2 Tracing Rules for Unstandardized Variables

If we are working with unstandardized variables, the tracing rules of the previous section are insufficient to derive expected correlations. However, in the absence of paths from dependent variables to other dependent variables, expected covariances, rather than correlations, may be derived with only slight modifications to the tracing rules (see Heise, 1975):

1. At any change of direction in a tracing route which is not a two-way arrow connecting different variables in the chain, the expected variance of the variable at the point of change is included in the product of path coefficients; thus, any path from an dependent variable to an independent variable will include the double-headed arrow from the independent variable to itself, unless it also includes a double-headed arrow connecting that variable to another independent variable (since this would violate the rule against passing through adjacent arrowheads)
2. In deriving variances, the path from an dependent variable to an independent variable and back to itself is only counted once

Perhaps a simpler approach to unstandardized path analysis is to make certain that all residual variances are included explicitly in the diagram with double-headed arrows pointing to the variable itself. Then the chains between two variables are formed simply if we

1. Trace backwards, change direction at a two-headed arrow, then trace forwards.

As before, the expected covariance is computed by multiplying all the coefficients in a chain and summing over all possible chains. We consider chains to be different if either a) they don't have the same coefficients, or b) the coefficients are in a different order. For a clear and thorough mathematical treatment, see the RAMPATH manual (McArdle and Boker, 1990).