Nonparametric covariance estimation in functional mapping of complex dynamic traits

One of the fundamental objectives in agricultural, biological and biomedical research is the identification of genes that control the developmental pattern of complex traits, their responses to the environment, and the way these genes interact in a coordinated manner to determine the final expression of the trait. More recently, a new statistical framework, called functional mapping, has been developed to identify and map quantitative trait loci (QTLs) that determine developmental trajectories by integrating biologically meaningful mathematical models of trait progression into a mixture model for unknown QTL genotypes. Functional mapping has emerged to be a powerful statistical tool for mapping QTLs controlling the responsiveness (reaction norm) of a trait to developmental and environmental signals. From a statistical perspective, functional mapping designed to study the genetic regulation and network of quantitative variation in dynamic complex traits is virtually a joint mean-covariance likelihood model. Appropriate choices of the model for the mean and covariance structures are of critical importance to statistical inference about QTL locations and actions/interactions. While a battery of statistical and mathematical models have been proposed for mean vector modeling, the analysis of covariance structure has been mostly limited to parametric structures like autoregressive one (AR(1)) or structured antedependence (SAD) model. In functional mapping of reaction norms that respond to two environmental signals, a model, expressed as a Kronecker product of two AR(1) structures, has been proposed to test differences of the genetic control of responses to different environments. For practical longitudinal data sets, parametric modeling may be too simple to capture the complex pattern and structure of the covariance. There is a pressing need to develop a robust approach for modeling any possible structure of longitudinal covariance, ultimately broadening the use of functional mapping. Our study proposes a nonparametric covariance estimator in functional mapping of quantitative trait locus. We adopt Huang et al.'s (2006) approach of invoking the modified Cholesky decomposition and converting the problem into modeling a sequence of regressions of responses. A regularized positive-definite covariance estimator is obtained using a normal penalized likelihood with an L2 penalty. This approach is embedded within the mixture likelihood framework of functional mapping by using a reparameterized version of the derivative of the log-likelihood. We extend the idea of functional mapping to model the covariance structure of interaction effects between the two environmental signals in a non-separable way. The extended model allows the quantitative test of several fundamental biological questions. Is there a pleiotropic QTL that regulates genotypic responses to different environmental signals? What is the difference in the timing and duration of QTL expression between environment-specific responsiveness? How is an environment-dependent QTL regulated by a development-related QTL? We performed various simulation studies to reveal the statistical properties of the new models and demonstrate the advantages of the proposed estimator. By analyzing real examples in genetic studies, we illustrated the utilization and usefulness of the methodology. The new methods will provide a useful tool for genome-wide scanning for the existence, distribution and interactions of QTLs underlying a dynamic trait important to agriculture, biology and health sciences.