Are There Two Regressions?

Abstract If, in sampling from an existent population, we wish to know the regression of u on v, a line is fitted by least squares, minimizing the sum of the squared residuals of v (or y). If there is no error in measuring the independent variate u, regardless of whether the dependent variate has been measured without error as v or with error as y, the regression is an estimate of the true regression. If, however, the independent variate is measured with error in x, the regression obtained is a biassed one. The bias will characterize the fitted line, and will be present whether the sample is a random one of the entire population or has been taken at preassigned selected values of x. If, however, the experiment is one in which one of the variates is a controlled observation, it differs from sampling from an existent population in that, 1), the line estimated by least squares, minimizing the sum of the squared residuals of the dependent uncontrolled variate, is the same, whether x or y is the controlled variate, that is, there is only one regression; and 2), the estimated line is not biassed by the existence of an error of observation in the independent controlled variate, despite our taking no account of it in the least squares fit.