Correlation Coefficients > Partial Correlation

## What is Partial Correlation?

Partial correlation **measures the strength of a relationship between two variables, while controlling for the effect of one or more other variables.** For example, you might want to see if there is a correlation between amount of food eaten and blood pressure, while controlling for weight or amount of exercise. It’s possible to control for multiple variables (called control variables or covariates). However, more than one or two is usually not recommended because the more control variables, the less reliable your test.

Partial correlation has one continuous independent variable (the x-value) and one continuous dependent variable (the y-value); This is the same as in regular correlation analysis. In the blood pressure example above, the independent variable is “amount of food eaten” and the dependent variable is “blood pressure”. The control variables — weight and amount of exercise — should also be continuous.

## Notation

A period in the subscript separates the correlated variables and the controlled for variables. For example, correlating caloric intake (X_{1}) against blood pressure (X_{2}), while controlling for weight (X_{3}), is written as:

r_{12.3}

Alternatively, a bar is used instead of a period and subscript: r(1,2|3).

## Running the Test

The correlation coefficient, r, is also used to show the results from partial correlation. Like the regular correlation coefficient, r_{partial} returns a value from -1 to 1.

Partial correlation is usually carried out by running multiple regression analysis. Some software programs include partial correlation. For example, in SPSS choose Analyze > Correlations > Partial.

## How to Interpret the Result

If the partial correlation, r_{12.3}, is smaller than the simple (two-variable) correlation r_{12}, but greater than 0, then variable 3 partly explains the correlation between X and Y.

## Semipartial Correlation

Semipartial correlation is almost the same as partial. With semipartial correlation, the third variable holds constant for *either *X or Y but not both; with partial, the third variable holds constant for both X and Y.

## References

Sharpa, J. (2007). Business Stats. Pearson Education India.

Weatherburn, C. (1949). A First Course Mathematical Stats. CUP Archive.

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