Multiple Correlation Coefficient

The multiple correlation coefficient is a numerical measure of how well a linear regression model fits a set of data $Y_i$.

Technically speaking, it is the simple correlation coefficient for dependent variable values $Y_i$ and the predicted values $\hat Y_i$ that are obtained with the least squares multiple linear regression

Mathematically,

$$R_{mult} =\frac{n \sum_{i=1}^n hat Y_i Y_i - \left(\sum_{i=1}^n \hat Y_i \right) \left(\sum_{i=1}^n Y_i \right) }{\sqrt{n \sum_{i=1}^n \hat Y_i^2 - \left( \sum_{i=1}^n \hat Y_i \right)^2} \sqrt{n \sum_{i=1}^n Y_i^2 - \left( \sum_{i=1}^n Y_i \right)^2} }$$

but it can also be computed $\sqrt{\frac{SSR}{SST}}$, where $SSR$ is the sum of regression squares and $SST$ is the total sum of squares, because that way is a bit simpler by following some (intensive) matrix calculations .

What are the limits of multiple correlation coefficient?

For the case of a simple linear regression, the correlation coefficient may range from -1 to 1. For the case of the multiple correlation coefficient, it ranges from 0 to 1.

Other associated calculators

If you need to estimate the regression model instead, you can use this multiple linear regression calculator .