Table of Contents
ISRN Probability and Statistics
Volume 2013, Article ID 496180, 22 pages
http://dx.doi.org/10.1155/2013/496180
Research Article

Linear Models with Response Functions Based on the Laplace Distribution: Statistical Formulae and Their Application to Epigenomics

1INMS, Massey University, Albany Campus, Private Bag 102-904, North Shore Mail Centre, Auckland 0745, New Zealand
2Liggins Institute, The University of Auckland, Private Bag 92019, Victoria Street West, Auckland 1142, New Zealand
3Gravida, National Centre for Growth and Development, The University of Auckland, Private Bag 92019, Victoria Street West, Auckland 1142, New Zealand
4Ruakara Research Centre, 10 Bisley Road, Private Bag 3115, Hamilton 3240, New Zealand

Received 20 June 2013; Accepted 24 July 2013

Academic Editors: A. A. Ding and J. López-Fidalgo

Copyright © 2013 C. Z. W. Hassell Sweatman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The statistical application considered here arose in epigenomics, linking the DNA methylation proportions measured at specific genomic sites to characteristics such as phenotype or birth order. It was found that the distribution of errors in the proportions of chemical modification (methylation) on DNA, measured at CpG sites, may be successfully modelled by a Laplace distribution which is perturbed by a Hermite polynomial. We use a linear model with such a response function. Hence, the response function is known, or assumed well estimated, but fails to be differentiable in the classical sense due to the modulus function. Our problem was to estimate coefficients for the linear model and the corresponding covariance matrix and to compare models with varying numbers of coefficients. The linear model coefficients may be found using the (derivative-free) simplex method, as in quantile regression. However, this theory does not yield a simple expression for the covariance matrix of the coefficients of the linear model. Assuming response functions which are except where the modulus function attains zero, we derive simple formulae for the covariance matrix and a log-likelihood ratio statistic, using generalized calculus. These original formulae enable a generalized analysis of variance and further model comparisons.