The limits of normal approximation for adult height

Abstract

Adult height inspired the first biometrical and quantitative genetic studies and is a test-case trait for understanding heritability. The studies of height led to formulation of the classical polygenic model, that has a profound influence on the way we view and analyse complex traits. An essential part of the classical model is an assumption of additivity of effects and normality of the distribution of the residuals. However, it may be expected that the normal approximation will become insufficient in bigger studies. Here, we demonstrate that when the height of hundreds of thousands of individuals is analysed, the model complexity needs to be increased to include non-additive interactions between sex, environment and genes. Alternatively, the use of log-normal approximation allowed us to still use the additive effects model. These findings are important for future genetic and methodologic studies that make use of adult height as an exemplar trait.

Fig. 1. Relation between parameters of the distribution of adult human height across populations.

Linear regression of standard deviation (A) and CV (B) of height on mean height of women from ref. [20]. The dashed line shows the overall mean. (C) Linear regression of mean male height on mean female height in populations from ref. [21]. Unweighted linear regression was used to estimate the trend (k), its standard error (SE), the adjusted R² and, in brackets, the significance of deviation of the regression coefficient from zero for A, B and from one for C (p < 0.001–***; p < 0.01–*; p > 0.05—ns) (shown at the top of each panel).

Fig. 2. Changes of SD with the mean height and log-height in UK Biobank.

Relation of standard deviation to mean of height (A) and log-height (B) for six groups of British individuals of white descent from UK Biobank, defined based on place of birth and split by sex, median polygenic score, and median residual predictor (48 groups in total). The size of a symbol is proportional to the regression weight, defined as twice the group size. Weighted linear regression was used to estimate the trend (k), its standard error (SE), the adjusted R² and, in brackets, the significance of deviation of the regression coefficient from zero (p < 0.001–***; p > 0.05—ns) (shown at the top of each panel).

Fig. 3. Changes of the effects of different factors with the mean height and log-height in UK Biobank.

Relation between the estimate of the effect size of sex (A, B), genotype (C, D; genotype was defined as a polygenic height score, PGHS), other factors (E, F; a linear residual predictor, RP, combining sociodemographic and study covariates) and mean height (A, C, E) and log-height (B, D, F) for six groups of British individuals of white descent from UK Biobank, defined based on place of birth. The six groups are additionally split by sex (C–F), median polygenic height score (A, B, E, F), and median residual predictor (A–D). The size of a symbol is proportional to the group size (used as the regression weight). Weighted linear regression was used to estimate the trend (k), its standard error (SE), the adjusted R² and, in brackets, the significance of deviation of the regression coefficient from zero (p < 0.001–***; p > 0.05—ns) (shown at the top of each panel).