Powerful Test of Heterogeneity in Two-Sample Summary-Data Mendelian Randomization.

Background: The success of a Mendelian randomization (MR) study critically depends on the validity of the assumptions underlying MR. We focus on detecting heterogeneity (also known as horizontal pleiotropy) in two-sample summary-data MR. A popular approach is to apply Cochran's $Q$ statistic method, developed for meta-analysis. However, Cochran's $Q$ statistic, including its modifications, is known to lack power when its degrees of freedom are large. Furthermore, there is no theoretical justification for the claimed null distribution of the minimum of the modified Cochran's $Q$ statistic with exact weighting ( $Q_{\min }$ ), although it seems to perform well in simulation studies.

Method: The principle of our proposed method is straightforward: if a set of variables are valid instruments, then any linear combination of these variables is still a valid instrument. Specifically, this principle holds when these linear combinations are formed using eigenvectors derived from a variance matrix. Each linear combination follows a known normal distribution from which a $p$ value can be calculated. We use the minimum $p$ value for these eigenvector-based linear combinations as the test statistic. Additionally, we explore a modification of the modified Cochran's $Q$ statistic by replacing the weighting matrix with a truncated singular value decomposition.

Results: Extensive simulation studies reveal that the proposed methods outperform Cochran's $Q$ statistic, including those with modified weights, and MR-PRESSO, another popular method for detecting heterogeneity, in cases where the number of instruments is not large or the Wald ratios take two values. We also demonstrate these methods using empirical examples. Furthermore, we show that $Q_{\min }$ does not follow, but is dominated by, the claimed null chi-square distribution. The proposed methods are implemented in an R package iGasso.

Conclusions: Dimension reduction techniques are useful for generating powerful tests of heterogeneity in MR.