Abstract
Since simple linear regression theory was established at the beginning of the 1900s, it has been used in a variety of fields. Unfortunately, it cannot be used directly for calibration. In practical calibrations, the observed measurements (the inputs) are subject to errors, and hence they vary, thus violating the assumption that the inputs are fixed. Therefore, in the case of calibration, the regression line fitted using the method of least squares is not consistent with the statistical properties of simple linear regression as already established based on this assumption. To resolve this problem, “classical regression” and “inverse regression” have been proposed. However, they do not completely resolve the problem. As a fundamental solution, we introduce “reversed inverse regression” along with a new methodology for deriving its statistical properties. In this study, the statistical properties of this regression are derived using the “error propagation rule” and the “method of simultaneous error equations” and are compared with those of the existing regression approaches. The accuracy of the statistical properties thus derived is investigated in a simulation study. We conclude that the newly proposed regression and methodology constitute the complete regression approach for univariate linear calibrations.
Highlights
Simple linear regression is a model with a single independent variable in which a regression line is fitted through n data points such that the sum of squared errors (SSE), i.e., the vertical distances between the data points and the fitted line, is as small as possible
In Suh’s measurement experiment, r(x, y) is 0.9964 (n = 5), the estimate derived via classical regression at the upper end of the calibration range is approximately 1.5% greater than that derived via inverse regression, and the estimate derived via reversed inverse regression is approximately 0.15% greater than that derived via inverse regression
We conducted a Monte Carlo simulation study to investigate the accuracy of the statistical properties derived using the error propagation rule and the method of simultaneous error equations based on the first-order Taylor approximation. var[b^], bias[b^] and E[mean squared error (MSE)] were the main targets of investigation because the accuracy of other properties, such as var[a^], bias[a^], var[^y], bias[^y] and var [prediction interval], depends on the accuracy of these three properties
Summary
Simple linear regression is a model with a single independent variable in which a regression line is fitted through n data points such that the sum of squared errors (SSE), i.e., the vertical distances between the data points and the fitted line, is as small as possible. In the second approach [6], called inverse regression, the standards (the x values) are treated as the response, the observed measurements (the y values) are treated as the inputs, and these values are used to fit a regression line of x on y. ( put, “fundamental solution for the univariate linear calibration problem” = “reversed inverse regression” + “new methodology for deriving the statistical properties of the regression”.) In the proposed regression approach, the observed measurements (the x values) are treated as the inputs, and the standards (the y values) are treated as the response; these values are used to fit a regression line of y on x. The statistical properties of this regression are derived using the “error propagation rule” and the “method of simultaneous error equations” In this regression approach, it is not necessary to measure any standards multiple times independently. The accuracy of the statistical properties derived using the new methodology is investigated in a Monte Carlo simulation study
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