关于Pearson相关系数r与回归R²平方根的一致性及表述合理性的技术问询

Hey there, these are great questions—super common points of confusion in basic stats, so let's break them down clearly:

1. Is the Pearson correlation coefficient r equal to the square root of R-squared from regression analysis? #

• Quick take: Only when you're doing simple linear regression (one predictor, one outcome variable).
• Full breakdown:
  • For simple linear regression (predicting Y using just X), the square root of R² will have the exact same absolute value as the Pearson r between X and Y. The sign will align with the slope of X in the regression model—positive if X and Y tend to increase together, negative if one goes up when the other goes down.
  • If you're doing multiple linear regression (two or more predictors), the square root of R² is called the multiple correlation coefficient (written as R). This measures how well all predictors together predict Y, not the pairwise relationship between a single X and Y. So it’s totally different from the Pearson r between any individual predictor and Y, which only looks at those two variables in isolation.

2. Is the statement "Testing the Pearson correlation coefficient r between independent variable X and dependent variable Y via regression analysis" statistically correct? #

• It depends entirely on what kind of regression you’re using:
  • Simple linear regression: Sort of, but it’s not the most precise wording. The significance test for the slope of X in simple linear regression is mathematically identical to the significance test for the Pearson r between X and Y—both are checking if there’s a non-zero linear relationship between the two variables. A better way to phrase it would be "Testing the linear relationship between X and Y using simple linear regression" or "Testing the significance of the Pearson r between X and Y".
  • Multiple linear regression: No, this is incorrect. When you run a multiple regression, the test for X is checking if X contributes to predicting Y after accounting for all other predictors in the model. This has nothing to do with testing the raw Pearson r between X and Y, which doesn’t consider any other variables. Using regression here to "test Pearson r" would be misleading and statistically inaccurate.

内容的提问来源于stack exchange，提问作者Vyas