  This paper introduces a new measure of heritability which relaxes the
classical assumption that the degree of heritability of a continuous trait can
be summarized by a single number.This measure can be used in situations where
the trait dependence structure between family members is nonlinear, in which
case traditional mixed effects models and covariance (correlation) based
methods are inadequate. Our idea is to combine the notion of a correlation
curve with traditional correlation based measures of heritability, such as the
formula of Falconer. For estimation purposes, we use a multivariate Gaussian
mixture, which is able to capture non-linear dependence and respects certain
distributional constraints. We derive an analytical expression for the
associated correlation curve, and investigate its limiting behaviour when the
trait value becomes either large or small. The result is a measure of
heritability that varies with the trait value. When applied to birth weight
data on Norwegian mother father child trios, the conclusion is that low and
high birth weight are less heritable traits than medium birth weight. On the
other hand, we find no similar heterogeneity in the heritability of Body Mass
Index (BMI) when studying monozygotic and dizygotic twins.
