A Flexible Estimating Equations Approach for Mapping Function-Valued Traits

Abstract

In genetic studies, many interesting traits, including growth curves and skeletal shape, have temporal or spatial structure. They are better treated as curves or function-valued traits. Identification of genetic loci contributing to such traits is facilitated by specialized methods that explicitly address the function-valued nature of the data. Current methods for mapping function-valued traits are mostly likelihood-based, requiring specification of the distribution and error structure. However, such specification is difficult or impractical in many scenarios. We propose a general functional regression approach based on estimating equations that is robust to misspecification of the covariance structure. Estimation is based on a two-step least-squares algorithm, which is fast and applicable even when the number of time points exceeds the number of samples. It is also flexible due to a general linear functional model; changing the number of covariates does not necessitate a new set of formulas and programs. In addition, many meaningful extensions are straightforward. For example, we can accommodate incomplete genotype data, and the algorithm can be trivially parallelized. The framework is an attractive alternative to likelihood-based methods when the covariance structure of the data is not known. It provides a good compromise between model simplicity, statistical efficiency, and computational speed. We illustrate our method and its advantages using circadian mouse behavioral data.

A Bayesian Nonparametric Approach for Mapping Dynamic Quantitative Traits

“…Note that the model (1) differs from some single-locus functional mapping approaches (Ma et al 2002;Xiong et al 2011). Those authors specify the model for marker j (j = 1, .…”

“…Yang et al 2009;R. Yang et al 2009;Xiong et al 2011;Gong and Zou 2012;Xing et al 2012). In addition, the residual terms of these models were often assumed to share a certain covariance structure such as an autoregressive process (Ma et al 2002).…”

“…Most of the existing frequentist approaches (Ma et al 2002;Lin and Wu 2006;Xiong et al 2011;Gong and Zou 2012) follow single-locus functional mapping, which maps the dynamic traits to each marker one at a time and uses either likelihood methods or least-squares-estimating equations approaches (with independent residual covariance structure). These approaches typically construct a test statistic (e.g., log-likelihood ratio or Wald statistic) to screen the important variables (QTL) through a multiple-testing procedure (e.g., adjusting the P-value by permutation or by Bonferroni correction).…”

“…First, with Bsplines, our focus is on automatic adaptive determination of the degree of smoothness (i.e., number of knots), which is a crucial problem in B-splines modeling. In the earlier works of Xiong et al (2011) and Gong and Zou (2012), the degree of smoothness of B-splines was chosen explicitly by cross-validation or the Akaike information criterion (AIC). While here we estimate the degree of smoothness implicitly by assigning second-order penalty priors (Fahrmeir and Kneib 2011) to the time-varying parameters.…”

“…The idea is similar to the well-known forward/backward selection (Hastie et al 2009;Segura et al 2012), but it differs substantially from the Bayesian variable selection procedures used currently for Bayesian QTL mapping (e.g., Yang et al 2011). The rest of the article is organized as follows: in Methods, the concept of the (functional) multitrait model and Bsplines is reviewed, and the prior settings, the variational Bayes algorithms, and the variable selection procedures are introduced; in Example analyses, we show the results of analyzing two types of simulated data sets (including data replicates) and one real mouse data set (Xiong et al 2011); and finally we summarize some key points of our functional approach in the Discussion.…”