# Derivation of the Shrinkage Estimates of Quantitative Trait Locus Effects

## Article excerpt

ABSTRACT

The shrinkage estimate of a quantitative trait locus (QTL) effect is the posterior mean of the QTL effect when a normal prior distribution is assigned to the QTL. This note gives the derivation of the shrinkage estimate under the multivariate linear model. An important lemma regarding the posterior mean of a normal likelihood combined with a normal prior is introduced. The lemma is then used to derive the Bayesian shrinkage estimates of the QTL effects.

(ProQuest: ... denotes formulae omitted.)

THE Bayesian shrinkage estimation of quantitative trait locus (QTL) effects was first introduced byXu (2003) and later formalized by Wang et al. (2005). The multivariate version of the shrinkage estimation of QTL effects was recently developed by Yang and Xu (2007). The main purpose of the shrinkage estimation is to avoid variable selection for mapping multiple QTL. Once a normal prior distribution for each regression coefficient is incorporated into the QTL mapping program, the method can handle substantially more QTL effects than the classical maximum-likelihood (ML) method. In addition, the shrinkage method produces much clearer signals ofQTLon the genome than theMLmethod. As a result, shrinkage mapping appears to have pointed to a new direction for future research in QTL mapping.

The key issue of shrinkage estimation is the normal prior distribution assigned to the regression coefficient (QTL effect). More importantly, different regression coefficients are assigned different normal priors. Because the variances in the prior distributions determine the degrees of shrinkage, assigning different prior variances to different regression coefficients allows the method to differentially shrink regressioncoefficients.A smaller prior variance will cause the regression coefficient to shrink more while a larger prior variance will lead to less shrinkage. This phenomenon is called selective shrinkage.

After incorporating the normal prior distribution into the likelihood function, we can derive the posterior distribution of the regression coefficient, which remains normal due to the conjugate nature of the normal prior. The posterior mean and posterior variance are used to generate a posterior sample of the regression coeffi- cient. Formulas for the posterior mean and posterior variance are mathematically attractive (see Xu 2003; Wang et al. 2005; Yang and Xu 2007). However, due to page limitations of these publications, derivation of the formulas was not provided in these articles.

Derivation of the univariate shrinkage estimation closely followed Box and Tiao's (1973, Appendix A1.1) combination of a univariate normal likelihood and a univariate normal prior. Derivation of themultivariate shrinkage estimation followed the general Bayesian linear model of Lindley and Smith (1972) and the best linear unbiased prediction (BLUP) ofRobinson (1991). The derivations presented by these authors were particularly targeted to statisticians and often difficult to understand by the audience of the genetics community. I have been regularly receiving e-mails and calls from readers asking for the derivation. These readers (almost all genetics professionals and students) are often interested in extending the shrinkage method to handle QTL mapping in different mapping populations. Understanding the derivation of these formulas is crucial to the development of new shrinkage methods. Simply pointing them to the above references often does not help too much because intermediate steps are needed to lead to the shrinkage estimate presented by Xu (2003). By doing this, I often give them an impression of irresponsibility. Therefore, I prepared a short note for the derivation and distributed the note to these interested readers. The note briefly summarizes the derivation using a language that is easy to understand by geneticists with basic statistical training. Given the increasing interest of the derivation from the QTL mapping community, it is more efficient to publish the note in Genetics where the very first shrinkage method (Xu 2003) was published. …

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