David A. Kenny
September 6, 2011

 

Illustration of Deriving Correlations in Terms of Path Coefficients

 
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.
                                
Variables
             PP -- physical punishment
             CR -- caretaker rejection
             CB -- child beliefs
             PA -- psychological adjustment
 
 
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.)
 
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
 
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
 
The results are the same, although the last one differs a bit due to rounding error.
 
See Kenny’s 1979 book, Correlation and Causality for more details. 










 
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