Genome-Wide Complex Trait Analysis (GCTA): Methods, Data Analyses, and Interpretations

Abstract

Estimating genetic variance is traditionally performed using pedigree analysis. Using high-throughput DNA marker data measured across the entire genome it is now possible to estimate and partition genetic variation from population samples. In this chapter, we introduce methods and a software tool called Genome-wide Complex Trait Analysis (GCTA) to estimate genomic relationships between pairs of conventionally unrelated individuals using genome-wide single nucleotide polymorphism (SNP) data, to estimate variance explained by all SNPs simultaneously on genomic or chromosomal segments or over the whole genome, and to perform a joint and conditional multiple SNPs association analysis using summary statistics from a meta-analysis of genome-wide association studies and linkage disequilibrium between SNPs estimated from a reference sample.

Appendix A

In Eq. 4, we have

$$ \mathbf{y}=\mathbf{Xb}+\sum\limits_{i-1}^r {{{\mathbf{g}}_i}} +\mathbf{e}\;\mathrm{ and} \operatorname {var}(\mathbf{y})=\mathbf{V}=\sum\limits_{i-1}^r {{{\mathbf{A}}_i}\sigma_i^2} +\mathbf{I}\sigma_{\mathrm{ e}}^2, $$

of which Eq. 2 is a special case with r = 1. By default in GCTA, we use the average information (AI) REML algorithm [31] to obtain the estimates the variance components $\sigma_i^2$ and $\sigma_{\mathrm{ e}}^2$ through iteration. In the tth iteration, ${{\mathbf{q}}^{(t) }}={{\mathbf{q}}^{(t-1) }}+{{({{\mathbf{H}}^{(t-1) }})}^{-1 }}\frac{{\partial L}}{{\partial \mathbf{q}}}|{{\mathbf{q}}^{(t-1) }}$, where $\mathbf{q}$ is a vector of the estimates of variance components ($\hat{\sigma}_1^2$, …, $\hat{\sigma}_r^2$ and $\hat{\sigma}_{\mathrm{ e}}^2$); L is the log likelihood function of the mixed linear model (ignoring the constant), $L=-1/2(\log |\hat{\mathbf{V}} |+\log |{\mathbf{X}}^{\prime}{{\hat{\mathbf{V}}}^{-1 }\bf X}|+\mathbf{y} \mathbf{^{\prime}\bf Py})$ with $\hat{\mathbf{V}} =\sum\limits_{i=1}^r {{{\mathbf{A}}_i}\hat{\sigma}_i^{2(t-1) }} +\mathbf{I}\hat{\sigma}_e^{2(t-1) }$ and $\mathbf{P}={{\hat{\mathbf{V}}}^{-1 }}-{{\hat{\mathbf{V}}}^{-1 }}\mathbf{X}{{({\mathbf{X}}^{\prime}{{\hat{\mathbf{V}}}^{-1 }}\mathbf{X})}^{-1 }}{\mathbf{X}}^{\prime}{{\hat{\mathbf{V}}}^{-1 }}$ ; H is the average of the observed and expected information matrices [22],

$$ \mathbf{H}=\frac{1}{2}\left\lfloor {\begin{array}{*{20}{c}} {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_1}\mathbf{P}{{\mathbf{A}}_1}\mathbf{P}\mathbf{y}} & \cdots & {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_1}\mathbf{P}{{\mathbf{A}}_r}\mathbf{P}\mathbf{y}} & {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_1}\mathbf{P}\mathbf{P}\mathbf{y}} \\ {\vdots } & \vdots & \vdots & \vdots \\ {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_r}\mathbf{P}{{\mathbf{A}}_1}\mathbf{P}\mathbf{y}} & \cdots & {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_r}\mathbf{P}{{\mathbf{A}}_r}\mathbf{P}\mathbf{y}} & {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_r}\mathbf{P}\mathbf{P}\mathbf{y}} \\ {\mathbf{y} \mathbf{^{\prime}PP}{{\mathbf{A}}_1}\mathbf{P}\mathbf{y}} & \cdots & {\mathbf{y} \mathbf{^{\prime}PP}{{\mathbf{A}}_r}\mathbf{P}\mathbf{y}} & {\mathbf{y} \mathbf{^{\prime}PPPy}} \\ \end{array}} \right\rfloor; $$

and $\frac{{\partial L}}{{\partial \mathbf{q}}}$ is a vector of first derivatives of the log likelihood function with respect to each variance component,

$$ \frac{{\partial L}}{{\partial \mathbf{q}}}=-\frac{1}{2}\left[ {\begin{array}{*{20}{c}} {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_1})-\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_1}\mathbf{Py}} \\ \vdots \\ {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_r})-\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_r}\mathbf{Py}} \\ {\mathrm{tr}(\mathbf{P})-\mathbf{y} \mathbf{^{\prime}PPy}} \\ \end{array}} \right] $$

