Fig. 1: MR-EILLS model.

The MR-EILLS model aims to infer the causal relationships between one or multiple exposures and one outcome, integrating multiple GWAS summary datasets from heterogeneous populations. There are different pleiotropic effects and IV strengths for the same IVs in heterogeneous populations. The objective function of the MR-EILLS model considers both correlated and uncorrelated pleiotropy and removes invalid IVs. Panels a–c show the results of the motivating simulation. Panel a shows the point plot for the absolute value of \({\hat{\varepsilon }}_{j}^{(e)}\) in different populations, where a larger point indicates a larger value of \(|{\hat{\varepsilon }}_{j}^{(e)}|\). As the pleiotropic effect increases, \(|{\hat{\varepsilon }}_{j}^{(e)}|\) increases; thus, the first part of the MR-EILLS model minimizes the pleiotropic effect between different populations. Panel b shows the correlation between \({\hat{\varepsilon }}_{j}^{(e)}\) and \({\hat{\theta }}_{p,j}^{(e)}\), which represents the correlated pleiotropic effect or the violation of the InSIDE assumption. As the correlated pleiotropic effect increases, this correlation becomes stronger. This corresponds to the second part of MR-EILLS, the empirical focused linear invariance regularizer, which discourages the selection of exposures with strong correlations between \({\theta }_{p,j}^{(e)}\) and \({\varepsilon }_{j}^{(e)}\) in some populations because this correlation would distort the causal effect estimation. Panel c shows the ridge plot of \({\sum }_{e\in {{\rm E}}}|{\varepsilon }_{j}^{(e)}|+{\sum }_{p\in P}{\sum }_{e\in {{\rm E}}}|{\hat{\theta }}_{p,j}^{(e)}{\varepsilon }_{j}^{(e)}|\) when there are different proportions of invalid IVs. When there are invalid IVs, the ridge plot has two peaks, whereas the ridge plot has only one peak when there is no invalid IV. The corresponding abscission value at the lowest point between the two peaks is the optimal \(\lambda \). The third part of the MR-EILLS model removes the invalid IVs by \(\lambda \).