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isdsapress:books:isdsa2019:isdsa2019c11 [2020/05/04 02:02] (current)
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 +**More Accurate Estimators of Multiple
 +Correlation Coefficient?​
 +**
  
 +Bingjiang Li and Lu Peng\\
 +Nanjing University of Posts and Telecommunications,​ China\\
 +1218084111@njupt.edu.cn
 +
 +Kentaro Hayashi\\
 +University of Hawaii at Manoa, USA
 +
 +Ke-Hai Yuan\\
 +University of Notre Dame, USA
 +
 +**Abstract**. The squared multiple correlation ($R^2$) is commonly used to measure ​
 +how well the outcome variable is linearly related to a set of predictors. Unfortunately,​ $R^2$ is biased for its population counterpart ($\rho^2$), and the bias increases as the number of variables ($p$) increases. Efforts have been made to modify $R^2$. The most notable result is the adjusted $R^2$ ($R_{adj}^2$),​ which incorporates the influence of the sample size ($N$) and $p$. However, $R_{adj}^2$ is still biased, and an unbiased estimator of $\rho^2$ does not exist. Using empirical modeling and statistical learning, this article develops new formulas for estimating the population $\rho$. The development involves obtaining formulas for the empirical bias of $R$ via Monte Carlo simulation across many conditions. Values of the empirical bias are then predicted by functions of $N$, $p$ and the observed values of the $R$. Best-subset regression are used to identify the best predictors for the empirical bias. Improved formulas for estimating $\rho$ are obtained via a bias correction to $R$. Results of cross validation show that empirically corrected estimators contain little bias and perform better than both $R$ and $R_{adj}$ in mean squared error and variance.
 +
 +
 +**Keywords**:​ Empirical modeling • Monte Carlo simulation • Bias correction • Best-subset regression.
 +
 +**DOI**: https://​doi.org/​10.35566/​isdsa2019c11

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