Mendelian randomization (MR) is a statistical method that utilizes genetic variants as instrumental variables (IVs) to investigate causal relationships between risk factors and outcomes. Although MR has gained popularity in recent years due to its ability to analyze summary statistics from genome-wide association studies (GWAS), it requires a substantial number of single nucleotide polymorphisms (SNPs) as IVs to ensure sufficient power for detecting causal effects. Unfortunately, the complex genetic heritability of many traits can lead to the use of invalid IVs that affect both the risk factor and the outcome directly or through an unobserved confounder. This can result in biased and imprecise estimates, as reflected by a larger mean squared error (MSE). In this study, we focus on the widely used two-stage least squares (2SLS) method and derive formulas for its bias and MSE when estimating causal effects using invalid IVs. Using those formulas, we identify conditions under which the 2SLS estimate is unbiased and reveal how the independent or correlated pleiotropic effects influence the accuracy and precision of the 2SLS estimate. We validate these formulas through extensive simulation studies and demonstrate the application of those formulas in an MR study to evaluate the causal effect of the waist-to-hip ratio on various sleeping patterns. Our results can aid in designing future MR studies and serve as benchmarks for assessing more sophisticated MR methods.

中文翻译：

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具有无效工具变量的孟德尔随机化中的偏差和均方误差

孟德尔随机化 (MR) 是一种利用遗传变异作为工具变量 (IV) 来研究风险因素与结果之间因果关系的统计方法。尽管 MR 近年来因其能够分析全基因组关联研究 (GWAS) 的汇总统计数据而受到欢迎，但它需要大量的单核苷酸多态性 (SNP) 作为 IV，以确保有足够的能力来检测因果效应。不幸的是，许多性状的复杂遗传性可能导致使用无效的 IV，从而直接或通过未观察到的混杂因素影响风险因素和结果。这可能会导致估计有偏差且不精确，正如较大的均方误差 (MSE) 所反映的那样。在本研究中，我们重点关注广泛使用的两阶段最小二乘法 (2SLS)，并推导出使用无效 IV 估计因果效应时其偏差和 MSE 的公式。使用这些公式，我们确定了 2SLS 估计无偏的条件，并揭示了独立​​或相关的多效性效应如何影响 2SLS 估计的准确性和精度。我们通过广泛的模拟研究验证了这些公式，并展示了这些公式在 MR 研究中的应用，以评估腰臀比对各种睡眠模式的因果影响。我们的结果可以帮助设计未来的 MR 研究，并作为评估更复杂的 MR 方法的基准。