Note on bias from averaging repeated measurements in heritability studies.

Ge et al. (1) consider the extension of Fisher’s classic model for heritability to the case where there are repeated measurements on subjects. One approach to analyzing repeated measurements is to average observations. The authors show empirically and via simulations that estimates of heritability derived from averaging repeated measurements lead to underestimates of heritability. Some may find the bias revealed by Ge et al. (1) to be surprising because averaging is commonly justified in other settings such as repeated measures ANOVA. In this letter we detail the bias that arises from conflating measurement error and unique environmental variance. This elucidates the authors’ empirical findings, which represent a case with large measurement error exacerbated by only two measurements per subject. Consider the model for additive genetic, common environmental, and unique environmental components. We use the mixed-model formulation as in ref. 2 but include measurement error. For conciseness, we assume no nuisance covariates. Let yijk be the kth measurement for k = 1, . . . , n (for simplicity, we assume the same n for all subjects) for the jth individual in the ith family. Let ai ∼Nð0, σAÞ denote the additive genetic component, and for dizygotics (DZs) and siblings we add aij as in ref. 2. Let ci ∼Nð0, σCÞ denote the common environmental component, eij ∼Nð0, σEÞ the unique environmental, and «ijk ∼Nð0, σMÞ the measurement error (as the authors note, «ijk can also include biological transients). For monozygotic twins (MZs),