Multiple Correlation coefficient

Multiple correlation was used in multiple linear regression to find the relationship between the dependent variable and the combined effect of independent variables on dependent variables in the model.

• It is nonnegative and varies from 0 to 1.
• When the multiple correlation coefficient was 1, then the relation was perfect, and regression residuals were zero.
• The multiple correlation coefficient was always greater than equal to any other combination variables 'simple correlation' in the model.
• If multiple correlations were zero, the dependent variable was uncorrelated with the variables in the model, and multiple regression failed to estimate the dependent variable when independent variables were known.

The formula and calculation procedure are as follows to calculate multiple correlation and R-square: W is the determinant of the correlation matrix of all the factors(dependent and independent) in the model. W11 is the cofactor of the dependent variable in the correlation matrix. y is the dependent variable, and x and z are independent variables in the model

In the above formula, R-square is the coefficient of determination used as a goodness of fit of the model to explain the variance in the model, R is a multiple correlation coefficient.

The following was the example data set, y is the dependent variable and x, z, a, and b were independent variables.

In the following correlation table, correlation between the variables is calculated and provided. Simple correlation was r(x, y). In this example, N=10 and k=5.

Determinant of the correlation matrix,

w  = 0.264723

The cofactor of y,

Multiple correlation coefficient, R(y, xzab) = sqrt(1-w/wyy)

                   R(y, xzab)   = sqrt(1-(0.264723/0.5267528))

                                     =  0.7052968

                    R-square =  0.4974436

Adjusted R-Square      = 1-(1-R-square)*(N-1)/(N-k)

= 1-(1-0.4974436)*(9/5)

  = 0.09539848

Multiple correlation coefficients can help understand the association between the combined effect of the dependent and independent variables. It provides some insights into the model as mentioned above. We can calculate the R-square and adjusted R-square measures using multiple correlation coefficients. And when you calculate the correlation matrix, it provides the idea of the multicollinearity between the variables. In the above example, even though the correlation of the model was 0.7, the performance of the model-adjusted R-square was 0.09. Even though the correlation was high, there was no guarantee that the model's performance would also be high. And need to be a better fit for the model.