The use of genetic correlations to evaluate associations between SNP markers and quantitative traits

Abstract

Open-pollinated progeny of Corymbia citriodora established in replicated field trials were assessed for stem diameter, wood density, and pulp yield prior to genotyping single nucleotide polymorphisms (SNP) and testing the significance of associations between markers and assessment traits. Multiple individuals within each family were genotyped and phenotyped, which facilitated a comparison of standard association testing methods and an alternative method developed to relate markers to additive genetic effects. Narrow-sense heritability estimates indicated there was significant additive genetic variance within this population for assessment traits (${\widehat{h}^{{2}}} = 0.{28}\;{\text{to}}\;0.{44}$) and genetic correlations between the three traits were negligible to moderate (r _G = 0.08 to 0.50). The significance of association tests (p values) were compared for four different analyses based on two different approaches: (1) two software packages were used to fit standard univariate mixed models that include SNP-fixed effects, (2) bivariate and multivariate mixed models including each SNP as an additional selection trait were used. Within either the univariate or multivariate approach, correlations between the tests of significance approached +1; however, correspondence between the two approaches was less strong, although between-approach correlations remained significantly positive. Similar SNP markers would be selected using multivariate analyses and standard marker-trait association methods, where the former facilitates integration into the existing genetic analysis systems of applied breeding programs and may be used with either single markers or indices of markers created with genomic selection processes.

Appendix A

Univariate test of association

The usual SMR approach taken for association testing partitions the genetic variance into two independent factors; one describes random polygenic effects and the other represents a fixed effect for the SNP marker of interest (Yu et al. 2006). This univariate mixed model has been presented as:

$$ y = Xb + S\alpha + Qv + Zu + e, $$

(4)

where y is the vector of observations, b is the vector of (non-SNP) fixed effects such as replication within the experiment (field trial), a is a vector of fixed SNP effects, v is a vector of fixed population effects, u is the vector of random polygene background effects associated with the relationship matrix Z, X is the incidence matrix of zeros and ones relating the y observations to the b fixed effects, Q and S are also incidence matrices relating observations to their associated effects, and e is the vector of residuals. Within the Program TASSLE, Q may be constructed from marker-based stratification of populations when there is significant population structure (Bradbury et al. 2007). While included for completeness, in this study the population effect (Qv) was not included in the model due to a lack of evident population structure.

The same univariate model was fitted using ASReml for comparative purposes. The same relationship matrix, Z, was used for all analyses; while the relationship matrix was generated externally and provided to TASSLE, the equivalent relationship matrix was constructed internally by ASReml using a pedigree file. Both TASSLE and ASReml produce p values to test the significance of the fixed marker effect.

Bivariate tests of association

The significance of tests of associations using the univariate SMR method were compared with alternative bivariate and a multivariate mixed model approaches that test the significance of genetic correlations. Notably, this alternative model classifies both phenotypic and genotypic observations as assessment traits:

$$ {{{\bf y}}_i} = {{{\bf X}}_i}{{{\bf b}}_i} + {{{\bf Z}}_{{{q_i}}}}{{{\bf q}}_i} + {{{\bf Z}}_{{{u_i}}}}{{{\bf u}}_i} + {{{\bf e}}_i}, $$

(5)

