{\sim }{\sigma}}_{e_3}^2\Big)\right)\right]\end{array}} $$
(A2.15a)

and

$$ {\displaystyle \begin{array}{l} Var\left({\overline{\sigma}}_{\mathrm{ee}}\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\left[ Var\left({\overset{\sim }{\sigma}}_{e_1{e}_2}\right)+ Var\Big(\right.{\overset{\sim }{\sigma}}_{e_1{e}_3}\Big)+ Var\left({\overset{\sim }{\sigma}}_{e_2{e}_3}\right)+\\ {}\ \left.2\left( Cov\left({\overset{\sim }{\sigma}}_{e_1{e}_2},{\overset{\sim }{\sigma}}_{e_1{e}_3}\right)+ Cov\Big({\overset{\sim }{\sigma}}_{e_1{e}_2},{\overset{\sim }{\sigma}}_{e_2{e}_3}\left)+ Cov\right({\overset{\sim }{\sigma}}_{e_1{e}_3},{\overset{\sim }{\sigma}}_{e_2{e}_3}\Big)\right)\right]\end{array}} $$
(A2.15b)

Expressions (A2.11) to (A2.15b) involve the estimation of sampling variances and covariances of several variances and covariance components. They can be summarized and described as follows. \( Var\left({\overset{\sim }{\sigma}}_w^2\right) \) and \( Var\left({\overset{\sim }{\sigma}}_S^2\right) \) are the sampling variances of the variance estimates of random hybrid effects and genetic group (nothospecies or genomic clusters), respectively; \( Var\left({\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right) \) and \( Cov\left({\overset{\sim }{\sigma}}_S^2,{\overset{\sim }{\sigma}}_w^2\right) \) are the sampling variance of the covariance estimate between intra-hybrid residual effects from two different moments and the covariance between variance components from hybrid effects and genetic groups, respectively; \( Cov\left({\overset{\sim }{\sigma}}_w^2,{\overset{\sim }{\sigma}}_{e{e}^{\prime }}\right) \) is the sampling covariance between the variance component of the clonal effect and the covariance component between residual effects from two different moments. Other sampling variances and covariances can be equally deducted.

Now, expressions in (A2.15a) and (A2.15b) can also be summarized as follows. \( Var\left({\overset{\sim }{\sigma}}_{e_1}^2\right) \) is the sampling variance of the variance of residual effects at time 1; \( Var\left({\overset{\sim }{\sigma}}_{e_1{e}_2}\right) \) is the sampling variance of the covariance between residual effects from time 1 to 2; \( Cov\left({\overset{\sim }{\sigma}}_{e_1}^2,{\overset{\sim }{\sigma}}_{e_2}^2\right) \) is the sampling covariance between variances of residual effects from time 1 to 2; and \( Cov\left({\overset{\sim }{\sigma}}_{e_1{e}_2},{\overset{\sim }{\sigma}}_{e_1{e}_3}\right) \) is the sampling covariance between residual effects from time 1 to 2 and covariance between residual effects from time 1 to 3.

The vector of parameters involved in the likelihood function L(θ| y) of our data is\( {\boldsymbol{\uptheta}}^{\prime }=\left({\boldsymbol{\upupsilon}}_S^{\prime }{\boldsymbol{\upupsilon}}_w^{\prime }{\boldsymbol{\upupsilon}}_{ST}^{\prime }{\mathbf{e}}^{\prime}\right) \), where the different components of each vector are defined above. A description of the likelihood function is fundamental to using the maximum likelihood estimation method (MLE), and useful property of the ML estimator \( \overset{\sim }{\boldsymbol{\uptheta}} \) is that its large-sample, or asymptotic (as N → ∞), dispersion matrix is known. For I(θ), known as the information matrix, and defined as

$$ \mathbf{I}\left(\boldsymbol{\uptheta} \right)=E\left(\frac{\partial L}{\partial \boldsymbol{\uptheta}}\frac{\partial L}{\partial {\boldsymbol{\uptheta}}^{\prime }}\right)=E{\left\{{}_m\frac{\partial L}{\partial {\theta}_i}\frac{\partial L}{\partial {\theta}_j}\right\}}_{i,j}, $$
(A2.16)

the asymptotic dispersion matrix is \( Var\left(\overset{\sim }{\boldsymbol{\uptheta}}\right)\simeq {\left[\mathbf{I}\left(\boldsymbol{\uptheta} \right)\right]}^{-1} \) (Searle et al. 1992), the inverse of the information matrix. One of the attractive features of ML estimation is that the asymptotic dispersion matrix of the estimators is always available. The PROC MIXED, from SAS 9.4, displayed the asymptotic covariance matrix of covariance parameters needed for our research by using the ASICOV option as part of the PROC MIXED statement (SAS Institute Inc., 2011). All the sampling variances needed in expressions (A2.11) to (A2.15b) were obtained from the diagonal of the asymptotic dispersion matrix obtained from running PROC MIXED, and the required sampling covariances were obtained from the off-diagonal elements of the same estimated dispersion matrix.

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Zamudio, F., Yañez, M., Guerra, F. et al. Comparative analysis of SNP data and hybrid taxa information by using a classificatory linear mixed model to study the genetic variation and heritability of initial height growth in selected poplar hybrids. Tree Genetics & Genomes 16, 69 (2020). https://doi.org/10.1007/s11295-020-01435-1

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