Table 1.
Summary of calculations for regression estimates.
Fig 1.
The effect of environmental correlation re between siblings on the R2 of PGS regression analyses from different study designs.
y-axis shows the ratio of R2 from between family design (red) or sibling difference design (blue) vs. the proportion of phenotypic variance due to the direct effect in the population. Each boxplot shows the simulation results of 200 repeats. In each repeat, we simulated the true SNP effect sizes (βdir and βind) from a bivariate normal distribution, and the environmental effect sizes for siblings from a bivariate normal distribution. We then calculated the phenotype, PGS, and run the linear regression analyses. In this figure, the PGS was calculated using (βdir+βind). The variance of direct effect size and the variance of the environmental effect size were fixed at 1 and 3, respectively. The indirect effect size and the correlation rg between direct and indirect effect were both set to 0. When re = 1, the environmental terms for two siblings become identical, thus their phenotypic difference becomes ΔPGSdir and their PGS difference is also ΔPGSdir since here we set βind = 0. Thus, its R2 is always 1 in each repeat whereas the proportion of the phenotypic variance by the direct effect is ¼. Therefore, the ratio is always 4 for the last setting as shown in the figure.
Fig 2.
The effect of indirect effect size variance on the R2 of PGS regression analyses from different study designs.
y-axis shows the ratio of R2 from between family design (red) or sibling difference design (blue) vs. the proportion of phenotypic variance due to the direct effect in the population. The yellow box shows the ratio of R2 from the sibling difference design based on PGSmix vs. that in a sibling difference design based on PGSdir. Each boxplot shows the simulation results of 200 repeats. In this figure, the variance of direct effect size and the variance of the environment effect size were fixed at 1 and 3, respectively. The correlation rg between direct and indirect effect sizes and the correlation re between two sibling’s environments were fixed at 0 and 0.5, respectively. When the indirect effect is 0 (both and βind are 0), the sibling difference analyses become identical regardless of whether the PGS is computed based on (βdir+βind) or βdir, thus the yellow box is fixed at 1 when
in this figure.
Fig 3.
The effect of correlation rg between the direct and indirect effect sizes on the R2 of PGS regression analyses from different study designs.
y-axis shows the ratio of R2 from between family design (red) or sibling difference design (blue) vs. the proportion of phenotypic variance due to the direct effect in the population. The yellow box shows the ratio of R2 from the sibling difference design based on PGSmix vs. that in a sibling difference design based on PGSdir. Each boxplot shows the simulation results of 200 repeats. In this figure, the variance of direct effect size and the variance of the environmental effect size were fixed at 1 and 3, respectively. The correlation between two sibling’s environments was fixed at 0.5. The two panels correspond to the results when the variance of the indirect effect size is 0.5 and 1, respectively.
Fig 4.
The effect of correlation rg between the direct and indirect effect sizes on PGS regression coefficients from different study designs.
y-axis shows the ratio of the PGS regression coefficients from between family design (red) or sibling difference design (blue) vs. 1, which is the effect size of the PGSdir (see Table 1). We used the same simulation settings as those used in Fig 3. Since we set the variance of the direct effect size , when the variance of the indirect effect size is also 1 and their correlation is -1, the direct and indirect effect sizes become exactly the opposite of each other, therefore PGSmix = 0 and the linear regression cannot be run under this scenario. Therefore, we do not include results when
and ρg = −1.