Regression Lines and the Functional Relation.

Abstract

This discussion is concerned with the linear functional relation r1 = A + BX1, whose variables are known only through observed values X and Y, both affected by errors of measurement. The object is (1) to find A and B, (2) to reduce the values X to the system of F (or vice versa), with results in both cases free from systematic error. A complete solution for (1) presents no particular difficulty. The solution for (2) requires the regression line Y~ = a + b~X, where V~ is a computed value and a and li~ are parame- ters which depend on the mean error of measurement. For observations of unequal weight, therefore, no single line suffices. This difficulty is most easily overcome by establishing a general expression for b~ as a function of B and the mean error of X. Incidentally, this relation gives at once all the characteristic prop- erties of the regression lines, among them the fact that the lines will be strictly linear only when the fre- quency distribution for X (and F) is gaussian