THEORY BUILDING AND THE STATISTICAL CONCEPT OF INTERACTION.

Assume that the (ij) elements of P are all non-zero, and that PA has the same structure as A. The ith equation may be linearly combined with the jth. Moreover, xi+, is causally dependent on xj+i: indeed, since the jth equation defines xj+i, the coefficient of xJ+I is not zero in this equation, whence a linear combination of the ith and jth equations will retain the structure of the former only if aj+, i+1 is not zero, i.e., if xi+1 is causally dependent on xi+,. Now, in equation (11), premultiplication of E(ex') by P, where pij is not zero, will substitute a linear combination of the i-th and j-th rows of E(ex') for the original i-th row. But, since xi+, is causally dependent on x + , the term of the (j+l)th column in the ith row of E(ex') i.e., E(ei+ixj+i) is zero, while the term of the same (j+1)th column in the jth row is not zero. Hence, premultiplication by P modifies the structure of the ith row, and it is impossible to find a non-diagonal matrix P such that ( 11 ) satisfies the structural conditions of (10). To illustrate this, let us go back to system (4). The corresponding causal structure (see Figure 1) implies E(e2x) =E(e3x1) =E (e3x2) =O. That is, using our substantive example, implicit factors acting on the extension of family group are not correlated with age, and implicit factors acting on suicide are correlated neither with age nor with extension of family group. Now, we see from (5) or (7) that we may find a non-diagonal matrix P with P12#0, such that PA has the same structure as A, and, of course, PAE (xx'), the same structure as AE(xx') (the same coefficients are zero). Now in the equation PAE(xx') =PE(ex'), premultiplication by P substitutes for E(e3x2) a quantity which we may designate E(e3'x2) and which is equal to E (PI2e2x2+e3x2) =p12E (e2X2) +(e3x2). But, since E(e2x2) is not zero, E(e3'x2) is zero if and only if P12=O. But we have assumed P12:#O. Thus, premultiplication by P violates the condition derived from the assumption that implicit factors are uncorrelated, according to which X2 and the implicit factors acting on X3 are independent.