Multiple regression for molecular-marker, quantitative trait data from large F2 populations

Abstract

Molecular marker-quantitative trait associations are important for breeders to recognize and understand to allow application in selection. This work was done to provide simple, intuitive explanations of trait-marker regression for large samples from an F2 and to examine the properties of the regression estimators. Beginning with a(- 1,0,1) coding of marker classes and expected frequencies in the F2, expected values, variances, and covariances of marker variables were calculated. Simple linear regression and regression of trait values on two markers were computed. The sum of coefficient estimates for the flanking-marker regression is asymptotically unbiased for an included additive effect with complete interference, and is only slightly biased with no interference and moderately close (15 cM) marker spacing. The variance of the sum of regression coefficients is much more stable for small recombination distances than variances of individual coefficients. Multiple regression of trait variables on coded marker variables can be interpreted as the product of the inverse of the marker correlation matrix R and the vector a of simple linear regression estimators for each marker. For no interference, elements of the correlation matrix R can be written as products of correlations between adjacent markers. The inverse of R is displayed and used to illustrate the solution vector. Only markers immediately flanking trait loci are expected to have non-zero values and, with at least two marker loci between each trait locus, the solution vector is expected to be the sum of solutions for each trait locus. Results of this work should allow breeders to test for intervals in which trait loci are located and to better interpret results of the trait-marker regression.

A Modified Algorithm for the Improvement of Composite Interval Mapping

“…Under the assumption of additivity of QTL effects, the genetic value G of an individual under an additive genetic model can be written in the following form: (Whittaker et al 1996). The expectation of QTL genotype g j is dependent on the position of the jth QTL on the chromosomal interval flanked by the jth and ( j 1 1)th markers and the length of the interval (Zeng 1993;Wright and Mowers 1994;Whittaker et al 1996); i.e.,…”

“…The coefficient of the jth marker is affected by QTL only on intervals ( j À 1, j) and ( j, j 1 1). If there are no QTL in the neighboring intervals of the current interval ( j, j 1 1), corresponding to the assumption of isolated QTL according to Whittaker et al (1996), the two coefficients b j and b j11 contain all the position and additive effect information of the QTL in the interval ( j, j 1 1), which provides the theoretical basis for mapping additive QTL in CIM (Zeng 1994) and other regression mapping methods (Wright and Mowers 1994;Whittaker et al 1996). Suppose that we have a sample of n individuals from a backcross population with observations on a quantitative trait of interest and m 1 1 ordered markers.…”

“…A number of statistical methods have been developed for QTL detection and effect estimation (Lander and Botstein 1989;Haley and Knott 1992;Jansen 1994;Wright and Mowers 1994;Zeng 1994;Satagopan et al 1996;Whittaker et al 1996;Piepho and Gauch 2001;Sen and Churchill 2001;Broman and Speed 2002;van den Oord and Sullivan 2003;Xu 2003;Bogdan et al 2004).…”

Interactions Between Markers Can Be Caused by the Dominance Effect of Quantitative Trait Loci

“…A number of statistical methods have been developed for QTL detection and effect estimation. For regression-based methods, see Haley and Knott (1992), Martinez and Curnow (1992), Haley et al (1994), Wright and Mowers (1994), Whittaker et al (1996), and Feenstra et al (2006); for maximumlikelihood-based methods, see Lander and Botstein (1989), Knott and Haley (1992), Zeng (1994), Kao et al (1999), and Li et al (2007Li et al ( , 2008; and for Bayesian model-based methods, see Satagopan et al (1996), Ball (2001), Sen and Churchill (2001), Sillanpää and Corander (2002), Yi et al (2003), and Bogdan et al (2004).…”

“…F 2 populations have been widely used in genetic studies of animals and plants since the rediscovery of Mendel's hybridization experiments. Relatively fewer methods have been developed on the basis of F 2 populations, and dominance has sometimes been ignored (Wright and Mowers 1994;Whittaker et al 1996;Jia and Xu 2007). Using similar principles in interval mapping (IM) as proposed by Lander and Botstein (1989), Knott and Haley (1992) investigated the maximumlikelihood methods for QTL mapping in F 2 populations using simulated data.…”