GSMR is based on summary data based MR (SMR). The estimated effect of IV isβ^XY={β^XY1,β^XY2,⋅⋅⋅β^XYj}and the variance–covariance matrix is: COV(β^XYi,β^XYj)≈rβXGiβXGjσβ^YGi2σβ^YGj2+βXYiβXYj[rσ
It is also assumed that the model matrix of the fixed effects is based on a general two-level fractional factorial design. The goal of this paper is to provide an analytic form of the covariance matrix of the generalized least squares estimators of the fixed factorial effects in the model, ...
i.e., analysis of variance (ANOVA), the analysis of covariance (ANCOVA), the analysis of heterogeneous covariance (ANHECOVA), the inverse probability weighting (IPW), the augmented inverse probability weighting (AIPW), and the overlap weighting (OW) as well as the augmented overlap...
Statistical Analysis We combined summary data across SNPs into a single instrument, using maximum likelihood to estimate the slope of the relationship between βGD and βGP and a variance-covariance matrix to make allowance for linkage disequilibrium between SNPs,28 where βGD is the change in dise...
{{\boldsymbol{\beta }}}_{{{\bf{X}}}_{{\boldsymbol{\gamma }}})}^{-1}\)is the inverse of the shrinkage covariance between the genetic associations of the risk factors. For a detailed derivation of the Bayes factor we refer to the Supplementary Methods in the Supplementary Information....
Multivariable IVW requires knowledge of the covariance between the effects of the genetic variants on each exposure, which we approximated from the phenotypic covariance matrix across exposure traits [59]. If genetic association data on cardiovascular diseases was available from two or more independent ...
(estimated) covariance matrix of the joint effect estimate (\({\hat{{{\boldsymbol{\beta }}}_{X}\)) as\({{{\mathbf{\Sigma }}}_{X}={{{\bf{R}}}_{X}^{-1}{{{\boldsymbol{\Omega }}}_{X}{{{\bf{R}}}_{X}^{-1}\), where\({{{\boldsymbol{\Omega }}}_{X}={{{\bf...