For three random variables X₁ , X₂ and X₃, the pairwise correlation coefficients are r₁₂ = r₁₃ = r₂₃ = r. The multiple correlation coefficient \(\rm R_{1.2.3}^2\) is

Key Points 

• The formula for the multiple correlation coefficient (R) = \(\rm \frac{2r^2}{1+r}\)
• In this formula, r represents the pairwise correlation coefficient between any two variables, such as r12 for X1 and X2, r13 for X1 and X3, and r23 for X2 and X3. 
• The multiple correlation coefficient measures the strength of the linear relationship between the dependent variable (Y) and the set of independent variables (X1, X2, X3) collectively. It indicates how well the independent variables, taken together, predict the variation in the dependent variable.
• The value of R ranges from 0 to 1, where 0 indicates no linear relationship and 1 indicates a perfect linear relationship between the variables. A higher value of R suggests a stronger correlation and implies that the independent variables jointly explain more of the variance in the dependent variable.
• The formula calculates the multiple correlation coefficient based on the pairwise correlation coefficients. It shows the relationship between the individual pairwise correlations (r) and the overall multiple correlations (R), highlighting the combined effect of the variables on the dependent variable. 

Additional Information

• The multiple correlation coefficient is commonly employed in regression analysis. It helps evaluate the overall goodness of fit of the regression model and assess the strength of the relationship between the independent variables and the dependent variable when multiple predictors are involved.
• While the formula assumes equal pairwise correlations (r), in practice, the actual values of the pairwise correlations can vary, and more advanced methods may be required to estimate the multiple correlation coefficient accurately.