Principal covariate regression (PCOVR) is a method for regressing a set of criterion variables with respect to a set of predictor variables when the latter are many in number and/or collinear. This is done by extracting a limited number of components that simultaneously synthesize the predictor variables and predict the criterion ones. So far, no procedure has been offered for estimating statistical uncertainties of the obtained PCOVR parameter estimates. The present paper shows how this goal can be achieved, conditionally on the model specification, by means of the bootstrap approach. Four strategies for estimating bootstrap confidence intervals are derived and their statistical behaviour in terms of coverage is assessed by means of a simulation experiment. Such strategies are distinguished by the use of the varimax and quartimin procedures and by the use of Procrustes rotations of bootstrap solutions towards the sample solution. In general, the four strategies showed appropriate statistical behaviour, with coverage tending to the desired level for increasing sample sizes. The main exception involved strategies based on the quartimin procedure in cases characterized by complex underlying structures of the components. The appropriateness of the statistical behaviour was higher when the proper number of components were extracted.

中文翻译：

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主协变量回归的 Bootstrap 置信区间

主协变量回归 (PCOVR) 是一种用于在一组预测变量数量众多和/或共线时对一组标准变量进行回归的方法。这是通过提取有限数量的组件来完成的，这些组件同时合成预测变量和预测标准变量。到目前为止，还没有提供用于估计获得的 PCOVR 参数估计的统计不确定性的程序。本文展示了如何通过引导方法在模型规范的条件下实现这一目标。推导出四种用于估计 bootstrap 置信区间的策略，并通过模拟实验评估它们在覆盖方面的统计行为。这种策略的特点是使用 varimax 和 quartimin 程序以及使用 Procrustes 旋转引导解决方案到样本解决方案。总的来说，这四种策略显示出适当的统计行为，覆盖率趋向于增加样本量所需的水平。主要的例外涉及基于组件复杂底层结构的情况下基于 quartimin 程序的策略。当提取出适当数量的成分时，统计行为的适当性更高。主要的例外涉及基于组件复杂底层结构的情况下基于 quartimin 程序的策略。当提取出适当数量的成分时，统计行为的适当性更高。主要的例外涉及基于组件复杂底层结构的情况下基于 quartimin 程序的策略。当提取出适当数量的成分时，统计行为的适当性更高。