| Summary: | The estimation of heritability has been an important question in statistical genetics. Due to the clear mathematical properties, the modified Haseman–Elston regression has been found a bridge that connects and develops various parallel heritability estimation methods. With the increasing sample size, estimating heritability for biobank-scale data poses a challenge for statistical computation, in particular that the calculation of the genetic relationship matrix is a huge challenge in statistical computation. Using the Haseman–Elston framework, in this study we explicitly analyzed the mathematical structure of the key term tr(KTK), the trace of high-order term of the genetic relationship matrix, a component involved in the estimation procedure. In this study, we proposed two estimators, which can estimate tr(KTK) with greatly reduced sampling variance compared to the existing method under the same computational complexity. We applied this method to 81 traits in UK Biobank data and compared the chromosome-wise partition heritability with the whole-genome heritability, also as an approach for testing polygenicity.
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