Semipartial Correlation Coefficient

A squared, semipartial correlation coefficient can be used in connection with multiple regression analysis to measure the strength of the association between the dependent and an independent variable, taking into account the relationships among all the variables. A squared semipartial correlation coefficient is also called a squared part correlation. To illustrate the squared semipartial correlation coefficient, consider data that include final mathematics grades (MA), student perception of teacher's academic support (TAS) in mathematics class, and positive affect (PA) in mathematics class. The sample size is N = 200: A multiple regression model for these variables is

where α denotes the intercept; β1 and β2 denote the regression coefficients (slopes) for PA and TAS, respectively; and ∊ denotes the residual. The sample squared multiple correlation coefficient for the model is R22 =: 169; where the subscript 2 indicates that there are two variables in the model and .169 means that 16.9% of the variance in MA is associated with the joint variability in PA and TAS. Suppose the researcher wants to know how much variance in MA is uniquely associated with TAS. The simple linear regression model

has R21 =: 121 and indicates that 12.1% of the variance in mathematics grades is associated with variability in PA. So starting with PA in the model, 12.1% of the total variance in mathematics grades is associated with variability in PA. When TAS is added to the model, 16.9% of the total variance in mathematics grades is associated with joint variability in PA and TAS. Consequently, 4.8% of the total variance in mathematics grades is uniquely associated with variability in TAS. The squared semipartial correlation coefficient for TAS is simply

where ΔR2 is read change in R2 and measures the strength of the association between MA and TAS taking into account the relationships among all three variables. The squared semipartial correlation coefficient for PA can also be computed. The squared correlation coefficient for the model [Page 1335]

is R21 = .107, and the squared semipartial correlation coefficient for PA is

When the goal is to determine how strongly TAS and MA are associated taking into account the relationships among all three variables, the roles of TAS and MA can be interchanged in Equations 1 and 2:

and Δ = .046, so 4.6% of the variance in TAS is uniquely associated with MA. The choice between reporting the 4.8% associated with Equation 1 and the 4.6% associated with Equation 4 should be based on whether MA or TAS is substantively considered the dependent variable.

A semipartial correlation can also be expressed as a correlation between the dependent variable and a residualized version of an independent variable. Let the predicted value for the regression of TAS on PA be denoted by

. The predicted value is

where a is the sample intercept and b is the sample slope for PA. The residual is TAS −

, which is uncorrelated with PA and can be interpreted as the part of TAS that is not associated with PA. The semipartial correlation between MA and TAS is the correlation between MA and TAS −, which is often written as rMA(TAS.PA), and can be interpreted as the correlation between MA and the part of TAS that is not associated with PA. In a similar fashion, rTAS(MA. PA) can be formulated based on