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##

1 Tracing Rules for Standardized Variables

The basic principle of tracing rules is
described by Sewall Wright (1934) with the
following words:
``*Any correlation between variables in a network of sequential
relations can be analyzed into contributions from all the paths
(direct or through common factors) by which the two variables are
connected, such that the value of each contribution is the product of
the coefficients pertaining to the elementary paths. If residual
correlations are present (represented by bidirectional arrows) one
(but never more than one) of the coefficients thus multiplied together
to give the contribution of the connecting path, may be a correlation
coefficient. The others are all path coefficients.''*

In general, the expected correlation between two variables in a path
diagram of standardized variables may be derived by tracing all
connecting routes (or ``chains'') between the variables, while
adhering to the following conditions. One may:
- Trace backward along an arrow and then forward, or simply
forwards from one variable to the other but
*never forward and
then back*
- Pass through each variable only once in each chain of paths
- Trace through at most one two-way arrow in each chain of paths

A corollary of the first rule is that one may *never pass through
adjacent arrowheads.*
The contribution of each chain traced between two variables to their
expected correlation is the *product* of its standardized
coefficients. The expected correlation between two variables is the
sum of the contributions of all legitimate routes between those two
variables. Note that these rules assume that there are no
feedback loops; i.e., that the model is
`recursive'.

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Jeff Lessem
2002-03-21