Inferring linear relationships lies at the heart of many empirical investigations. A measure of linear dependence should correctly evaluate the strength of the relationship as well as qualify whether it is meaningful for the population. Pearson's correlation coefficient (PCC), the \textit{de-facto} measure for bivariate relationships, is known to lack in both regards. The estimated strength $r$ maybe wrong due to limited sample size, and nonnormality of data. In the context of statistical significance testing, erroneous interpretation of a $p$-value as posterior probability leads to Type I errors -- a general issue with significance testing that extends to PCC. Such errors are exacerbated when testing multiple hypotheses simultaneously. To tackle these issues, we propose a machine-learning-based predictive data calibration method which essentially conditions the data samples on the expected linear relationship. Calculating PCC using calibrated data yields a calibrated $p$-value that can be interpreted as posterior probability together with a calibrated $r$ estimate, a desired outcome not provided by other methods. Furthermore, the ensuing independent interpretation of each test might eliminate the need for multiple testing correction. We provide empirical evidence favouring the proposed method using several simulations and application to real-world data.
翻译:推断线性关系是许多经验性调查的核心。 线性依赖度的尺度应该正确评估这种关系的强度,并证明这种关系是否对人口有意义。 Pearson的双轨关系相关系数(PCC),即双轨关系中的基于机器学习的预测数据校准方法(PCC),在这两个方面已知都缺乏。 估计的强度美元可能是错误的,因为抽样规模有限,数据不具有正常性。 在统计意义测试方面,错误地解释美元价值作为后继概率导致I类错误 -- -- 这个问题具有重大意义的测试,延伸至PCC。在同时测试多个假设时,这种错误会加剧。为了解决这些问题,我们建议一种基于机器学习的预测数据校准方法,基本上为预期线性关系的数据样本提供条件。 使用校准数据计算得出的校准美元价值,可以被解释为事后概率,加上校准的美元估计数,这是其他方法没有提供的预期结果。 此外,随后对每次测试进行的独立解释可能会消除对多重测试的需要。 我们提供实验性证据, 使用模拟方法来模拟模拟数据。