---
title: "Partial correlation"
method_id: SM.REL.COR.PAR.PARTIAL
family: Relationship
version: 1.1.0
date_modified: 2026-06-18
canonical: https://clarus.ofrencber.com/library/partial_correlation.md
source: Clarus method library
---

# Partial correlation

> A classical statistical method in the Clarus library (Relationship family).
> Clarus selects, assumption-checks and runs this method on your data with a
> deterministic rule engine and real numerical libraries (scipy / statsmodels);
> it never generates numbers, citations or results.

**Also known as:** partial correlation coefficient; first-order/higher-order partial correlation; correlation controlling for covariates; r_xy.z.

## Hypotheses

- **Null (H0):** rho_{xy.z} = 0 (no linear association between the two variables after removing the linear influence of the control variable(s))
- **Alternative (H1):** rho_{xy.z} != 0 (a linear association remains after controlling)

## When to use it

Use this method when your goal is to:
- quantify linear association controlling for other variables
- test relationship net of confounders

## Data it expects

- **primary_pair:** 2 continuous variables of interest
- **controls:** 1+ continuous covariate(s) to hold constant
- **pairing:** paired (every case has all measures)

## Assumptions Clarus checks

- **linearity** (severity: critical; on violation: reroute)
- **normality** (gate: shapiro_wilk; severity: warning; on violation: reroute)

## Effect size reported

- **correlation r** (Cohen (1988); Cohen, Cohen, West & Aiken (2003))

## Honest limitations

- Controlling for a measured variable does NOT establish causation and does NOT remove the influence of variables you did not measure (unmeasured confounding).
- Partial correlation removes only the LINEAR influence of the control(s); if a control relates non-linearly to a target, it is incompletely partialled and the result is biased.
- If control variables overlap heavily (collinearity), the partial correlation is unstable — small data changes can move it substantially.
- Over-controlling can be misleading: holding constant a variable that lies on the causal path between the two targets (a mediator) can remove a real relationship.
- A partial r near zero means no LINEAR residual association — it does not prove independence.
- Results generalize only to the sampled population.

## Primary sources

- Yule, G. U. (1907). On the theory of correlation for any number of variables, treated by a new system of notation. Proc. R. Soc. Lond. A, 79(529), 182-193. https://doi.org/10.1098/rspa.1907.0028
- Fisher, R. A. (1924). The distribution of the partial correlation coefficient. Metron, 3, 329-332.
- Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.). Routledge.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.).
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.), Ch.7.

## How to cite this

To cite this Clarus method page (the page itself — for the method's own primary sources, see above):

**Plain text**

Clarus, "Partial correlation", version 1.1.0, Clarus method library, 2026. https://clarus.ofrencber.com/library/partial_correlation.md

**APA**

Clarus. (2026). Partial correlation (Version 1.1.0) [Statistical method, Clarus method library]. Retrieved from https://clarus.ofrencber.com/library/partial_correlation.md

**BibTeX**

```bibtex
@misc{clarus-partial-correlation,
  author       = {Clarus},
  title        = {Partial correlation},
  howpublished = {Clarus method library},
  version      = {1.1.0},
  year         = {2026},
  url          = {https://clarus.ofrencber.com/library/partial_correlation.md}
}
```

_No DOI is minted for Clarus method pages yet; this citation uses the canonical URL, version and date. Cite the primary sources above for the method's scientific provenance._

---

METHOD SM.REL.COR.PAR.PARTIAL · VERSION 1.1.0 · UPDATED 2026-06-18 · SOURCE Clarus method library