Overview

Note that this command is under construction. Conditional dependencies between quantities and collections of quantities can be explored by making various transformations of their joint variance-covariance matrices. The transformations essentially reveal certain features of the underlying linear relationships between the variables, and these features can be represented on diagrams.

Suppose that a number of collections together form a collection of elements whose variance matrix is . Firstly we transform to correlation form by determining , where D is the diagonal matrix whose entries are . Next we determine the Moore-Penrose generalised inverse of R as follows. The pseudo-inverse obtained is , where is the diagonal matrix of ordered eigenvalues of R, and has the non-zero diagonal elements inverted; and where Q is the matrix of corresponding orthonormal eigenvectors. Next we transform this generalised inverse itself into correlation form by calculating , where is the diagonal matrix whose entries are . However, we perform this final process only on the off-diagonal entries of , leaving alone the diagonal entries. Finally we multiply each off-diagonal entry by -1. The matrix G so constructed has the following properties.

1. the diagonal values are such that

   

   where R is the multiple correlation coefficient between and the remaining elements.

2. the off-diagonal values are , representing the partial correlation coefficient between and given the remaining elements.

In [B/D] we make available G given by using the PCDIAG:  command in combination with the pcdest  control, which sets the belief store in which these results are to be stored. The belief store containing the original variance matrix V can be switched using the pcsource  control. The partial correlation diagram summarising the linear relationships is obtained by using the PCDIAG:  command in combination with the pcdiag  option. A title may be defined for this diagram by using the PCTITLE:  command. Linear relationships (partial) are shown as arcs on the diagram. The number of arcs shown depends upon the value of the pcarc  control: arcs corresponding to partial correlations less than this value are supressed. The diagram itself needs to be designed beforehand, using the GRID:  or, more conveniently, the GRID0:  command.