Calculate the partial correlation between x and y controlling for z from three Pearson correlations, with a significance test and semipartial values.

Use Pearson correlations from the same observations. Adjustment describes linear association, not causation.

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Partial Correlation Formula

The following formula is used to calculate the partial correlation between two variables, controlling for a third variable.

rxy.z = (rxy − rxz × ryz) / √((1 − rxz2) × (1 − ryz2))

Variables:

  • rxy.z is the partial correlation between x and y, controlling for z
  • rxy is the correlation between x and y
  • rxz is the correlation between x and z
  • ryz is the correlation between y and z

To calculate the partial correlation, subtract the product of the correlation between x and z and the correlation between y and z from the correlation between x and y. Then, subtract the square of the correlation between x and z from 1, and do the same for the correlation between y and z. Multiply these two results together and take the square root. Finally, divide the first result by the second result.

What is a Partial Correlation?

Partial correlation is a statistical measure that describes the association between two variables while controlling for the effects of one or more additional variables. It is used when researchers want to determine the relationship between two variables without the influence of other variables. This is particularly useful in multivariate analysis where multiple variables may be interrelated. The partial correlation coefficient ranges from -1 to +1, with -1 indicating a perfect negative linear association, +1 indicating a perfect positive linear association, and 0 indicating no remaining linear association. All correlations must come from the same observations and form a valid correlation matrix. A perfect correlation with the control variable leaves zero residual variance, making this formula undefined. Adjustment does not establish causation.