Abstract
Calculations of isochrons by different methods are compared for the case when both variables are subject to error. Least-squares operations of the kind given by York (1969) and linear methods are compared. The advantage of the linear methods is demonstrated for the case of data reported by Housner and Brennan (1948), which we have modified for nonreplicated sampling. The sampling distribution of the slope is obtained and confidence intervals for it are given. The confidence intervals for the intercept on the ordinate and its distribution are obtained for fixed values of the slope. Examples of application of the method are given, based on data from the literature compiled in Appendix A.
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References
Bartlett, M. S., 1949, Fitting a Straight Line When Both Variables Are Subject to Error: Biometrics, v. 5, p. 207.
Brooks, C., Hart, S. R., and Wendt, I., 1972, Realistic Use of Two-Error Regression Treatment as Applied to Rubidium-Strontium Data: Reviews of Geophysics and Space Physics, v. 10, p. 551–578.
Cramer, H., 1946, Mathematical Methods of Statistics: Princeton University Press (Russian Translation, 1948, 631 p.).
Dorff, M., and Gurland, Z., 1961, Estimation of the Parameters of a Linear Function Relation: J. Roy. Stat. Soc., Ser. B, v. 23, p. 160–170.
Dorff, M., and Gurland, Z., 1961a, Small Sample Behaviour of Slope Estimation in a Linear Functional Relation: Biometrics, v. 17, p. 283.
Deming, W. E., 1943, Statistical Adjustment of Data: John Wiley & Sons, New York, 261 p.
Faure, G., and Powell, J. L., 1972, Strontium Isotope Geology: Springer-Verlag, New York, 188 p.
Faure, G., Personal communication (letter to A. B. Vistelius, September 11, 1989).
Gnedenko, B. V., 1965, Theory of Probability, four edition: Moscow, Nauka Press, 400 p. (in Russian).
Gray, C. M., and Compston, W., 1978. A Rubidium-Strontium Chronology of the Metamorphism and Prehistory of Central Australian Granulites: Geochim. Cosmochim. Acta, v. 42, p. 1735–1748.
Hensel, H. D., McCulloch, M. T., and Chappell, B. W., 1985, The New England Batholith: Constraints of Its Derivation from Nd and Sr Isotopic Studies of Granitoids and Country Rocks: Geochim. Cosmochim. Acta, v. 49, p. 369–384.
Housner, G. W., and Brennan, J. F., 1948, The Estimation of Linear Trends: Ann. Math. Stat., v. 19, p. 380–389.
Jones, T. A., 1979, Fitting Straight Lines When Both Variables Are Subject to Error. I. Maximum Likelihood and Least Squares Estimation: Math. Geol., v. 11, p. 1–26.
Kendall, M. G., and Stuart, A., 1967, The Advanced Theory of Statistics, vol. 2, Inference and Relationship, second edition: London, Ch. Griffin, 690 p.
Kralik, M., 1984, Effect of Cation Exchange Treatment and Acid Leaching on the Rb-Sr System of Illite from Fithian, Illinois: Geoch. Cosm. Acta, v. 48, p. 527–528.
Legendre, A. M., 1806, Nouvelles methodes pour la determination des orbites de cometes. 2. Appendice sur la metodes des moindre carres: Paris, 1806.
McIntyre, G. A., Brooks, C., Compston, W., and Turek, A., 1966, The Statistical Assessment of Rb-Sr Isochron: Jour. Geoph. Res., v. 71, p. 5459–5468.
Papanastassiou, D. A., and Wasserburg, G. J., 1969, Initial Strontium Isotopic Abundances and the Resolution of Small Time Differences in the Formation of Planetary Objects: Earth and Planetary Science Letters, v. 5, p. 361–376.
Vistelius, A. B., 1980, Principles of Mathematical Geology: Leningrad, Nauka Press, 389 p. (in Russian).
Vistelius, A. B., Gerling, E. C., and Gorokhovsky, B. M., 1982, On the Geochemistry of40Ar in Aplitic Rocks of Muroto Cape, Shikoky Island, Japan: Geochemistry, v. 6, p. 875–884 (in Russian).
Vugrinovich, R. G., 1981, A Distribution-Free Alternative to Least-Squares Regression and Its Application to Rb/Sr Isochron Calculation: Math. Geol., v. 13, p. 443–454.
Wald, A., 1940, The Fitting of Straight Lines If Both Variables Are Subject to Error: Ann. Math. Stat., v. 11, p. 284.
Walker, R. J., Shirey, S. B., and Stecher, O., 1987, Comparative Re-Os, Sm-Nd, and Rb-Sr Isotope and Trace Elements Systematics for Archean Komatiite Flews from Munro Township Abitibi Belt, Ontario: Cited from Carnegie Institution Year Book 86, The Presidents Report, 1986–1987.
Wilks, S. S., 1962, Mathematical Statistics: John Wiley & Sons, New York (Russian edition, 1968).
York, D., 1966, Least Square Fitting of a Straight Line: Can. Journ. Physics. v. 44, p. 1079–1087.
York, D., 1967, The Best Isochron: Earth and Planetary Science Letters, v. 2, p. 479–483.
York, D., 1969, Least Square Fitting of Straight Line with Correlated Errors: Earth and Planetary Science Letters, v. 5, p. 320–325.
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Vistelius, A.B., Faas, A.V. On the precision of age measurements by Rb-Sr isochrons in nontrivial cases. Math Geol 23, 999–1044 (1991). https://doi.org/10.1007/BF02066109
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DOI: https://doi.org/10.1007/BF02066109