A Fisher scoring algorithm for the weighted regression method of QTL mapping

Quantitative trait loci influencing forking defects in an outbred pedigree of loblolly pine

Background

The use of wood as an industrial raw material has led to development of plantation forestry, in which trees are planted, managed, and harvested as crops. The productivity of such plantations often exceeds that of less-intensively-managed forests, and land managers have the option of choosing specific planting stock to produce specific types of wood for industrial use. Stem forking, or division of the stem into two or more stems of roughly equal size, is a character trait important in determining the quality of the stem for production of solid wood products. This trait typically has very low individual-tree heritability, but can be more accurately assessed in clonally-replicated plantings where each genotype is represented by several individual trees. We report results from a quantitative trait mapping experiment in a clonally-replicated full-sibling family of loblolly pine (Pinus taeda L.).

Results

Quantitative trait loci influencing forking defects were identified in an outbred full-sibling family of loblolly pine, using single-nucleotide polymorphism markers. Genetic markers in this family segregated either in 1:2:1 (F2 intercross-like segregation) or 1:1 ratio (backcross-like segregation). An integrated linkage map combining markers with different segregation ratios was assembled for this full-sib family, and a total of 409 SNP markers were mapped on 12 linkage groups, covering 1622 cM. Two and three trait loci were identified for forking and ramicorn branch traits, respectively, using the interval mapping method. Three trait loci were detected for both traits using multiple-trait analysis.

Conclusions

The detection of three loci for forking and ramicorn branching in a multiple-trait analysis could mean that there are genes with pleiotropic effects on both traits, or that separate genes affecting different traits are clustered together. The detection of genetic loci associated with variation in stem quality traits in this study supports the hypothesis that marker-assisted selection can be used to decrease the rate of stem defects in breeding populations of loblolly pine.

Electronic supplementary material

The online version of this article (doi:10.1186/s12863-016-0446-6) contains supplementary material, which is available to authorized users.

Generalized linear model for interval mapping of quantitative trait loci

We developed a generalized linear model of QTL mapping for discrete traits in line crossing experiments. Parameter estimation was achieved using two different algorithms, a mixture model-based EM (expectation-maximization) algorithm and a GEE (generalized estimating equation) algorithm under a heterogeneous residual variance model. The methods were developed using ordinal data, binary data, binomial data and Poisson data as examples. Applications of the methods to simulated as well as real data are presented. The two different algorithms were compared in the data analyses. In most situations, the two algorithms were indistinguishable, but when large QTL are located in large marker intervals, the mixture model-based EM algorithm can fail to converge to the correct solutions. Both algorithms were coded in C++ and interfaced with SAS as a user-defined SAS procedure called PROC QTL.

An expectation-maximization algorithm for the Lasso estimation of quantitative trait locus effects

The least absolute shrinkage and selection operator (Lasso) estimation of regression coefficients can be expressed as Bayesian posterior mode estimation of the regression coefficients under various hierarchical modeling schemes. A Bayesian hierarchical model requires hyper prior distributions. The regression coefficients are parameters of interest. The normal distribution assigned to each regression coefficient is a prior distribution. The variance parameter in the normal prior distribution is further assigned a hyper prior distribution so that the variance parameter can be estimated from the data. We developed an expectation-maximization (EM) algorithm to estimate the variance parameter of the prior distribution for each regression coefficient. Performance of the EM algorithm was evaluated through simulation study and real data analysis. We found that the Jeffreys' hyper prior for the variance component usually performs well with regard to generating the desired sparseness of the regression model. The EM algorithm can handle not only the usual regression models but it also conveniently deals with linear models in which predictors are defined as classification variables. In the context of quantitative trait loci (QTL) mapping, this new EM algorithm can estimate both genotypic values and QTL effects expressed as linear contrasts of the genotypic values.

PROC QTL-A SAS Procedure for Mapping Quantitative Trait Loci

Statistical analysis system (SAS) is the most comprehensive statistical analysis software package in the world. It offers data analysis for almost all experiments under various statistical models. Each analysis is performed using a particular subroutine, called a procedure (PROC). For example, PROC ANOVA performs analysis of variances. PROC QTL is a user-defined SAS procedure for mapping quantitative trait loci (QTL). It allows users to perform QTL mapping for continuous and discrete traits within the SAS platform. Users of PROC QTL are able to take advantage of all existing features offered by the general SAS software, for example, data management and graphical treatment. The current version of PROC QTL can perform QTL mapping for all line crossing experiments using maximum likelihood (ML), least square (LS), iteratively reweighted least square (IRLS), Fisher scoring (FISHER), Bayesian (BAYES), and empirical Bayes (EBAYES) methods.