where, y _i is the vector of observations that is indexed (i) by the phenotypic (DBH, KPY, i = 1 and n = 3) and genic (SNP (i = 2 and n = 65)) traits; ${{{\bf y}}_i} = \left[ {\matrix{ {{{{\bf y}}_{{{\text{Phenotypi}}{{\text{c}}_{{1}}}}}}} \\ {{{{\bf y}}_{{{\text{Geni}}{{\text{c}}_{{1}}}}}}} \\ }<!end array> } \right] = \left[ {\matrix{ {{{{\bf y}}_1}} \\ {{{{\bf y}}_2}} \\ }<!end array> } \right],$ b _i is the vector of fixed effects and X _i is the incidence matrix relating the y _i observations to the b _i fixed effects where X _i is conditioned on the trait so that field trial and replications within the trials are modeled for the phenotypic trait and not modeled for the genic trait; ${{{\bf X}}_i}{{{\bf b}}_i} = \left[ {\matrix{ {{{{\bf X}}_1}} &{{\bf 0}} \\ {{\bf 0}} &{{{{\bf X}}_2}} \\ }<!end array> } \right]\left[ {\matrix{ {{{{\bf b}}_1}} \\ {{{{\bf b}}_2}} \\ }<!end array> } \right]$, 0 is the null matrix,q _i is the vector of random provenance effects that approximate a multivariate normal distribution ($\sim {\text{MVN}}\left( {{{\bf 0}},{{\bf L}} \otimes {{{\bf I}}_l}} \right)$)where ${{\bf L}} = \left[ {\matrix{ {\sigma_{{{\text{Po}}{{\text{p}}_{{1}}}}}^2} &{{\sigma_{{{\text{Po}}{{\text{p}}_{{{1,2}}}}}}}} \\ {{\sigma_{{{\text{Po}}{{\text{p}}_{{1,2}}}}}}} &{\sigma_{{{\text{Po}}{{\text{p}}_2}}}^2} \\ }<!end array> } \right]$, $\otimes$ is the Kronecker product and I _l is an identity matrix of size equal to the number of populations, $\sigma_{{{\text{Po}}{{\text{p}}_i}}}^2$ is the population variance and ${\sigma_{{{\text{Po}}{{\text{p}}_{{{1,2}}}}}}}$ is the between-trait population covariance,u _i is the vector of random individual-tree effects $\sim {\text{MVN}}\left( {{{\bf 0}},{{\bf G}} \otimes {{{\bf A}}_i}} \right)$, where ${{\bf G}} = \left[ {\matrix{ {\sigma_{{{{\text{f}}_{{1}}}}}^{{2}}} &{{\sigma_{{{{\text{f}}_{{{1,2}}}}}}}} \\ {{\sigma_{{{{\text{f}}_{{{1,2}}}}}}}} &{\sigma_{{{{\text{f}}_{{2}}}}}^{{2}}} \\ }<!end array> } \right]$ differentiates genetic variances $\sigma_{{{f_i}}}^2$ for each trait and between-trait genetic covariances ${\sigma_{{{f_{{1,2}}}}}}$, and A _i is the relationship matrix that is constructed using the population’s pedigree, e _i is the vector of random residual terms $\sim {\text{MVN}}\left( {{{\bf 0}},R \otimes I} \right)$ where residuals were assumed to be heterogeneous across traits, $R = \left[ {\matrix{ {\sigma_{{{E_1}}}^2} &{{\sigma_{{{E_{{1,2}}}}}}} \\ {\sigma_{{{E_{{1,2}}}}}} &{\sigma_{{{E_2}}}^2} \\ }<!end array> } \right]$, $\sigma_{{{E_i}}}^2$is the error variance with the between-trait error covariance ${\sigma_{{{E_{{1,2}}}}}}$ is fixed at zero, and I _i is the identity matrix of dimension equal to the number of observations of each trait. As was the case for the aforementioned univariate model, the lack of population structure led to the removal of the (${{{\bf Z}}_{{{q_i}}}}{{{\bf q}}_i}$) term from the model.

For all multivariate models, restricted maximum likelihood derived variance and covariance estimates were constrained to fall within the theoretical bounds; variance components estimates were constrained to be greater than zero, while covariance estimates were constrained so that correlation estimates ranged from −1 to +1 (Eq. 5). Two separate analyses were used to test for the significance of the genetic correlation between the marker and the phenotype, where the difference between the log-likelihood estimates of (1) a (complete) model that includes a genetic covariance and (2) a (reduced) model that fixed the genetic covariance to zero is used within a two degree of freedom chi-squared test to estimate p values (Wilkes 1987).

