Algorithm (Partial Correlation Coefficient)

Partial correlation coefficient is used to describe the relation between two variables in the presence of controlling variables.

Partial Correlation Coefficient

For a set of n_y random variables Y and n_x controlling variables X, combine two sets of variables X and Y, its variance-covariance matrix can be expressed as:

\Sigma_{xx} & \Sigma_{xy} \\
\Sigma_{yx} & \Sigma_{yy}

The variance-covariance matrix of Y variables for controlling variables X is given by:

\Sigma_{y|x} = \Sigma_{yy}-\Sigma_{yx}\Sigma_{xx}^{-1}\Sigma_{xy}

The partial correlation coefficient matrix is calculated by:

\text{diag}(\Sigma_{y|x})^{-1/2} \Sigma_{y|x} \text{diag}(\Sigma_{y|x})^{-1/2}.

Significance of Partial Correlation Coefficient

A t-Test can be used to test the hypothesis that a partial correlation coefficient is zero.

The degrees of freedom are:


where n is the number of observations in the calculation of the full correlation. For pairwise deletion of missing values, in the calculation of partial correlation of two variables Y_i, Y_j given controlling variables X, n is the minimum number of observations in the pairs of (Y_i, Y_j), (Y_i, X), (Y_j, X) and pairs in X.

t Statistic is:

t = |r| \sqrt{ \frac {df} {1-r^2} }

where r is the partial correlation coefficient.

The two-tailed significance level \text{Prob}>|t| can be calculated as:

2(1 - \text{tcdf}(t, df)).


  1. Morrison, D. F. (1976), Multivariate Statistical Methods, Second Edition, New York: McGraw-Hill.
  2. nag_partial_corr (g02byc)