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

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. 1. Introduction and Motivation This work arose in a biological context, in epigenomics, namely, the modelling of the distribution of errors in the proportions of chemical modification (methylation) on DNA, measured at specific genomic sites (CpG sites). It was observed that this error distribution may be suitably modelled by a truncated Laplace distribution perturbed by a Hermite polynomial. This error distribution was first noticed in Sequenom measurements but has wider application. A survey of data generated by measurements on the Infinium, Illumina, Affymetrix, and MeDIP2 machines showed similar characteristics to that of the Sequenom, where such an amended Laplace distribution was required to properly describe the probability density function. It is thought that variation in the scattering angle of light in the measurement processes common to all of these platforms is responsible for the frequencies in the tails of the measurement distributions not conforming to a simple Laplace density and requiring our proposed amendment. Without the amendment the Laplace density gives tail probabilities for the deviations that are too high, potentially leading to an incorrect failure to reject a null hypothesis. Because the observed frequency distribution of