Can the coefficient be negative?

Yes; its sign describes the direction of the residual predictor's association with Y.

Can several controls be removed?

Yes, by residualizing the predictor against their joint linear model.

Does swapping which variable is residualized preserve the result?

Not generally; specify which side is adjusted.

AI Glossary / Statistics and probability / Advanced

Semi-partial correlation ( r 0(1.2) )

Definition

A semi-partial correlation measures the association between an outcome and the residual part of a predictor after specified other predictors have been linearly accounted for in that predictor. The outcome remains unadjusted. It is also called a part correlation. In the supplied notation, variable 0 is the outcome, while variable 1 is adjusted for variable 2.

What it means

Suppose an analyst wants to know how a predictor relates to an outcome beyond information shared with another predictor. Semi-partial correlation isolates the first predictor's residual variation, then relates it to the original outcome.

In ordinary least-squares regression with an intercept, its square gives the increase in ordinary R-squared when that single predictor is added after the specified controls, using the same observations. This is a contribution to model fit, not proof of a causal contribution.

How it works

Regress predictor X on control Z, including an intercept, and retain the residuals. Calculate the Pearson correlation between those residuals and outcome Y. With several controls, regress X on all of them together.

For one control, the coefficient is [r(Y,X) - r(Y,Z)r(X,Z)] divided by sqrt[1 - r(X,Z)^2]. The denominator must be nonzero: a predictor perfectly explained by the controls has no residual variation to correlate.

Partial correlation is different. It removes the controls from both X and Y before correlating their residuals. The two procedures use different outcome variation in their denominators.

Three practical examples

1. Service analysis: examine whether a workflow measure adds information about response time beyond workload, using a specified linear model and consistently measured observations.

2. AI evaluation: assess the additional linear contribution of an evaluation metric to an outcome after another metric is already included. This does not establish that changing the metric would change the outcome.

3. Workforce research: quantify the extra fitted variance associated with training hours beyond prior experience, while separately considering selection, measurement and other potential confounding.

Think of it this way

Imagine first setting aside the part of a predictor that a control variable can linearly predict. Then compare what remains with the original outcome. Partial correlation takes an additional step by also setting aside the control's prediction of the outcome.

Worked example

Consider fictional correlation summaries for an outcome Y, predictor X and control Z. Suppose r(Y,X) = 0.6, r(Y,Z) = 0.5 and r(X,Z) = 0.4. These correlations form a valid positive-definite correlation matrix; they are stipulated for this arithmetic example.

The numerator of the semi-partial formula is 0.6 - 0.5 × 0.4 = 0.4. The denominator is sqrt(1 - 0.4^2) = sqrt(0.84), approximately 0.9165.

The semi-partial correlation is therefore 0.4/sqrt(0.84), approximately 0.4364. Squaring gives 0.16/0.84 = 4/21, approximately 0.190476.

To interpret that square, first fit the ordinary least-squares model predicting Y from Z with an intercept. Its R-squared is 0.5^2 = 0.25. Adding X increases R-squared by approximately 0.190476, producing a full-model R-squared of approximately 0.440476.

Thus, the additional predictor accounts for about 19.05% of the total variation in Y beyond Z in this model. Equivalently, explained variance rises from 25% to about 44.05%, a difference of about 19.05 percentage points. It would be incorrect to call 0.4364 itself the percentage of variance added.

Now calculate the partial correlation. Its numerator is still 0.4, but its denominator is sqrt[(1 - 0.5^2)(1 - 0.4^2)] = sqrt(0.63). The partial correlation is approximately 0.5040, and its square is 0.16/0.63, approximately 0.253968.

That square describes the fraction of Y's remaining variation, after Z, associated with adding X. Since the reduced model leaves 75% of Y's total variation unexplained, 0.253968 × 0.75 = 0.190476. Both calculations describe the same improvement using different denominators.

The larger partial coefficient does not mean it reveals a stronger causal effect. It reflects the residual outcome scale. Neither calculation identifies whether workload, motivation or an omitted variable causes the observed relationships.

All model comparisons here use the same observations, an intercept and ordinary R-squared. Changing cases because of missing values can invalidate a direct comparison. The identity should not be transferred without justification to adjusted R-squared, a logistic pseudo-R-squared or a predictive score measured on new data.

Illustrative case study

Fictional Cedar Evaluation compares the worked results. One analyst reports that X explains 43.64% of outcome variance, while another reports 25.40%. Neither statement correctly identifies the additional share of total outcome variation.

The reviewer separates the coefficient, its square and the denominator. Cedar reports approximately 19.05% additional total variation accounted for, conditional on the specified linear model and control. It then assesses uncertainty and performance on new observations separately. This invented case illustrates interpretation, not evidence that a particular feature should be deployed or changed.

Common misconceptions and mistakes

“Semi-partial and partial correlation are interchangeable.” One residualizes only the predictor; the other residualizes both variables.

“The coefficient is a variance percentage.” The squared coefficient has the stated variance interpretation.

“Unique means causal.” It means additional fit relative to the included controls, not isolation of a causal mechanism.

“Controlling always weakens a relationship.” Suppression can produce a different pattern; the result depends on the correlation structure.

Business use

Name the outcome, predictor and control set explicitly. Report whether the coefficient is semi-partial or partial and explain the denominator behind any variance percentage. Verify that compared models use identical cases and compatible definitions.

For AI-assisted analysis, distinguish incremental fit from out-of-sample predictive value. A predictor may add little beyond correlated alternatives without being unimportant operationally. Conversely, a sizable fitted contribution can be unstable or reflect leakage. Inspect measurement quality, collinearity and the rationale for controls before turning the statistic into a feature-ranking or intervention decision.

Check your understanding

Why is the semi-partial denominator sqrt(0.84)?

Answer: The residual predictor variance is proportional to one minus 0.4 squared.

Why does full-model R-squared reach about 0.440476?

Answer: It adds 0.190476 to the reduced value 0.25.

Why is partial squared correlation larger here?

Answer: It divides the improvement by the remaining 75% of outcome variation.

Related terms

Partial correlation
Residual
Multiple regression
R-squared
Collinearity
incremental validity
suppression
control variable

Learn next

Keep the outcome correlations fixed and set r(X,Z) to zero. Recalculate both coefficients and explain why removing nothing from X does not imply that residualizing Y has no effect.

Sources

David C. Howell, Statistical Methods for Psychology glossary, semi-partial correlation: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html

Effectsize package documentation, Semi-Partial (Part) Correlation Squared, model comparison interpretation: https://search.r-project.org/CRAN/refmans/effectsize/html/r2_semipartial.html

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