Optimal line fitting by computer

Abstract

Current methods of line fitting used in laboratories are in many cases approximations made to simplify the arithmetic. Now that computing time is readily available to most laboratories, more accurate, though more complex, methods can be used. For instance, calibration curves for clotting factor assays are usually drawn on double logarithmic paper. This is not out of any theoretical considerations, but simply because the points always seem to lie fairly close to a straight line on this sort of paper. There is no justification for employing logarithmic functions when using a computer for this sort of calibration; we have found that other methods can give better results. The problem of fitting straight lines where both variables x and y are subject to error is common in evaluations of new tests. The usually employed method for determining the coefficients a and b of the straight line y = ax + b is that of choosing values a and b which minimize the sum of the squares of the deviations of the y's. It is well known that treating y as an independent variable and minimizing the sum of the squares of the deviations of the xs gives a different straight line as best fit. If both variables are subject to error, there is no reason to prefer one of these regression lines to the other. In fact, neither equation gives the best structural relationship between x and y, but the true line lies somewhere between. In the evaluation of new methods in our laboratory we use a computer to calculate the ‘best’ structural relationship, rather than fit a line by simple linear regression. Current methods of line fitting used in laboratories are in many cases approximations made to simplify the arithmetic. Now that computing time is readily available to most laboratories, more accurate, though more complex, methods can be used. For instance, calibration curves for clotting factor assays are usually drawn on double logarithmic paper. This is not out of any theoretical considerations, but simply because the points always seem to lie fairly close to a straight line on this sort of paper. There is no justification for employing logarithmic functions when using a computer for this sort of calibration; we have found that other methods can give better results. The problem of fitting straight lines where both variables x and y are subject to error is common in evaluations of new tests. The usually employed method for determining the coefficients a and b of the straight line y = ax + b is that of choosing values a and b which minimize the sum of the squares of the deviations of the y's. It is well known that treating y as an independent variable and minimizing the sum of the squares of the deviations of the xs gives a different straight line as best fit. If both variables are subject to error, there is no reason to prefer one of these regression lines to the other. In fact, neither equation gives the best structural relationship between x and y, but the true line lies somewhere between. In the evaluation of new methods in our laboratory we use a computer to calculate the ‘best’ structural relationship, rather than fit a line by simple linear regression.