text: The theoretical underpinnings of standard least squares regression analysis are based on the assumption that the independent variable (often labelled as x) is measured without error as a design variable. The dependent variable (labeled y) is modeled as having uncertainty or error. Both independent and dependent measurements may have multiple sources of error. Therefore, the underlying least squares regression assumptions can be violated. Reduced major axis (RMA) regression is specifically formulated to handle errors in both the x and y variables. If the estimate of the ratio of the error variance of y to the error variance of x is denoted by , then the reduced major axis method assumes that  can be approximated by the ratio of the total variances of x and y. RMA minimizes both vertical and horizontal distances of the data points from the predicted line (by summing areas) rather than the least squares sum of squared vertical (yaxis) distances.
  In Dents 1935 paper on linear regression, entitled "", she admitted that when the variances in the x and y' variables are unknown, "we cannot hope to find the true positions of the observed points, but only their most probable positions." However, by treating the probability of the errors in terms of Gaussian error functions, she contended that this expression may be regarded as "a function of the unknown quantities", or the likelihood function of the data distribution. Furthermore, she argued that maximising this function to obtain the maximum likelihood estimation, subject to the condition that the points are collinear, will give the parameters for the line of best fit. She then deduced formulae for the errors in estimating the centroid and the line inclination when the data consists of a single (unrepeated) observation.
  Maurice Kendall and Alan Stuart showed that the maximum likelihood estimator of a likelihood function, depending on a parameter , satisfies the following quadratic equation:
  :where x and y are the  and  vectors in a covariance matrix giving the covariance between each pair of x and y variables.
  Using the quadratic formula to solve for the positive root (or zero) of ():
  Inspection of () shows that as  tends to plus infinity, the positive root tends to:
  Correspondingly, as  tends to zero, the root tends to:
  Dent had solved the maximum likelihood estimator in the case where the covariance matrix is not known. Dent's maximum likelihood estimator is the geometric mean of  and , equivalent to:
  Dennis Lindley repeated Dents analysis and stated that Dents geometric mean estimator is not a consistent estimator for the likelihood function, and that the gradient of the estimate will have a bias, and this remains true even if the number of observations tends to infinity. Subsequently, Theodore Anderson pointed out that the likelihood function has no maximum in this case, and therefore, there is no maximum likelihood estimator. Kenneth Alva Norton, a former consulting engineer with the then National Bureau of Standards, responded to Lindley, stating Lindley's own methods and assumptions lead to a biased prediction. Furthermore, Albert Madansky, late H. G. B. Alexander professor of business administration at University of Chicago Booth School of Business, noted that Lindley took the wrong root for the quadratic in () for the case where  is negative.
  Richard J. Smith has stated that Dent was the first to develop a RMA regression method for line fitting that built on the work of Robert Adcock in "" (1878) and Charles Kummell in "" (1879). It is now believed that she was the first to propose what is often called the geometric mean functional relationship estimator of slope, and that her essential arguments can be generalised to any number of variables. Moreover, although her solution has its theoretical limitations, it is of practical importance, as it likely represents the best a priori estimate if nothing is known about the true error distribution in the model. It is generally much less reasonable to assume that all the error, or residual scatter, is attributable to one of the variables.