I had a question regarding R^2 values. I remember reading that you can get R^2 values from correlation coefficients from linear models with a single independent variable. I’m doing question 20 on page 388 (Chapter 10 – Multiple Linear Models).
The Multiple R^2 value is given as 0.36. The answer and interpretation of the 0.36 is -à correlation between predicted and actual values of the dependent variable is 0.60 (Square root of 0.36).
Since the (Correlation Coefficient)^2 does NOT equal the Coefficient of determination in multiple linear regression, how can this be the answer?
In a simple (one independent variable) regression, there is one correlation coefficient (ρ) between the independent variable and the dependent variable, and R² = ρ².
In a multiple (more than one independent variable), there are several correlation coefficients between independent variables and the dependent variable, so R² isn’t the square of any of those individual correlation coefficients. However, there is only one correlation coefficient (ρ) between the predicted value of the dependent variable and the actual value of the dependent variable, and R² = ρ².
R^2 in multiple linear regression (and simple lienar regression for that matter) is the percent of total variation that is explained by the variation of the independent variables.