Overview of LASSO-related penalized regression methods for quantitative trait mapping and genomic selection

Abstract

Quantitative trait loci (QTL)/association mapping aims at finding genomic loci associated with the phenotypes, whereas genomic selection focuses on breeding value prediction based on genomic data. Variable selection is a key to both of these tasks as it allows to (1) detect clear mapping signals of QTL activity, and (2) predict the genome-enhanced breeding values accurately. In this paper, we provide an overview of a statistical method called least absolute shrinkage and selection operator (LASSO) and two of its generalizations named elastic net and adaptive LASSO in the contexts of QTL mapping and genomic breeding value prediction in plants (or animals). We also briefly summarize the Bayesian interpretation of LASSO, and the inspired hierarchical Bayesian models. We illustrate the implementation and examine the performance of methods using three public data sets: (1) North American barley data with 127 individuals and 145 markers, (2) a simulated QTLMAS XII data with 5,865 individuals and 6,000 markers for both QTL mapping and genomic selection, and (3) a wheat data with 599 individuals and 1,279 markers only for genomic selection.

Appendix: Coordinate descent algorithm

Initially, the marker data are assumed to be standardized and phenotype data to be centered so that $\frac{1}{n}\sum\nolimits_{{i = 1}}^{n} {x_{{ij}} } = 0,$ $\sum\nolimits^n_{i=1} x^2_{ij}=1$ for j = 1, …, p, and $\frac{1}{n}\sum\nolimits^n_{i=1} y_i=0.$ The Elastic net problem (LASSO: α = 1, Ridge regression: α = 0) can be specified as

$$ \hat{\varvec{\beta}}=\arg \min_{\varvec{\beta}}\left\{\frac{1}{2N}\sum^n_{i=1}\left(y_i-\sum^{p}_{j=1}x_{ij}\beta_j\right)^2 +\lambda[(1-\alpha)\frac{1}{2}\sum^{p}_{j=1}\beta^2_j+\alpha\sum^{p}_{j=1}|\beta_j|]\right\}. $$

(16)

The principle of the coordinate descent is that when minimizing the Elastic net target function, the algorithm updates each component β _j successively in the direction giving the largest decrease of the objective function by fixing all other components. Assuming the current estimate of β _j is β ⁽⁰⁾ _j , and we have already updated the estimate of $\beta_1, \beta_2, \ldots, \beta_{j-1}$ as $\beta^{(1)}_1, \beta^{(1)}_2, \ldots, \beta^{(1)}_{j-1},$ the estimate of β ⁽¹⁾ _j can be updated as

$$ \beta^{(1)}_j(\lambda)=\frac{S(\beta^{(0)}_j+\frac{1}{N}\sum^n_{i=1}x_{ij}r_i,\lambda\alpha)}{1+\lambda(1-\alpha)} , $$

(17)

where S(a, b) is the thresholding function defined as

$$ S(a,b) = \hbox {sign}(a)\cdot\max(|a|-b,0), $$

(18)

and $r_i=y_i-\sum^{p}_{j}x_{ij}{\beta_j}$ for $i=1,\ldots,n$ is the residual, which should be updated as $r_i=r_i+x_{ij}(\beta^{(0)}_j-\beta^{(1)}_j)$ when $\beta^{(1)}_{j}$ is ready. The algorithm updates each component of ${\varvec{\beta}}$ in a cyclic manner as $1, 2, \ldots, p, 1, 2, \ldots, p, \ldots ,$ until the solutions converge.

The coordinate descent algorithm can be used for Adaptive LASSO as well. In each iteration, we use the update function:

$$ \beta^{(1)}_j(\lambda)=S\left(\beta^{(0)}_j+\frac{1}{N}\sum^n_{i=1} x_{ij}r_i,\frac{\lambda}{|\hat{\beta}_{{\rm init},j}|}\right) , $$

(19)

where $\hat{\beta}_{{\rm init},j}$ are certain initial estimates, for example, from OLS or standard LASSO.

For more information, see Friedman et al. (2007) and (2010).