Higher-Order Least Squares: Assessing Partial Goodness of Fit of Linear Causal Models

We introduce a simple diagnostic test for assessing the overall or partial goodness of fit of a linear causal model with errors being independent of the covariates. In particular, we consider situations where hidden confounding is potentially present. We develop a method and discuss its capability to distinguish between covariates that are confounded with the response by latent variables and those that are not. Thus, we provide a test and methodology for <i>partial</i> goodness of fit. The test is based on comparing a novel higher-order least squares principle with ordinary least squares. In spite of its simplicity, the proposed method is extremely general and is also proven to be valid for high-dimensional settings. Supplementary materials for this article are available online.

本文提出一种简便的诊断检验方法，用于评估误差项与协变量独立的线性因果模型（linear causal model）的整体或局部拟合优度。特别地，我们考虑了潜在存在隐藏混杂（hidden confounding）的场景。我们开发了相应方法，并探讨了其区分两类协变量的能力：一类是与响应变量存在潜变量（latent variables）混杂的协变量，另一类则未受此类混杂影响。据此，我们提供了针对**局部**拟合优度的检验方法与研究体系。该检验基于将一种新颖的高阶最小二乘准则与普通最小二乘法（ordinary least squares）进行对比。尽管该方法原理简洁，却具备极强的普适性，且已被证明在高维场景（high-dimensional settings）下依然有效。本文的补充材料可在线获取。