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##

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):
- 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)
- 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
- 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).

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Jeff Lessem
2002-03-21