Nonparametric Functional Mapping of Quantitative Trait Loci

Abstract

Functional mapping is a useful tool for mapping quantitative trait loci (QTL) that control dynamic traits. It incorporates mathematical aspects of biological processes into the mixture model-based likelihood setting for QTL mapping, thus increasing the power of QTL detection and the precision of parameter estimation. However, in many situations there is no obvious functional form and, in such cases, this strategy will not be optimal. Here we propose to use nonparametric function estimation, typically implemented with B-splines, to estimate the underlying functional form of phenotypic trajectories, and then construct a nonparametric test to find evidence of existing QTL. Using the representation of a nonparametric regression as a mixed model, the final test statistic is a likelihood ratio test. We consider two types of genetic maps: dense maps and general maps, and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples.

A Flexible Estimating Equations Approach for Mapping Function-Valued Traits

“…Here the analyst specifies the form of the mean function (say logistic, for modeling growth curves) and the error function (say autoregressive Gaussian errors). If there is not enough information about the form of the mean function, one may model the mean function nonparametrically using different basis function families: Legendre polynomials , orthogonal polynomials (Yang et al 2006), B-splines (Yang et al 2009;Yap et al 2009), and wavelets (Zhao et al 2007) have all been used in the past. The approaches of Yang et al (2009) and Yap et al (2009) allow for an unstructured form of the variancecovariance matrix of the errors and use a multivariate Gaussian distribution.…”

“…If there is not enough information about the form of the mean function, one may model the mean function nonparametrically using different basis function families: Legendre polynomials , orthogonal polynomials (Yang et al 2006), B-splines (Yang et al 2009;Yap et al 2009), and wavelets (Zhao et al 2007) have all been used in the past. The approaches of Yang et al (2009) and Yap et al (2009) allow for an unstructured form of the variancecovariance matrix of the errors and use a multivariate Gaussian distribution. However, when the number of measurements per individual exceeds the number of samples and the empirical covariance matrix is thus singular, additional procedures such as wavelet dimension reduction (Zhao et al 2007) or regularized estimation of the covariance matrix must be employed.…”

Varying Coefficient Models for Mapping Quantitative Trait Loci Using Recombinant Inbred Intercrosses

“…The marker genotypes were simulated using R/qtl (Broman et al 2003). We set m(t), the mean temporal growth function for QTL genotype Aa to 10=ð115e 20:1t Þ, which is a logistic growth curve (Ma et al 2002; Yang et al 2009). We randomly generated t i from (0, 60) for each subject.…”

“…They used growth curve data as an example of functional traits, and the genetic effect was modeled by a parametric function such as sigmoidal or logistic function . While the parametric nature of functional mapping offers tremendous biological and statistical advantages, a reliance on the availability of mathematical functions limits its applicability (Yang et al 2009).…”

“…One advantage of B-splines over some other nonparametric approaches, like smoothing splines, is that the smoother matrix is independent of the responses. Yang et al (2009) proposed a nonparametric functional mapping framework for genetic mapping of QTL controlling for a dynamic trait, implemented with B-splines.…”