Marginal Versus Conditional Association in Gaussian Triples

Why Marginal and Conditional Association Can Differ

Two variables can move together because they are directly associated, because both are related to a third variable, or because conditioning on a third variable changes the pattern of their variation. These possibilities are not interchangeable. The marginal association of \(X\) and \(Y\) describes their joint distribution without conditioning on \(Z\); their conditional association describes how they vary together when \(Z\) is held fixed. In a Gaussian triple, both can be summarized by correlations, but the two correlations need not agree in magnitude or sign.

This distinction matters when interpreting adjusted analyses. A positive marginal correlation does not guarantee a positive association at fixed values of a covariate, and a zero marginal correlation does not guarantee conditional independence. The formulas below make those differences explicit.

Positive definiteness ensures that the conditional variances of \(X\) and \(Y\) given \(Z\) are positive, so the conditional correlation is defined. Write \(\rho_{XZ}\) and \(\rho_{YZ}\) for the other two marginal correlations. The central formula is

The denominator is positive. Thus the sign of the conditional association is determined by \(\rho_{XY}-\rho_{XZ}\rho_{YZ}\), not by \(\rho_{XY}\) alone.

Deriving the Partial-Correlation Formula

The derivation uses residuals after linear prediction from \(Z\). Let \(\sigma_{AB}=\operatorname{Cov}(A,B)\) and \(\sigma_{AA}=\operatorname{Var}(A)\). Define

These are the parts of \(X\) and \(Y\) left after subtracting their linear predictions from \(Z\). They are jointly Gaussian with \(Z\). Each has zero covariance with \(Z\), so the Gaussian independence property implies that the pair \((R_X,R_Y)\) is independent of \(Z\). Consequently, their covariance and variances are the conditional covariance and variances of \(X\) and \(Y\) given \(Z\).

For example, expanding the covariance of the residuals gives

The same expansion with \(Y=X\) gives the stated conditional variance for \(X\); exchanging \(X\) and \(Y\) gives the other. Independence of the residual pair from \(Z\) means these are also the conditional moments at any \(Z=z\). Dividing the conditional covariance by the square root of the product of conditional variances gives the conditional correlation. Substituting \(\sigma_{AB}=\rho_{AB}\sqrt{\sigma_{AA}\sigma_{BB}}\) into that ratio and cancelling the positive marginal standard deviations yields the displayed partial-correlation formula. \(\square\)

When Does Conditioning Reverse the Sign?

Because the denominator in the partial-correlation formula is positive, the marginal and conditional associations have opposite signs precisely when \(\rho_{XY}\) and \(\rho_{XY}-\rho_{XZ}\rho_{YZ}\) have opposite signs. If the marginal correlation is positive, reversal occurs when \(\rho_{XZ}\rho_{YZ}>\rho_{XY}\). If the marginal correlation is negative, reversal occurs when \(\rho_{XZ}\rho_{YZ}<\rho_{XY}\). A zero marginal correlation is not a sign reversal, but conditioning can still reveal a nonzero conditional association.

A zero partial correlation has a particularly strong interpretation in a Gaussian model. This equivalence depends on joint Gaussianity; for general distributions, zero conditional covariance or correlation need not imply conditional independence.

To prove the result, fix \(Z=z\). The conditional distribution of \((X,Y)\) is bivariate normal, with covariance \(\operatorname{Cov}(X,Y\mid Z)=\sigma_{XY}-\sigma_{XZ}\sigma_{YZ}/\sigma_{ZZ}\) and positive conditional variances. If \(\rho_{XY\cdot Z}=0\), the formula for partial correlation implies that this conditional covariance is zero. A bivariate normal density with zero covariance and positive variances factors into the product of its two univariate normal densities. Hence \(X\) and \(Y\) are conditionally independent given \(Z=z\); this holds for almost every \(z\), so they are conditionally independent given \(Z\).

Conversely, conditional independence implies that the conditional covariance is zero for almost every \(z\), since the conditional variables have finite second moments. The conditional covariance is constant in \(z\) for a Gaussian triple, so it is zero. The positive conditional variances then make the conditional correlation, and therefore the partial correlation, zero. \(\square\)

Interpretation and a Common Pitfall

Marginal and conditional association answer different questions. The marginal correlation summarizes the joint behavior of \(X\) and \(Y\) across the distribution of \(Z\). The partial correlation summarizes their residual association after their linear relationships with \(Z\) have been removed. Neither quantity, by itself, establishes a causal relationship. In applications, conditioning is meaningful only in relation to the scientific question and the chosen variables; a mathematical change in correlation is not automatically evidence of a causal mechanism.

• Common mistake: “The marginal correlation is positive, so the adjusted association must also be positive.” This ignores the term \(\rho_{XZ}\rho_{YZ}\).
• Correction: Compute the partial-correlation numerator \(\rho_{XY}-\rho_{XZ}\rho_{YZ}\). Its sign, not the marginal sign alone, determines the conditional sign.
• Common mistake: “A zero correlation proves independence in any model.” Zero correlation implies independence for a bivariate Gaussian distribution, and zero partial correlation implies conditional independence for a nonsingular Gaussian triple. Those implications are not general facts for arbitrary distributions.

When interpreting a reported association, state explicitly whether it is marginal or conditional, and identify the variable being conditioned on. That distinction prevents a numerical adjustment from being mistaken for a contradiction.