By Ian Birnbaum
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Extra info for An Introduction to Causal Analysis in Sociology
L Standardised Unear Models Let us confme our attention initially to a single population. l, we see that P4 1 , for instance, can be interpreted as the number of a 4 's by which X4 will change for a change of a 1 in X 1 , all other causes constant. Hence,p4 1 , P42 and P4J are standardised path regression coefficients. What is the point of standardising, however? To see this, we must return to the basic problems of measurement. Measure:nent of any quantity can only take place against some standard.
Assume, without any loss of generality, that only X 1 is exogenous. Then X 1 has no parameters on which it depends, and no parameters to estimate. •• , Xm given X 1 over all parameters 0 2 , 0 3 , •• , Om which we write as Om, 2 • Now, Pom, 2(X2, ... , Xm I Xt) =Pom,3(X3, ... 6 where 0 m, 3 is the parameter set 0 3, . . , Om. 6, ending up finally with Pom,2(X2, ... ,XmiXt)= Pom(Xm I X,, ... Xm -t)Pom _,(Xm-t I X1, ... 7 Hence the log-likelihood of the term on the left can be written as the sum of the log-likelihoods of those on the right.
If we have terms in Y1 , X 1 and this may well happen, and thus care must be taken with interactive models. In any case, some care needs to be taken with linear models too, though it is often difficult to deal with in practice, and we must then be aware of the drop of precision in estimation implied. Actually, in the cross-sectional data that characterise much of the causal analysis described here, the situation is relatively rare, for typically there is considerable residual variation. The coefficients b4 1 , b4 2 and b4 3 are, of course, just regression coefficients but the causal interpretation we have been able to give them depends on the additional causal assumptions specified above.