Simulation studies are used to understand the properties of statistical methods. A key luxury in many simulation studies is knowledge of the true value (i.e. the estimand) being targeted. With this oracle knowledge in-hand, the researcher conducting the simulation study can assess across repeated realizations of the data how well a given method recovers the truth. In causal inference simulation studies, the truth is rarely a simple parameter of the statistical model chosen to generate the data. Instead, the estimand is often an average treatment effect, marginalized over the distribution of confounders and/or mediators. Luckily, these variables are often generated from common distributions such as the normal, uniform, exponential, or gamma. For all these distributions, Gaussian quadratures provide efficient and accurate calculation for integrands with integral kernels that stem from known probability density functions. We demonstrate through four applications how to use Gaussian quadrature to accurately and efficiently compute the true causal estimand. We also compare the pros and cons of Gauss-Hermite quadrature to Monte Carlo integration approaches, which we use as benchmarks. Overall, we demonstrate that the Gaussian quadrature is an accurate tool with negligible computation time, yet is underused for calculating the true causal estimands in simulation studies.

翻译：模拟研究用于理解统计方法的特性。许多模拟研究中的一个关键优势在于知晓所针对的真实值（即估计目标）。借助这种先验知识，进行模拟研究的研究人员可以评估给定方法在数据的重复实现中恢复真实值的程度。在因果推断模拟研究中，真实值很少是用于生成数据的所选统计模型的简单参数。相反，估计目标通常是平均处理效应，在混杂变量和/或中介变量的分布上进行边缘化处理。幸运的是，这些变量通常来自常见分布，如正态分布、均匀分布、指数分布或伽马分布。对于所有这些分布，高斯求积法为源自已知概率密度函数的积分核的被积函数提供了高效且准确的计算方法。我们通过四个应用案例展示了如何使用高斯求积法准确高效地计算真实的因果估计目标。我们还比较了高斯-埃尔米特求积法与蒙特卡洛积分方法的优缺点，后者被用作基准。总体而言，我们证明了高斯求积法是一种计算时间可忽略不计的精确工具，但在模拟研究中用于计算真实因果估计目标的应用仍显不足。