By Ian Birnbaum
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Extra resources for An Introduction to Causal Analysis in Sociology
Notice that there has been no need to make any causal ordering amongst X 1 , X 2 and X 3 themselves. 1 only, then we say that X 1 , X 2 and X 3 are all exogenous variables, because none of them is accounted for in any way by any other explicit variable in the system. By contrast, X 4 , which is so accounted for, is called an endogenous variable. It is usual, however, to want to investigate what are called indirect effects, and to this end further orderings are made. Let us suppose, then, that X 1 and X 2 are causally prior to X 3.
E. those transformations that preserve the defining characteristics of the variables). It is this that makes the use of the mean in cardinal variables mathematically valid, quite apart from whether or not it is socially meaningful. (The standard deviation does not change its value with a change of origin, but this is because it is a second order measure concerned with the distances between variables, which, of course, are independent of origin. ) The mean is not invariant to the transformations preserving the relationships on ordinal variables.
3 ;~: We know that k -I E(XkiX1, ... ,xk-d= l: h;X;+ck, i= I where Ck =E( Uk ), a constant. Note that ck consists of the effect of explicit variables when they are zero plus the average effect of implicit variables. We assume that for every configuration of X 1 , •.. , Xk _ 1 , Uk is normally distributed with the same mean, ck, and the same variance, which will be Var( Uk). 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' .