By Ian Birnbaum

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The most intuitive approach to estimation, however, is to estimate the population values by minimising the sum of the residuals in the sample. e4 2 is the minimum possible for the sample, and b41, b42, b43 and 4 are the corresponding values of the coefficients for this minimum. We note that e4 is not the same as ~4 since the sample regression coefficients will not, in general, have the same value as the population regression coefficients. What we have done here, in effect, is to·apply constrained linear regression to the sample.

The reason for this should be clear. If the mean of a set of cardinal variables is m and we change the origin by an amount x then the mean becomes m + x. It thus preserves the interval level relationship of distance. Similarly, a change of scale gives sm, say, again preserving the interval level relationship of simple ratio. 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 Theory of Causal Analysis 21 Since Var ~4 is the minimum possible, Var Lis the maximum possible. 8) and hence minimising £(~ 4 2) is equivalent to maximising the pmcc. IO p 2 (X4 , L) is called the multiple co"elation coefficient and is usually designated by P4( 12 3) 2 (or in a sample by R4 ( 12 3) 2). It measures the variance in X4 accounted for by the joint variation of the best linear functionofX 1,X2 andX3. 3 still holds. This can be verified by the reader by using the same technique as on p.