Two different goals in fitting straight lines to data are to estimate a “true” linear relation (physical law) and to predict values of the dependent variable with the smallest possible error. Regarding the first goal, a Monte Carlo study indicated that the structural-analysis (SA) method of fitting straight lines to data is superior to the ordinary least-squares (OLS) method for estimating “true” straight-line relations. Number of data points, slope and intercept of the true relation, and variances of the errors associated with the independent (X) and dependent (Y) variables influence the degree of agreement. For example, differences between the two line-fitting methods decrease as error in X becomes small relative to error in Y. Regarding the second goal—predicting the dependent variable—OLS is better than SA. Again, the difference diminishes as X takes on less error relative to Y. With respect to estimation of slope and intercept and prediction of Y, agreement between Monte Carlo results and large-sample theory was very good for sample sizes of 100, and fair to good for sample sizes of 20. The procedures and error measures are illustrated with two geologic examples.
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Anderson, T. W., 1976, Estimation of Linear Functional Relations: Approximate Distributions and Connections with Simultaneous Equations in Econometrics: J. Roy. Stat. Soc. Ser. B, v. 38, p. 1–36.
Bray, D. I., 1979, Estimating Average Velocity in Gravel-Bed Rivers: J. Hydraulics Division, Am. Soc. Civ. Eng., v. 105, p. 1103–1122.
Fuller, W. A., 1987, Measurement Error Models: Wiley, New York, 440 p.
Guy, H. P., Simons, D. B., and Richardson, E. V., 1966, Summary of Alluvial Channel Data from Flume Experiments, 1956–61: U.S. Geol. Surv. Prof. Pap. 462-I, 96 p.
Kennedy, J. F., 1961, Stationary Waves and Antidunes in Alluvial Channels: W. M. Keck Lab. of Hydraulics and Water Resources, Calif. Inst. Technology, Pasadena, Report no. KH-R-2, 146 p.
Limerinos, J. T., 1970, Determination of the Manning Coefficient from Measured Bed Roughness in Natural Channels: U.S. Geol. Surv. Water-Supply Pap. 1898-B, 47 p.
Madansky, A., 1959, The Fitting of Straight Lines When Both Variables Are Subject to Error: J. Am. Stat. Assoc., v. 54, p. 173–205.
Mark, D. M., and Church, M., 1977, On the Misuse of Regression in Earth Science: Math. Geol., v. 9, p. 63–75.
Mejia, J. M., and Rodriguez-Iturbe, I., 1974, Correlation Links Between Normal and Log Normal Processes: Water Resour. Res., v. 10, p. 689–690.
Osterkamp, W. R., McNellis, J. M., and Jordan, P. R., 1978, Guidelines for the Use of Structural Versus Regression Analysis in Geomorphic Studies: U.S. Geol. Surv. Water-Resources Investigations 78–135, 22 p.
Richardson, D. H., and Wu, D.-M., 1970, Least-Squares and Grouping Method Estimates in the Errors-in-Variables Model: J. Am. Stat. Assoc., v. 65, p. 724–748.
Robertson, C. A., 1974, Large-Sample Theory for the Linear Structural Relation: Biometrika, v. 61, p. 353–359.
Troutman, B. M., and Williams, G. P., 1987, Fitting Straight Lines in the Earth Sciences,in Size, W. B. (Ed.), Use and Abuse of Statistical Methods in the Earth Sciences: Oxford University Press, New York, p. 107–128.
Williams, G. P., 1970a, Manning Formula—A Misnomer?: J. Hydraulics Div. Am. Soc. Civ. Eng., v. 96, p. 193–200.
Williams, G. P., 1970b, Flume Width and Water Depth Effects in Sediment-Transport Experiments: U.S. Geol. Surv. Prof. Pap. 562-H, 37 p.
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Williams, G.P., Troutman, B.M. Comparison of structural and least-squares lines for estimating geologic relations. Math Geol 22, 1027–1049 (1990). https://doi.org/10.1007/BF00890122
- structural analysis
- least squares
- parameter estimation
- Monte Carlo