Fig. 7.

Estimated variance components and correlations from MRNM for BMI-NEU analysis. Var(τ₀): Estimated residual variance for BMI as the main outcome. Var(α₀): Estimated genetic variance for BMI as the main outcome. Var(ε): Estimated residual variance for NEU as the covariate. Var(β): Estimated genetic variance for NEU as the covariate. r_e: Estimated residual correlation between BMI and NEU. r_g: Estimated genetic correlation between BMI and NEU. Error bars are 95% confidence interval. R_e matrix is the residual (co)variance structure between different covariate levels (see Eq. 4), which is derived based on the estimated random regression coefficients and polynomial matrix as R_e = ΦM_yΦ′. Φ is the matrix of polynomials evaluated at given covariate values, where entries of the first column are all 1s and the second column is the standardised covariates of respective individuals. M_y is the variance–covariance matrix of random regression coefficients estimated from MRNM as ${\mathbf{M}}_{\mathbf{y}}=\left[\begin{array}{cc}\hfill \mathrm{var}\left({\mathbf{\tau }}_0\right)\hfill & \hfill \mathrm{cov}\left({\mathbf{\tau }}_0,{\mathbf{\tau }}_1\right)\hfill \\ \hfill \mathrm{cov}\left({\mathbf{\tau }}_0,{\mathbf{\tau }}_1\right)\hfill & \hfill \mathrm{var}\left({\mathbf{\tau }}_1\right)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 16.45\left(\mathrm{SE}0.21\right)\hfill & \hfill 0.54\left(\mathrm{SE}0.11\right)\hfill \\ \hfill 0.54\left(\mathrm{SE}0.11\right)\hfill & \hfill 0.06\left(\mathrm{SE}0.18\right)\hfill \end{array}\right]$. V_g matrix in is the genetic (co)variance structure between different covariate levels (see Eq. 2), which is derived based on the estimated random regression coefficients and polynomial matrix as V_g = ΦK_yΦ′. Φ is the matrix of polynomials evaluated at given covariate values, where entries of the first column are all 1s and the second column is the standardised covariates of respective individuals. K_y is the variance–covariance matrix of random regression coefficients estimated from MRNM as ${\mathbf{K}}_{\mathbf{y}}=\left[\begin{array}{cc}\hfill \mathrm{var}\left({\mathbf{\alpha }}_0\right)\hfill & \hfill \mathrm{cov}\left({\mathbf{\alpha }}_0,{\mathbf{\alpha }}_1\right)\hfill \\ \hfill \mathrm{cov}\left({\mathbf{\alpha }}_0,{\mathbf{\alpha }}_1\right)\hfill & \hfill \mathrm{var}\left({\mathbf{\alpha }}_1\right)\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 4.94\left(\mathrm{SE}0.17\right)\hfill & \hfill 0.38\left(\mathrm{SE}0.11\right)\hfill \\ \hfill 0.38\left(\mathrm{SE}0.11\right)\hfill & \hfill 0.28\left(\mathrm{SE}0.13\right)\hfill \end{array}\right]$