Partial/Semipartial Correlation

Why are partial and semipartial correlations used?

They allow statistical control when third variables cannot be controlled experimentally, which is a common problem in observational studies.

What does a semipartial correlation represent in terms of linear relationships?

The unique contribution of one predictor to Y, controlling for another variable

In a partial correlation between X and Y controlling for Z, which variables are residualized?

Both X and Y

When would a researcher prefer a semipartial correlation over a partial correlation?

When estimating the unique effect of one predictor on an outcome in a regression context

What is a spurious correlation?

A correlation that is entirely or partly explained by a third variable instead of the relationship between the two focal variables.

What is statistical control?

A method for holding third variables constant statistically by removing their linear effects from variables of interest.

What is a residual?

The difference between the observed and the predicted value.

What does residualizing X with respect to Z mean?

Regress X on Z and retain the residuals; these residuals represent the part of X that cannot be explained by Z.

How should regression residuals be interpreted?

They denote the part of a variable that cannot be explained by another variable used to predict it.

What is a key property of residuals after regressing X on Z?

The residualized part of X is uncorrelated with Z.

What are the two cases that must be distinguished?

Partial correlation and semipartial correlation.

What is a partial correlation?

The correlation between two variables after the effect of a third variable has been removed from both.

How is a partial correlation obtained with residuals?

Regress both focal variables on the control variable and correlate the two resulting residual variables.

What is the main memory rule for partial correlation?

Partial correlation = correlation between two residualized variables.

What does r_XY·Z mean?

The correlation between X and Y when Z is partialled out from both X and Y.

What is the main purpose of partial correlation?

To remove the effect of one variable on two others when assessing their correlation.

How can partial correlation be used to detect confounding variables?

If two variables are no longer correlated after controlling a third variable, that third variable may explain the original correlation.

What is a confounding variable?

A third variable associated with both focal variables that may generate or inflate their observed correlation.

Can partial correlation prove a causal relationship?

No. It may help detect possible causal explanations, but statistical control alone does not establish causality.

What is a first-order partial correlation?

A partial correlation in which one variable is controlled.

What is a higher-order partial correlation?

A partial correlation in which two or more variables are controlled.

What is the first-order partial-correlation formula?

r_XY·Z = (r_XY − r_XZ·r_YZ) / sqrt[(1 − r_XZ²)(1 − r_YZ²)].

What is in the numerator of the partial-correlation formula?

The correlation between X and Y minus the product of their correlations with Z.

What is the role of the denominator in the partial-correlation formula?

It standardizes the remaining association by the residual standard deviations of X and Y after controlling Z.

What is the general residual-based definition of partial correlation?

The partial correlation is the bivariate correlation between two residualized variables.

What does the squared partial correlation represent?

The proportion of variance in one focal variable not explained by the control variable that is explained by the other focal variable.

What is a semipartial correlation?

The correlation between one original variable and the part of another variable that does not overlap with the control variable.

How is a semipartial correlation obtained with residuals?

Obtain the residuals of one focal variable after predicting it from the control variable, then correlate those residuals with the other variable in its original form.

What is the main memory rule for semipartial correlation?

Semipartial correlation = correlation between one residualized variable and one raw variable.

Why is it called semipartial correlation?

Because the control variable is partialled out from only one of the two focal variables rather than from both.

What is the main purpose of semipartial correlation?

To express the specific or unique contribution of a predictor in explaining a dependent variable.

What question does semipartial correlation answer?

What is the unique contribution of this predictor, above and beyond the other predictors, in terms of explained variance?

What does the specific or unique part of a predictor mean?

The portion of a predictor that is not shared with other predictor variables.

What is incremental variance?

Variance contributed to Y by a predictor above and beyond the variance explained by one or more other predictors.

How is semipartial correlation related to multiple linear regression?

It has a close mathematical relationship with regression weights and expresses the unique contribution of a predictor in the context of the other predictors.

What does the squared semipartial correlation represent?

The unique proportion of total variance in Y explained by a predictor above and beyond the other predictors.

How does squared semipartial correlation relate to R²?

It equals the increase in R² attributable uniquely to a predictor when that predictor is added after the other predictors.

What does r_Y(X·Z) mean?

The correlation between Y and a residualized X, where X has been adjusted for Z.

What is the general formulation of semipartial correlation?

The semipartial correlation is the correlation between a variable Y and a residual variable X, whereby X was adjusted by Z.

What is the first-order semipartial-correlation formula when Z is removed from X?

r_Y(X·Z) = (r_XY − r_XZ·r_YZ) / sqrt(1 − r_XZ²).

How does the semipartial formula differ from the partial-correlation formula?

Only the residualized variable is adjusted in the denominator; the other variable remains in its original form.

Why is the semipartial correlation often smaller than the corresponding partial correlation?

Because it is expressed relative to the total variance of the unaltered outcome rather than only its residual variance.

Does it matter whether Z is removed from X or from Y in a semipartial correlation?

Yes. The semipartial correlation changes depending on which variable is residualized.

What is a higher-order semipartial correlation?

A semipartial correlation in which more than one variable is partialled out from only one focal variable.

What is the central difference between partial and semipartial correlation?

Partial correlation removes the control variable from both focal variables; semipartial correlation removes it from only one.

Which correlation is primarily used to detect confounding?

Partial correlation.

Which correlation is primarily used to assess a predictor's unique contribution?

Semipartial correlation.

What does squared partial correlation use as its denominator?

The variance in Y that remains after the control variable has been partialled out.

What does squared semipartial correlation use as its denominator?

The total variance in Y.

Why do squared partial and squared semipartial correlations have the same numerator?

Both refer to the same unique overlap between the focal predictor and Y.

Why do squared partial and squared semipartial correlations have different denominators?

Squared partial correlation uses only residual outcome variance, whereas squared semipartial correlation uses total outcome variance.

Which squared statistic directly expresses unique explained variance in total Y?

The squared semipartial correlation.

If Z is partialled out from both X and Y, which statistic is used?

Partial correlation.

If Z is partialled out from X only and residualized X is correlated with raw Y, which statistic is used?

Semipartial correlation.

If a semipartial correlation is .30, how much total variance in Y is uniquely explained?

.30² = .09, or 9%.

If a partial correlation is .30, does this mean 9% of total Y variance is explained?

No. It means 9% of the variance remaining in Y after control is explained.

What is the safest interpretation of statistical control?

It removes modeled linear effects of measured control variables but does not guarantee that all confounding has been eliminated.