Second Generalisation: Functions Other Than the Scalar Product

Abstract

A function f which is investigated in the base situation 0, giving the value $f\left( {{{\vec x}^0}} \right)$ , and in the observed situation 1, giving the value $f\left( {{{\vec x}^1}} \right)$ . An additive causal analysis of the difference $f\left( {{{\vec x}^1}} \right) - f\left( {{{\vec x}^0}} \right)$ − $f\left( {{{\vec x}^0}} \right)$ is performed in section 6.1 and a multiplicative causal analysis of the ratio $f\left( {{{\vec x}^1}} \right)/f\left( {{{\vec x}^0}} \right)$ / $f\left( {{{\vec x}^0}} \right)$ is performed in section 6.2. Then follows an example from health insurance. — The function f is treated as a “black box”: It may be very complicated, contrary to the scalar product of the price and the quantity vectors in the context of price indices. To do the causal analysis one only needs 2(n+1) special values of the unknown function f. Readers may first read the beginning of section 6.3 and see, if they solve the problem described there in the same way as the authors.

If an instance in which the phenomenon under investigation occurs and an instance in which it does not occur have every circumstance in common save one, that one occurring only in the former, the circumstance in which alone the two instances differ is the effect, on the cause, or an indispensable part of the cause, of the phenomenon.

John Stuart Mill (1806–1873)