  In the analysis of complex traits, genetic effects are frequently modelled as
either fixed or random effects. Such assumptions serve as a foundation of
defining heritability and relatedness using genome-wide single nucleotide
polymorphism (SNP) markers. In this work, I propose a genealogical framework
connecting the two assumptions conditional on the ancestral recombination graph
(ARG). It turns out that the reference time point in which the probability is
defined determines whether the effect of a variant to behave as either a fixed
or random effect. This lays a connection between the PC regression and linear
mixed model (LMM) for genetic association study. The framework induces a
genetic relatedness matrix (GRM) in which the elements are a function of the
time to the recent common ancestor. Subsequently, a novel trait variance
decomposition respect to the regions sharing a common genealogy is followed.
The variance decomposition then provides a natural means to define heritability
and a potential method for gene mapping.
