Abstract
Linear mixed-effect models with two variance components are often used when variability comes from two sources. In genetics applications, variation in observed traits can be attributed to biological and environmental effects, and the heritability coefficient is a fundamental quantity that measures the proportion of total variability due to the biological effect. We propose a new inferential model approach which yields exact prior-free probabilistic inference on the heritability coefficient. In particular we construct exact confidence intervals and demonstrate numerically our method's efficiency compared to that of existing methods.
Highlights
Normal linear mixed effects models are useful in a variety of biological, physical, and social scientific applications with variability coming from multiple sources; see Khuri and Sahai (1985) and Searle, Casella and McCulloch (1992)
Given that “a central question in biology is whether observed variation in a particular trait is due to environmental or biological factors” (Visscher, Hill and Wray, 2008), the heritability coefficient, ρ = σα2 /(σα2 + σε2), which represents the proportion of phenotypic variance attributed to variation in genotypic values, is a fundamentally important quantity
The inferential model (IM) method proposed here gives exact confidence intervals for the heritability coefficient ρ, as well as the variance ratio ψ, and numerical results suggest increased efficiency compared to existing methods
Summary
Normal linear mixed effects models are useful in a variety of biological, physical, and social scientific applications with variability coming from multiple sources; see Khuri and Sahai (1985) and Searle, Casella and McCulloch (1992). Where Y is a n-vector of response variables, X and Z are design matrices for the fixed and random effects of dimension n × p and n × a, respectively, β is a p-vector of unknown parameters, α is a normal random a-vector with mean 0 and covariance matrix σα A, and ε is a normal random n-vector with mean 0 and covariance matrix σε2In. Here, σ2 = (σα , σε2) is the pair of variance components. The unknown parameters in this model are the p fixed-effect coefficients β and the two variance components σ2, so the parameter space is (p + 2)-dimensional. The quantities α and ε in (1) denote the genetic and environmental effects, respectively. Mixed-effect models and inference on the heritability coefficient has been applied recently in genome-wide association studies (Golan and Rosset, 2011; Yang et al, 2010); see Section 5 for more on these applications
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