Suppose that a data analyst wishes to report the results of a least squares linear regression only if the overall null hypothesis, $H_0^{1:p}: β_1= β_2 = \ldots = β_p=0$, is rejected. This practice, which we refer to as F-screening (since the overall null hypothesis is typically tested using an $F$-statistic), is in fact common across a number of applied fields. Unfortunately, it poses a problem: standard guarantees for the inferential outputs of linear regression, such as Type 1 error control of hypothesis tests and nominal coverage of confidence intervals, hold unconditionally, but fail to hold conditional on rejection of the overall null hypothesis. In this paper, we develop an inferential toolbox for the coefficients in a least squares model that are valid conditional on rejection of the overall null hypothesis. We develop selective p-values that lead to tests that are consistent and control the selective Type 1 error, i.e., the Type 1 error conditional on having rejected the overall null hypothesis. Furthermore, they can be computed without access to the raw data, i.e., using only the standard outputs of a least squares linear regression, and therefore are suitable for use in a retrospective analysis of a published study. We also develop confidence intervals that attain nominal selective coverage, and point estimates that account for having rejected the overall null hypothesis. We derive an expression for the Fisher information about the coefficients resulting from the proposed approach, and compare this to the Fisher information that results from an alternative approach that relies on sample splitting. We investigate the proposed approach in simulation and via re-analysis of two datasets from the biomedical literature.

翻译：假设数据分析师仅当整体原假设$H_0^{1:p}: β_1= β_2 = \ldots = β_p=0$被拒绝时，才希望报告最小二乘线性回归的结果。这种做法（我们称为F-筛选，因为整体原假设通常使用$F$-统计量进行检验）实际上在许多应用领域中十分常见。然而，这带来一个问题：线性回归推断输出的标准保证（如假设检验的第一类错误控制和置信区间的名义覆盖率）在无条件情况下成立，但在以拒绝整体原假设为条件时却不再成立。本文针对最小二乘模型中的系数开发了一套推断工具，这些工具在条件于整体原假设被拒绝时仍然有效。我们构建了选择性p值，由此得到的检验具有一致性，并能控制选择性第一类错误（即条件于已拒绝整体原假设的第一类错误）。此外，这些p值无需访问原始数据即可计算（仅需使用最小二乘线性回归的标准输出），因此适用于对已发表研究进行回顾性分析。我们还开发了能达到名义选择性覆盖率的置信区间，以及考虑了整体原假设被拒绝的的点估计。我们推导了由所提方法得到的系数费希尔信息的表达式，并将其与依赖样本分割的替代方法所得费希尔信息进行比较。通过仿真和来自生物医学文献中两个数据集的重新分析，我们对所提方法进行了研究。