Peter H. Schonemann
A note on Holzinger's heritability coefficient h2
Chinese Journal of Psychology, 1993, 35, 59-65
It is shown that the still widely used heritability estimate h2 developed by Holzinger (1937) is not valid because Holzinger's derivation of it was unsound: The variance component model he used to derive h2, together with his claim that it estimates the genetic variance ratio, imply the counterfactual assertion that dizygotic twins share no genes.
While the competing coefficient, Nichols' HR does indeed follow from the conventional variance component model, the necessary conditions it rests on lack empirical face validity.
Both coefficients were developed to estimate narrow heritability from MZT/DZT twin data (which are more abundant than MZT/DZT data). For a long time, Holzinger's nonsensical estimate was more widely used, probably because it appears more resaonable on the surface in producing fewer inadmissible values. However, on checking Holzinger's derivations one finds that it does not contain any environmental variance at all, since it cancels in the numerator and denominator. Hence, it cannot very well estimate the ratio of genetic over systematic total (= genetic + environmental) variance.
Instead, one finds that Nichols' HR turns out to be the mathematically correct estimates, if one accepts the conventional simplistic variance component model both Holzinger and Nichols started with. However, on checking the literature, one also finds that this coefficient, HR, produces an inordinate proportion of inadmissable values, usually exceeding unity. This means that the underlying assumptions are not met by the data.
One usually finds that the correlations for MZTs are too high relative to those for DZTs, i.e. MZTs are in practice more similar than the models allows for. Specifically, the underlying variance compontent model predicts the chain
r(MZT)/2 < r(DZT) < r(MZT).
While the right pair of this ineqality is usually satisfied by the data, the left side is often violated, producing HR's > 1.