ABSTRACT
It has long been recognized that the least-squares estimation method of fitting the best straight line to data points having normally distributed errors yields identical results for the slope and intercept of the line as does the method of maximum likelihood estimation. We show that, contrary to previous understanding, these two methods also give identical results for the standard errors in slope and intercept, provided that the least-squares estimation expressions are evaluated at the least-squares-adjusted points rather than at the observed points as has been done traditionally. This unification of standard errors holds when both and observations are subject to correlated errors that vary from point to point. All known correct regression solutions in the literature, including various special cases, can be derived from the original York equations. We present a compact set of equations for the slope, intercept, and newly unified standard errors.
- 1. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge U.P., New York, 1992), 2nd ed. Google Scholar
- 2. Derek York, “Least-squares fitting of a straight line,” Can. J. Phys. 44, 1079–1086 (1966). Google ScholarCrossref
- 3. B. Cameron Reed, “Linear least-squares fits with errors in both coordinates,” Am. J. Phys. 57, 642–646 (1989). Google ScholarScitation
- 4. D. York, “Least squares fitting of a straight line with correlated errors,” Earth Planet. Sci. Lett. 5, 320–324 (1969). Google ScholarCrossref
- 5. K. A. Kermackand J. B. S. Haldane, “Organic correlation and allometry,” Biometrika 37 (1/2), 30–41 (1950). Google ScholarCrossref
- 6. Karl Pearson, “On lines and planes of closest fit to systems of points in space,” Philos. Mag. 2 (6), 559–572 (1901). Google ScholarCrossref
- 7. C. Brooks, I. Wendt, and W. Harre, “A two-error regression treatment and its application to Rb-Sr and initial ratios of younger Variscan granitic rocks from the Schwarzwald Massif, Southwest Germany,” J. Geophys. Res. 73, 6071–6084 (1968). Google ScholarCrossref
- 8. B. Cameron Reed, “Linear least-squares fits with errors in both coordinates. II. Comments on parameter variances,” Am. J. Phys. 60 (1), 59–62 (1992). Google ScholarScitation
- 9. D. Michael Titteringtonand Alex N. Halliday, “On the fitting of parallel isochrons and the method of maximum likelihood,” Chem. Geol. 26, 183–195 (1979). Google ScholarCrossref
- 10. W. Edwards Deming, Statistical Adjustment of Data (Wiley, New York, 1943). Google Scholar
- 11. J.-F. Minster, L.-P. Ricard, and C. J. Allègre, “ chronology of enstatite meteorites,” Earth Planet. Sci. Lett. 44, 420–440 (1979). Google ScholarCrossref
- 12. Grenville Turner, “ ages from lunar maria,” Earth Planet. Sci. Lett. 11, 169–191 (1971). Google ScholarCrossref
- 13. J. De Basabe Delgado, “Regresión lineal con incertidumbres en todas las variables: Aplicaciones en geocronologı́a al cálculo de isocronas,” M.Sc. thesis, CICESE, 2002. Google Scholar
- 14. F. Albarède, Introduction to Geochemical Modeling (Cambridge U.P., Cambridge, 1995). Google Scholar
- © 2004 American Association of Physics Teachers.
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