David A. Kenny
September 6, 2011

Illustration of Deriving Correlations in Terms of Path Coefficients

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`Study`
Rohner, R. P., Kean, K. J., & Cournoyer, D. E.  (1991).  Effects of corporal punishment, perceived caretaker warmth, and cultural beliefs on the psychological adjustment of children in St. Kitts, West Indies.  Journal of Marriage and the Family, 53, 681-693.
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`Variables`
`             PP -- physical punishment`
`             CR -- caretaker rejection`
`             CB -- child beliefs`
`             PA -- psychological adjustment`
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`Standardized Structural Equations`
`             CR = .470PP + .883U`
`             CB = .023PP + 1.000V`
`             PA = .752CR + .172PP + .240CB + .610W`
```             rUV = -.477

(Note that because all the variables are standardized, the disturbances have coefficients less than one.)```
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Tracing Rule```

The correlation between Xi and Xj equals the sum of the product of all the standardized paths obtained from each of the possible tracings between variable i and j. The set of tracings includes all the possible routes from Xi to Xj given that (a) the same variable is not entered twice and (b) a variable is not entered through an arrowhead and left through an arrowhead.

`             rCR,PP = .470`
`             rCB,PP = .023`
`             rCR,CB = (.470)(.023) + (.883)(1.000)(-.477) = -.410`
`             rPA,PP = (.470)(.752) + (.023)(.240) + .172 = .531`
`             rPA,CR = .752 + (.470)(.172) + (.023)(.470)(.240) + (.883)(1.000)(.477)(.240) = .734`
`             rCB,PA = .240 + (.023)(.172) + (.023)(.470)(.752) + (.883)(1.000)(-.477)(.752) = -.065`
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First Law```

Given a standardized structural model, the correlation between Y and Z where Y is endogenous is equal: rYZ = ΣpYXirXiZ where p­ is the standardized path or causal parameter from variable Xi to Y, rXiZ is the correlation between Xi and Z, and the set of Xi variables are all the causes of the variable Y.

`             rCR,PP = .470`
`             rCB,PP = .023`
`             rCR,CB = (.470)(.023) + (.883)(-.477)(1.000)  = -.410`
`             rPA,PP = (.470)(.752) + (.023)(.240) + .172 = .531`
`             rPA,CR = .752 + (.172)(.470) + (.240)(-.410) = .734`
`             rCB,PA = .240 + (.752)(-.410) + (.172)(.023) = -.064`
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`The results are the same, although the last one differs a bit due to rounding error.`
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```See Kenny’s 1979 book, Correlation and Causality for more details.
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