  In the field of genetics, the concept of heritability refers to the
proportion of variations of a biological trait or disease that can be explained
by genetic factors. Quantifying the heritability of a disease is a fundamental
challenge in human genetics, especially when the causes are plural and not
clearly identified. Although the literature regarding heritability estimation
for binary traits is less rich than for quantitative traits, several methods
have been proposed to estimate the heritability of complex diseases. However,
to the best of our knowledge, the existing methods are not supported by
theoretical grounds. Moreover, most of the methodologies do not take into
account a major specificity of the data coming from medical studies, which is
the oversampling of the number of patients compared to controls. We propose in
this paper to investigate the theoretical properties of the method developed by
Golan et al. (2014), which is very efficient in practice, despite the
oversampling of patients. Our main result is the proof of the consistency of
this estimator. We also provide a numerical study to compare two approximations
leading to two heritability estimators.