We also provide in GCTA two optional algorithms to estimate the variance components, which we call the direct REML and EM-REML. For the direct REML algorithm, the variance components in the tth iteration are estimated as

$$ {{\mathbf{q}}^{(t) }}={{\left[ {\begin{array}{*{20}{c}} {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_1}\mathbf{P}{{\mathbf{A}}_1})} & \cdots & {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_1}\mathbf{P}{{\mathbf{A}}_r})} & {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_1}\mathbf{P})} \\ \vdots & \vdots & \vdots & \vdots \\ {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_r}\mathbf{P}{{\mathbf{A}}_1})} & \cdots & {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_r}\mathbf{P}{{\mathbf{A}}_r})} & {\mathrm{tr}(\mathbf{P}{{\mathbf{A}}_r}\mathbf{P})} \\ {\mathrm{tr}(\mathbf{P}\mathbf{P}{{\mathbf{A}}_1})} & \cdots & {\mathrm{tr}(\mathbf{P}\mathbf{P}{{\mathbf{A}}_r})} & {\mathrm{tr}(\mathbf{P}\mathbf{P})} \\ \end{array}} \right]}^{-1 }}\left[ {\begin{array}{*{20}{c}} {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_1}\mathbf{P}\mathbf{y}} \\ \vdots \\ {\mathbf{y} \mathbf{^{\prime}P}{{\mathbf{A}}_r}\mathbf{P}\mathbf{y}} \\ {\mathbf{y} \mathbf{^{\prime}PPy}} \\ \end{array}} \right] $$

The direct REML algorithm is generally more robust but computationally less efficient than AI-REML. For the EM-REML algorithm, each variance component is estimated as

$$ \sigma_i^{2(t) }=[\sigma_i^{4(t-1)}\mathbf{y} \mathbf{^{\prime}\bf P}{{\mathbf{A}}_i}\mathbf{P}\mathbf{y}+\mathrm{ tr}(\sigma_i^{2(t-1)}\mathbf{I}-\sigma_i^{4(t-1)}\mathbf{P}{{\mathbf{A}}_i})]/n $$

The EM-REML is robust, which guarantees increased likelihood after each iteration, but is extremely slow to converge. We therefore do not recommend choosing EM-REML in GCTA unless we know that the starting values are very close to the estimates. The GCTA option for choosing different REML algorithm is --reml-alg with the input value 0 for AI-REML (default), 1 for the direct REML algorithm and 2 for EM-REML. At the beginning of the iteration process, all the variance components are initialized by an arbitrary value, i.e., $\sigma_i^{2(0) }=\sigma_{\mathrm{ P}}^2/(r+1)$, which is subsequently updated by the EM-REML algorithm $\sigma_i^{2(1) }=[\sigma_i^{4(0)}\mathbf{y} \mathbf{^{\prime}\bf P}{{\mathbf{A}}_i}\mathbf{P}\mathbf{y}+\mathrm{ tr}(\sigma_i^{2(0)}\mathbf{I}-\sigma_i^{4(0)}\mathbf{P}{{\mathbf{A}}_i})]/n$. The EM-REML algorithm is used as an initial step to determine the direction of the iteration updates because it is robust to poor starting values. We also provide options (--reml-priors and --reml-priors-var) in GCTA for users to specify starting values. After one EM-REML iteration, GCTA switches to the AI-REML algorithm (or the other two algorithms) for the remaining iterations until the iteration converges with the criteria of L ^(t) − L ^(t−1) < 10⁻⁴ where L ^(t) is the log likelihood of the tth iteration. By default, any variance component that escapes from the parameter space (i.e., its estimate is negative) will be set to ${10^{-6 }} \times \sigma_{\mathrm{ P}}^2$. If a component keeps escaping from the parameter space, it will be constrained at ${10^{-6 }} \times \sigma_{\mathrm{ P}}^2$. There is an option in GCTA (--reml-no-constrain) that allows the estimates of variance components to be negative. This is justified because if a parameter is zero, an unbiased estimate of this parameter will have half chance being negative. In practice, however, a negative variance component is usually difficult to interpret. We also provide an option (--reml-maxit) for users to specify the maximum number of iterations at which the iteration process will stop without convergence.