Partial and Semipartial Correlations

Why Residualizing Changes the Question

Two variables can be associated partly because they share variation with other variables. A partial correlation asks how strongly they are associated after removing the linear contribution of a specified set of controls from both variables. A semipartial correlation removes that contribution from one variable only. Because these procedures use different denominators, their numerical values need not agree.

The distinction matters in regression. A partial correlation describes the association between the two variables after accounting for controls on both sides. A one-sided semipartial correlation retains the original scale of the variable that was not residualized. The direction of residualization therefore needs to be stated, not left implicit.

Definitions Using Linear Projection Residuals

Let \(Y\) be a response, \(X\) a second variable, and \(Z\) a vector of control variables. Assume these variables have finite second moments, that the covariance matrix of \(Z\) is nonsingular, and that the residual variances defined below are positive. Include an intercept in each regression, or equivalently work with centered variables. Write \(Y\cdot Z\) for the residual from the population linear regression of \(Y\) on \(Z\), and \(X\cdot Z\) for the residual from the population linear regression of \(X\) on \(Z\).

For centered variables, the projection of \(Y\) on \(Z\) has coefficient vector \(\Sigma_{ZZ}^{-1}\operatorname{Cov}(Z,Y)\), where \(\Sigma_{ZZ}=\operatorname{Cov}(Z,Z)\). Thus \(Y\cdot Z=Y-\operatorname{Cov}(Y,Z)\Sigma_{ZZ}^{-1}Z\); the corresponding expression defines \(X\cdot Z\). Each residual is uncorrelated with every linear combination of the controls. This orthogonality is the key fact behind the relationships below.

In multiple regression, “semipartial correlation” or “part correlation” often refers specifically to residualizing a predictor and correlating it with the unadjusted response. The response-side version is also a well-defined one-sided residual correlation, but it answers a different question. Giving the formula avoids ambiguity across conventions.

How the Three Correlations Are Related

The residualized response \(Y\cdot Z\) is uncorrelated with the part of \(X\) predicted by \(Z\). Consequently, its covariance with raw \(X\) is the same as its covariance with \(X\cdot Z\). Their correlations can still differ, because raw \(X\) has more variance than its residual unless the controls explain none of it.

Proof. Let \(\widehat X_Z\) denote the linear projection of \(X\) on \(Z\), so \(X=\widehat X_Z+(X\cdot Z)\). Since \(Y\cdot Z\) is orthogonal to the controls, it is orthogonal to \(\widehat X_Z\). Therefore, \[ \operatorname{Cov}(Y\cdot Z,X) =\operatorname{Cov}(Y\cdot Z,\widehat X_Z+X\cdot Z) =\operatorname{Cov}(Y\cdot Z,X\cdot Z). \] Divide this common covariance first by \(\sqrt{\operatorname{Var}(Y\cdot Z)\operatorname{Var}(X)}\), then by \(\sqrt{\operatorname{Var}(Y\cdot Z)\operatorname{Var}(X\cdot Z)}\). The first correlation is the second multiplied by \(\sqrt{\operatorname{Var}(X\cdot Z)/\operatorname{Var}(X)}\). Interchanging the roles of \(X\) and \(Y\) gives the other identity. \(\square\)

Each square-root factor is at most one: a linear projection and its residual are orthogonal, so \(\operatorname{Var}(X)=\operatorname{Var}(\widehat X_Z)+\operatorname{Var}(X\cdot Z)\), and likewise for \(Y\). These factors change magnitudes, not signs. The formulas also show why the two one-sided correlations generally differ from one another: they retain different variables in their original, unresidualized form.

Connection to Multiple Regression

In a sample, the same definitions use ordinary least-squares residuals from regressions that include an intercept. Correlations are computed from the resulting centered vectors. In a regression of \(Y\) on controls \(Z\) and a single additional predictor \(X\), the predictor-side semipartial correlation has a particularly direct interpretation: its square is the increase in \(R^2\) when \(X\) is added to the controls. The partial correlation instead measures the fraction of the response variation left after fitting the controls that is accounted for by the added predictor.

Derivation. After removing the controls, the additional least-squares slope for \(X\cdot Z\) is \(S_{Y\cdot Z,X\cdot Z}/S_{X\cdot Z,X\cdot Z}\). The reduction in residual sum of squares is therefore \(S_{Y\cdot Z,X\cdot Z}^2/S_{X\cdot Z,X\cdot Z}\). Dividing that reduction by the residual sum of squares from the controls-only model gives \(r_{\text{partial}}^2\). Dividing it by the total sum of squares \(S_{YY}\) gives the increase in \(R^2\); that expression is exactly \(r_{\text{semi},X}^2\). \(\square\)

This derivation also explains why the response-side semipartial correlation does not, in general, have a squared value equal to the increase in \(R^2\). Its denominator contains the total variation of raw \(X\), rather than the total variation of \(Y\). The predictor-side version is the one whose square corresponds to the added predictor’s incremental \(R^2\).

Common Pitfalls and Why the Distinction Matters

• Calling every residual correlation partial. A partial correlation residualizes both variables. If only one is residualized, specify the direction and call it semipartial.
• Assuming partial and semipartial correlations have the same magnitude. Their covariance numerators agree in the corresponding formulas, but their variance denominators differ.
• Interpreting a squared correlation without checking its definition. The squared predictor-side semipartial correlation equals the increase in \(R^2\) in the one-predictor addition described above. The squared partial correlation uses the controls-only residual variation as its reference.
• Treating either correlation as a regression coefficient. Correlations are standardized associations; regression coefficients depend on the measurement scales of the variables.

These measures are useful for different questions. Use a partial correlation when asking whether two variables remain linearly associated after removing the controls from both. Use a semipartial correlation when the unadjusted scale of one variable is substantively important. In regression reporting, state which variable was residualized and what denominator is being used; this makes the interpretation of the number precise.