Part and Partial Correlations

Multiple Regression Framework

In MLR, the goal is to predict, knowing the measurements collected on N subjects, a dependent variable Y from a set of K independent variables denoted

We denote by X the N × (K + 1) augmented matrix collecting the data for the independent variables (this matrix is called augmented because the first column is composed only of ones), and by y the N × 1 vector of observations for the dependent variable. These two matrices have the following structure:

The predicted values of the dependent variable Ŷ are collected in a vector denoted ŷ and are obtained using MLR as

The quality of the prediction is evaluated by computing the multiple coefficient of correlation, denoted R2Y.1,…,K. This coefficient is equal to the coefficient of correlation between the dependent variable (Y) and the predicted dependent variable (Ŷ).

Partial Regression Coefficient as Increment in Explained Variance

When the independent variables are pairwise orthogonal, the importance of each of them in the regression is assessed by computing the squared coefficient of[Page 737] correlation between each of the independent variables and the dependent variable. The sum of these squared coefficients of correlation is equal to the squared multiple coefficient of correlation. When the independent variables are correlated, this strategy overestimates the contribution of each variable because the variance that they share is counted several times; and therefore, the sum of the squared coefficients of correlation is not equal to the multiple coefficient of correlation anymore. In order to assess the importance of a particular independent variable, the partial regression coefficient evaluates the specific proportion of variance explained by this independent variable. This is obtained by computing the increment in the multiple coefficient of correlation obtained when the independent variable is added to the other variables.

For example, consider the data given in Table 1, where the dependent variable is to be predicted from the independent variables X and T. The prediction equation (using Equation 3) is

it gives a multiple coefficient of correlation of R2Y.XT = .9866. The coefficient of correlation between X and T is equal to rX.T = .7500, between X and Y is equal to rY.X = .8028, and between T and Y is equal to rY.T = .9890. The squared partial regression coefficient between X and Y is computed as