For two-sample MR models that require inverse variance weighting, a variety of different weights have been proposed that reduce the impact of violations in the NOME assumption.
In two-sample MR analyses, if the NOME assumption is violated (usually, when using weak instrumental variables (IVs) with a low F-statistic) and the causal effect estimate is non-zero, bias can be induced depending on the type of weights being used in the models that require inverse variance weighting. Using "first-order" weights in the models can inflate the heterogeneity across individual IV-level causal effect estimates but using "second-order" weights can lead to a failure in detecting this heterogeneity, even when it exists. Proposed modified second-order weights outperform both first- and second-order weights by accurately quantifying heterogeneity, detecting outliers in the presence of weak IVs and removing the effect of regression dilution bias. The "iterative" modified weights iteratively generate an optimum weight term when estimating the causal effect of an exposure on an outcome, and is helpful for assessing outlier status of individual IVs. The "exact" modified weights directly minimizes the Cochran's Q statistic used to estimate heterogeneity in the causal effect estimates across IVs, where the weight is a function of the causal effect estimate. See NOME assumption.