Partial correlation: what it means and how to read it

In the report: Correlation — The correlation of the two columns with the controls held constant: of what is left of both after the linear effect of the controls is removed. Empty when a control is constant or repeats another, or when there are fewer than k + 3 complete rows (k — the number of controls).

How to read it

Answers "do the two still go together when the controls are the same?". If it is much weaker than the ordinary correlation, the controls explain part of the relation.

In a paper: r(47) = .09, p = .540, controlling for age.

In plain words

A researcher found that older children read better. But older children have also spent more years in school, and she could not tell whether age itself goes with reading. She asked for the partial correlation, which looks at age and reading among children with the same years in school. It came out 0.1, close to zero. Now she can say that among children with the same years in school, age and reading hardly go together; most of the link went with the years in school. If the second column repeats the first or stays the same in every row, the cell stays empty.

How it is computed

r of the residuals of x and of y, each regressed on the controls

Example

The cell with Correlation is selected in each report; point at a label to see its note.

— the sample data (50 rows) the examples use
— x1 and x2 with age held constant =STATS_PARTIAL(Data!A1:T51, "x1,x2", "age")

Where you see it

STATS_PARTIAL — Partial correlation: the Pearson correlation of two variables with the controls held constant (SPSS PARTIAL CORR). Two variables give one correlation, three or more a matrix, as STATS_PEARSON.

Cohen's d — The effect size: the difference in standard deviations, so it does not depend on the units of the data. Empty when there is no spread.
Phi — The strength of the relation, from the chi-square: for a 2 × 2 table with the sign of the relation (the sign depends on the order of the categories), for a larger table it equals the square root of χ² / N.
Cramér's V — The strength of the relation of two categorical variables, from 0 (none) to 1 (one determines the other).
Eta squared (η²) — The share of the total variance explained by the groups: 0 — none, 1 — all.
Omega squared (ω²) — The share of variance explained by the groups, corrected for sample size: a less biased η², slightly smaller. Can be below 0 when there is no effect.

Get it in your sheet

One formula in a cell gives the whole report, Partial correlation included. ALS Stats is a free add-on for Google Sheets™.

Install ALS Stats from Google Workspace Marketplace (opens in a new tab)