17.1.12.2 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:

\[ \begin{pmatrix} \Sigma_{xx} & \Sigma_{xy} \\ \Sigma_{yx} & \Sigma_{yy} \end{pmatrix} \]

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:

\[df=n-n_x-2\]

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:

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

References

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