By Ian Birnbaum
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Extra info for An Introduction to Causal Analysis in Sociology
It follows that for each configuration of X I ' . . 'xk - I ' xk is normally distributed with mean k-1 L bk;X; + Ck and variance Var( Uk)• i= 1 Thus, for a particular unit of analysisj, the probability that the value taken on xk is Xkj given that the values on X 1' . . 'xk -1 are X1j, ... e. 4 There are (k + 1) parameters here, bk 1' . . 'bk k -1 'Ck and Var( uk) all pertaining to Xk. Let us write these parameters as 0 k, for short. 5 (since for any value T, log exp(7) = T, when the log is natural).
The variance interpretation of path coefficients, however, is not to be recommended. M Using Discrete Variables in Causal Analysis If an exogenous variable is discrete cardinal then the discussion is just as in Section G and no problem arises. Again, if an exogenous variable is noncardinal the discussion in Section G applies, with the conclusion that we should generally make an analysis in terms of sub-populations for each category of the non-cardinal variable. A conceptual problem arises, however, if an endogenous variable is discrete cardinal and one or more of the determining variables is continuous.
1 on e2. 2l The Theory of Ozuml Analysis 23 Now the pmcc between e4 . ;::c-'--~ or Var e4 . 1 b4 2. 1 2 Vare2 Var €4. 22. Since e2 is the residual of X 2 after we subtract out the linear function of X 1 , we see that the partial correlation coefficient is the pmcc between the residuals of X4 and X 2 after the linear effect of X 1 is subtracted out. This is the reason for the term 'partial'. Notice that, in general, a partial correlation coefficient like P4 2. 1 2 is not the pmcc between X 2 and X 4, for X 1 constant.