/math-88449ca5365fc85a4051a4f39cf72e31.png?v=0 "n_y")
random variables Y and /math-d037fcd5fa24c8673763af9c9253cdb1.png?v=0 "n_x")
controlling variables X, combine two sets of variables X and Y, its variance-covariance matrix can be expressed as:
Partial correlation coefficient is used to describe the relation between two variables in the presence of controlling variables.
For a set of random variables Y and controlling variables X, combine two sets of variables X and Y, its variance-covariance matrix can be expressed as:
/math-90b8ce2774c5073b1124c7e9a4b41957.png?v=0 "\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:
/math-a497642b29af9eda7964583875b85002.png?v=0 "\Sigma_{y|x} = \Sigma_{yy}-\Sigma_{yx}\Sigma_{xx}^{-1}\Sigma_{xy}")
The partial correlation coefficient matrix is calculated by:
/math-dc013e6067313a0ed2ca5041a49ac91b.png?v=0 "\text{diag}(\Sigma_{y|x})^{-1/2} \Sigma_{y|x} \text{diag}(\Sigma_{y|x})^{-1/2}.")
A t-Test can be used to test the hypothesis that a partial correlation coefficient is zero.
The degrees of freedom are:
/math-0613eb5b6527b5b309de243d94b08a47.png?v=0 "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 /math-cca0d5b689e524b619b30538d3432266.png?v=0 "Y_i, Y_j")
given controlling variables X, n is the minimum number of observations in the pairs of /math-9f58a707e9db8318aac1bb009020990d.png?v=0 "(Y_i, Y_j), (Y_i, X), (Y_j, X)")
and pairs in X.
t Statistic is:
/math-14450f33ce8531f5a46e17c259af12b7.png?v=0 "t = |r| \sqrt{ \frac {df} {1-r^2} }")
where r is the partial correlation coefficient.
The two-tailed significance level /math-550e9bcd0b294b7167dd9b8a1472af23.png?v=0 "\text{Prob}>|t|")
can be calculated as:
/math-704674c897463ad5d94a0842cbdadd58.png?v=0 "2(1 - \text{tcdf}(t, df)).")