Abstract

Ezekiel’s adjusted R² is widely used in linear regression analysis. The present study examined the statistical properties of Ezekiel’s measure through a series of Monte Carlo simulations. Specifically, we examined the bias and root mean squared error (RMSE) of Ezekiel’s adjusted R² relative to (a) the sample R² statistic, and (b) the sample R² minus the expected value of R². Simulation design factors consisted of sample sizes (N = 50, 100, 200, 400), number of predictors (2, 3, 4, 5, 6), and population squared multiple correlations (ρ² = 0, .10, .25, .40, .60). Factorially crossing these design factors resulted in 100 simulation conditions. All populations were normal/Gaussian, and for each condition, we drew 10,000 Monte Carlo samples. Regarding systematic variation (bias), results indicated that with few exceptions, Ezekiel’s adjusted R² demonstrated the lowest bias. Regarding unsystematic variation (RMSE), the performance of Ezekiel’s measure was comparable to the other statistics, suggesting that the bias-variance tradeoff is minimal for Ezekiel’s adjusted R². Additional findings indicated that sample size-to-predictor ratios of 66.67 and greater were associated with low bias and that ratios of this magnitude were accompanied by large sample sizes (N = 200 and 400), thus suggesting that researchers using Ezekiel’s adjusted R² should aim for sample sizes of 200 or greater in order to minimize bias when estimating the population squared multiple correlation coefficient. Overall, these findings indicate that Ezekiel’s adjusted R² has desirable properties and, in addition, these findings bring needed clarity to the statistical literature on Ezekiel’s classic estimator.

摘要

Ezekiel的调整后R ²广泛用于线性回归分析。本研究通过一系列的蒙特卡洛模拟检验了以西结测量的统计特性。具体地，我们研究结的调整的偏压和均方根误差（RMSE）- [R ²相对于（a）将样品- [R ²统计量，和（b）将样品- [R ²减去的预期值- [R ²。模拟的设计因素包括样本大小（的Ñ  = 50，100，200，400），预测器（2，3，4，5，6）的数目，和人口平方多次相关（ρ ²= 0，.10，.25，.40，.60）。跨越这些设计因素导致了100个仿真条件。所有人口均为正常人/高斯人，对于每种情况，我们抽取了10,000个蒙特卡洛样本。关于系统偏差（偏差），结果表明，除少数例外，以西结的调整后的R ²表现出最低的偏差。关于非系统变异（RMSE），以西结的测度性能与其他统计数据相当，这表明对于以西结的调整后的R ²，偏差-方差的权衡是最小的。其他发现表明，样本大小与预测子之比为66.67或更大与低偏倚相关，并且此数量级的比率伴随着大样本量（N = 200和400），因此建议使用Ezekiel调整后的R ^2的研究人员应针对200或更大的样本量，以便在估计总体平方相关系数时将偏差最小化。总体而言，这些发现表明以西结的调整后的R ²具有理想的特性，此外，这些发现为以西结的经典估计量的统计文献带来了必要的清晰度。