Fitting of Straight Lines and Prediction When Both Variables are Subject to Error

Abstract Define the linear structural relation with parameters α, β, by μ = α +βv. This paper considers the problem of confidence region estimation of α and β based on a sample of independent pairs, (y i , x i ), i = 1, 2, ···, with Ey i = μi, Ex i = v i . The pairs (y i , x i ) are assumed to have a common (unknown) covariance matrix and to follow the bivariate normal distribution. Point estimates subject only to and that the choice of the l i be independent of the actual observations are shown to lead to an exact confidence region for (α, β). The physical size of the region and whether or not the region is closed depend strongly on the choice of the li . Using as a criterion the maximization of the probability that the region is closed, it is shown that the optimum l i require knowledge of the vi; however, the region is at the required level even if the l i are sheer guesses. Of course, if one has a priori information about the v i levels (presumably not completely accurate) it can and should be used. The use of auxiliary experimentation to improve the choice of the l i is briefly discussed and it is shown that near optimum regions can be expected by such a device providing measurement error is small relative to the variation in the v i . The question of error control in predicting a number of new v i 's from corresponding new y i 's on the basis of a single estimate of (α, β) is also considered. It is suggested that instead of controlling the proportion of errors, one control instead the probability that the proportion of errors will not be more than some specified amount; intervals based on this principle are derived for several different assumptions about the kind of information available.