Multivariate tests of association

An expanded multivariate mixed model was used to simultaneously evaluate the three phenotypic traits (DBH, density, and KPY, i = 1 to 3) and each genic trait—one at a time (SNP markers, i = 4, n = 65):

$$ {{{\bf y}}_i} = {{{\bf X}}_i}{{{\bf b}}_i} + {{{\bf Z}}_{{{u_i}}}}{{{\bf u}}_i} + {{{\bf e}}_i} $$

(6)

where, y _i is the vector of observations that is indexed (i) by trait; ${{{\bf y}}_{{{\bf i}}}} = \left[ {\matrix{ {{{{\bf y}}_{{{\text{DB}}{{\text{H}}_{{1}}}}}}} \\ {{{{\bf y}}_{{{\text{DE}}{{\text{N}}_1}}}}} \\ {{{{\bf y}}_{{{\text{KP}}{{\text{Y}}_{{1}}}}}}} \\ {{{{\bf y}}_{{{\text{SN}}{{\text{P}}_n}}}}} \\ }<!end array> } \right] = \left[ {\matrix{ {{{{\bf y}}_1}} \\ {{{{\bf y}}_2}} \\ {{{{\bf y}}_3}} \\ {{{{\bf y}}_4}} \\ }<!end array> } \right],$ b _i is the vector of fixed effects representing the trial mean or replicate within trial for the phenotypic effects but not the genic ef of random individual-tree effects $\sim {\text{MVN}}\left( {{{\bf 0}},G \otimes {A_i}} \right)$ where $G = \left[ {\matrix{ {\sigma_{{{f_1}}}^2} &{{\sigma_{{{f_{{1,2}}}}}}} &{{\sigma_{{{f_{{1,3}}}}}}} &{{\sigma_{{{f_{{1,4}}}}}}} \\ {{\sigma_{{{f_{{1,2}}}}}}} &{\sigma_{{{f_2}}}^2} &{{\sigma_{{{f_{{2,3}}}}}}} &{{\sigma_{{{f_{{2,4}}}}}}} \\ {{\sigma_{{{f_{{1,3}}}}}}} &{{\sigma_{{{f_{{2,3}}}}}}} &{\sigma_{{{f_3}}}^2} &{{\sigma_{{{f_{{3,4}}}}}}} \\ {{\sigma_{{{f_{{1,4}}}}}}} &{{\sigma_{{{f_{{2,4}}}}}}} &{{\sigma_{{{f_{{3,4}}}}}}} &{\sigma_{{{f_4}}}^2} \\ }<!end array> } \right]$ is composed of genetic variances $\sigma_{{{f_i}}}^2$ and between-trait genetic covariances ${\sigma_{{{f_{{i,j}}}}}}$ for each trait (i and j),e _i is the vector of random residual terms $\sim {\text{MVN}}\left( {{{\bf 0}},R \otimes I} \right)$ where residuals were assumed to be heterogeneous across traits, $R = \left[ {\matrix{ {\sigma_{{{E_1}}}^2} &{{\sigma_{{{E_{{1,2}}}}}}} &{{\sigma_{{{E_{{1,3}}}}}}} & 0 \\ {{\sigma_{{{E_{{1,2}}}}}}} &{\sigma_{{{E_2}}}^2} &{{\sigma_{{{E_{{2,3}}}}}}} & 0 \\ {{\sigma_{{{E_{{1,3}}}}}}} &{{\sigma_{{{E_{{2,3}}}}}}} &{\sigma_{{{E_3}}}^2} & 0 \\ 0 & 0 & 0 &{\sigma_{{{E_4}}}^2} \\ }<!end array> } \right]$, $\sigma_{{Ei}}^2$is the error variance and ${\sigma_{{{E_{{i,j}}}}}}$ represents the between-trait error covariances with other terms defined in the bivariate model above. Between-trait error covariances are fixed at zero for all covariances between genic and phenotypic traits. Log-likelihood significance tests were estimated as above for the bivariate model with a reduced model fixing the genetic covariance between the phenotypic and genic trait of interest to zero.