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Furthermore, in tracing paths in more complex diagrams to reduce them to sets of equations, it is easy to make errors--for example, to overlook some indirect path that connects point A and point 8, or to include a path twice. Is there any way of mechanizing the construction of path equations, as well as their solution? In fact, there are such procedures, which allow the expression of the equations of a path diagram as the product of several matrices. Not only does such an approach allow one to tum a path diagram into a set of path equations with less risk of error, but in fact one need not explicitly write down the path equations at all--one can carry out the calculation of implied correlations directly via operations on the matrices.
It has one downstream variable, hence one structural equation: The score of a person on variable C is an additive function of his scores on A, 8, and X. If the variables are obtained in standard-score form for a set of subjects, the values of the weights a, b, and d required to give a best fit to the data in a least squares sense turn out to be just the standardized partial regression coefficients, or path coefficients, discussed earlier. 18 gives a slightly more complex example, based on the earlier Fig.
It is helpful (though not algebraically necessary) to group together source variables and downstream variables--the source variables A and B are given first in the Table 2-2 matrices, and the downstream variables C and D last. Curved arrows and variances are represented in matrix S. The top left hand part contains the correlation matrix among the source variables, A and B. The diagonal in the lower right-hand part contains the residual variances of the 41 Chapter 2: Table 2-2 Fining Path Models Matrix formulation of a path diagram by the McArdle-McDonald procedure A B 0 0 B 0 0 C a c D a 0 A c 0 0 0 0 A and d.