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:

1. Trace backward along an arrow and then forward, or simply forwards from one variable to the other but never forward and then back
2. Pass through each variable only once in each chain of paths
3. 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'.