University of Alberta, Canada The Chinese University of Hong Kong, Shatin, Hong Kong, The People's Republic of China.
Br J Math Stat Psychol. 2011 Nov;64(3):367-87. doi: 10.1348/2044-8317.002007. Epub 2010 Dec 6.
The standard Pearson correlation coefficient, r, is a biased estimator of the population correlation coefficient, ρ(XY) , when predictor X and criterion Y are indirectly range-restricted by a third variable Z (or S). Two correction algorithms, Thorndike's (1949) Case III, and Schmidt, Oh, and Le's (2006) Case IV, have been proposed to correct for the bias. However, to our knowledge, the two algorithms did not provide a procedure to estimate the associated standard error and confidence intervals. This paper suggests using the bootstrap procedure as an alternative. Two Monte Carlo simulations were conducted to systematically evaluate the empirical performance of the proposed bootstrap procedure. The results indicated that the bootstrap standard error and confidence intervals were generally accurate across simulation conditions (e.g., selection ratio, sample size). The proposed bootstrap procedure can provide a useful alternative for the estimation of the standard error and confidence intervals for the correlation corrected for indirect range restriction.
标准的皮尔逊相关系数 r 是有偏估计量，当预测变量 X 和准则变量 Y 受到第三个变量 Z（或 S）的间接范围限制时，它是总体相关系数 ρ(XY) 的有偏估计量。已经提出了两种校正算法，即 Thorndike（1949 年）的案例 III 和 Schmidt、Oh 和 Le（2006 年）的案例 IV，以校正偏差。然而，据我们所知，这两种算法并没有提供估计相关系数的标准误差和置信区间的程序。本文建议使用自举程序作为替代方法。进行了两项蒙特卡罗模拟，以系统地评估所提出的自举程序的经验性能。结果表明，自举标准误差和置信区间在模拟条件下（例如，选择率、样本大小）通常是准确的。所提出的自举程序可以为间接范围限制校正的相关系数的标准误差和置信区间的估计提供有用的替代方法。
