Multiple Correlation

In the problems of multiple correlations we are dealing with a situation that includes three of more variables. For e.g., we may consider the association between the yield of rice per acre and both the amount of rainfall & the average daily temperature. We are trying to make estimates of the value of one of the variables based on the values of all others. The variables whose value we are trying to estimate are termed as the dependent variable and the other variables. Thus in the age, height and weight problem, if we are trying to estimate men's weight we might denote.

X₁---> weight in lbs.

X₂--> height in inches.

X₃---> age in years.

The coefficient of multiple correlations can be expressed in terms of r ₁₂, r ₁₃ and r ₂₃ follows:

r_1.23 = r²₁₂ + r²₁₃ - 2 r₁₂ r₁₃ r₂₃ / 1 - r²₂₃

r_2.13 = r²₁₂ + r²₂₃ - 2 r₁₂ r₁₃ r₂₃ / 1 - r²₁₃

r_3.12 = r²₁₃ + r²₂₃ - 2 r₁₂ r₁₃ r₂₃ / 1 - r²₁₂

It should be noted that R_1.23 is the same as R_1.32.

An alternate formula for obtaining the value of R_1.23 is as follows;

r_1.23 = r²₁₂ + r_13.2 (1 - r²₁₃)

r² _1.12 = r²₁₂ +r²_13.2 (1 - r²₁₃)

Or similarly

r_1.24 = r²₁₂ + r²₁₄ - 2 r₁₂ r₁₄ r₂₄ / 1 - r²₂₄

Or r_1.24 = r²₁₂ + r²_14.2 (1 - r²₁₂)

And,  r_1.34 = r²₁₃ + r²₁₄ - 2 r₁₃ r₁₄ r₃₄ / 1 - r²₃₄

Or R_1.34 = r²₁₂ + r²_14.2 (1 - r²₁₃)

To determine a multiple coefficient with three independent variables, the following formula shall be used:

r_12.34 = 1 - (1 - r²₁₄) (1 - r²_13.4) (1 - r²_12.34)

Illustration:-the following zero-order correlation coefficients are given

r₁₂ = 0.98, r13 = 0.44 and r₂₃ = 0.54.

r_1.23 = r²₁₂ + r²₁₃ - 2 r₁₂ r₁₃ r₂₃ / 1 - r²₂₃

Substituting the given values

r_1.23 = (.98)² + (.44)² - 2 (.98) (.44) (.54) / 1 - (.54)²

= .9604 +.1936 - .4657 / .7084 = 0.986.