The Fitting of Straight Lines when Both Variables are Subject to Error

Abstract

Abstract Consider the situation where X and Y are related by Y = α + βX, where α and β are unknown and where we observe X and Y with error, i.e., we observe x = X + u and y = Y + v. Assume that Eu = Ev = 0 and that the errors (u and v) are uncorrelated with the true values (X and F). We survey and comment on the solutions to the problem of obtaining consistent estimates of α and β from a sample of (x, y)'s, (1) when one makes various assumptions about properties of the errors and the true values other than those mentioned above, and (2) when one has various kinds of “additional information” which aids in constructing these consistent estimates. The problems of obtaining confidence intervals for β and of testing hypotheses about β are not discussed, though approximate variances of some of the estimates of β are given.