Principal covariates regression (PCovR) allows one to deal with the interpretational and technical problems associated with running ordinary regression using many predictor variables. In PCovR, the predictor variables are reduced to a limited number of components, and simultaneously, criterion variables are regressed on these components. By means of a weighting parameter, users can flexibly choose how much they want to emphasize reconstruction and prediction. However, when datasets contain many criterion variables, PCovR users face new interpretational problems, because many regression weights will be obtained and because some criteria might be unrelated to the predictors. We therefore propose PCovR2, which extends PCovR by also reducing the criteria to a few components. These criterion components are predicted based on the predictor components. The PCovR2 weighting parameter can again be flexibly used to focus on the reconstruction of the predictors and criteria, or on filtering out relevant predictor components and predictable criterion components. We compare PCovR2 to two other approaches, based on partial least squares (PLS) and principal components regression (PCR), that also reduce the criteria and are therefore called PLS2 and PCR2. By means of a simulated example, we show that PCovR2 outperforms PLS2 and PCR2 when one aims to recover all relevant predictor components and predictable criterion components. Moreover, we conduct a simulation study to evaluate how well PCovR2, PLS2 and PCR2 succeed in finding (1) all underlying components and (2) the subset of relevant predictor and predictable criterion components. Finally, we illustrate the use of PCovR2 by means of empirical data.

主协变量回归 (PCovR) 允许人们处理与使用许多预测变量运行普通回归相关的解释和技术问题。在 PCovR 中，预测变量被减少到有限数量的组件，同时，标准变量在这些组件上回归。通过一个加权参数，用户可以灵活地选择他们想要强调多少重建和预测。然而，当数据集包含许多标准变量时，PCovR 用户面临新的解释问题，因为将获得许多回归权重，并且因为某些标准可能与预测变量无关。因此，我们提出了 PCovR2，它通过将标准减少到几个组件来扩展 PCovR。这些标准分量是基于预测器分量来预测的。PCovR2 加权参数可以再次灵活地用于专注于预测器和标准的重建，或过滤掉相关的预测器组件和可预测的标准组件。我们将 PCovR2 与其他两种基于偏最小二乘法 (PLS) 和主成分回归 (PCR) 的方法进行比较，这两种方法也减少了标准，因此被称为 PLS2 和 PCR2。通过模拟示例，我们表明，当旨在恢复所有相关的预测器组件和可预测的标准组件时，PCovR2 的性能优于 PLS2 和 PCR2。此外，我们进行了一项模拟研究，以评估 PCovR2、PLS2 和 PCR2 在发现 (1) 所有基础组件和 (2) 相关预测变量和可预测标准组件的子集方面的成功程度。最后，我们通过经验数据来说明 PCovR2 的使用。