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Atkinson, A. C. (1985). Plots, Transformations, and Regression: An Introduction to Graphical Methods of Diagnostic Regression Analysis. Oxford University Press, Oxford. Barnett, Y., and T. Lewis (1984). Outliers in Statistical Data, 2nd ed. Wiley, Chichester. Beckman, R J., and R D. Cook, (1983). Outiier.. ........ s. Technometrics, vol. 25, pp. 119-149. Belsley, D. A., E. Kuh, and R E. Welsch (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Wiley, New York. Box, G. E. P., W. G. Hunter, and J. S. Hunter (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. Wiley, New York. Carroll, R J., and D. Ruppert (1988). Transformation and Weighting in Regression. Chapman and Hall, New York. Cook, R. D., and S. Weisberg (1982). Residuals and Influence in Regression. Chapman and Hall, New York. Draper, N. R, and H. Smith (1981). Applied Regression Analysis, 2nd ed. Wiley, New York. Hampel, F. R, E. M. Ronehctti, P. J. Roussceuw, and W. A. Stahel (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York. Hawkins, D. M. (1980). Identification of Outliers. Chapman and Hall, London.
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McKean, J. W., S. J. Sheather, and T. P. Hettmansperger (1990). Regression diagnostics for rank-based methods. Journal of the American Statistical Association, vol. 85, pp. 1018-1028. Myers, R. H. (1990). Classical and Modern Regression with Applications. PWS-Kent, Boston. Rousseeuw, P. J., and A. M. Leroy (1987). Robust Regression and Outlier Detection. Wiley, New York. Sheather, S. J., and J. W. McKean (1992). The interpretation of residuals based on L1 estimation. In: Y. Dodge (ed.), L 1-Statistical Analysis and Related Methods. North-Holland, New York. Weisberg, S. (1985). Applied Linear Regression, 2nd cd. Wiley, New York.
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The most commonly used regression method is the method of least squares. It was discovered independently by Carl Friedrich Gauss in Germany around 1795 and by Adrien Marie Legendre in France around 1805. Early applications of the method were to astronomic and geodetic data. Its first published appearance was in 1805 in an appendix to a book by Legendre on determining the orbits of comets.
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3.2 AN EXAMPLE OF SIMPLE REGRESSION An experiment was conducted to find the relationship between two procedures for assessing the acid content of a chemical. The two procedures measure somewhat different but related quantities. The organic acid content of a sample of chemical can be determined by a method of extraction and weighing, which is expensive, but a relatively cheap titration method is available for determining the acid number. It was hoped that, by using regression, the cheap method could be used instead of the expensive method to measure organic acid content. Both procedures were used on 20 samples of chemical. The data are displayed in Table 3.1. Using these data we would like to obtain an equation that expresses the organic acid content measurement as an approximate function of the acid number measurement. For notation, let
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Yi = expensive organic acid content measurement of the ith chemical sample
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Xi =
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The data points (x"y,), ... ,(X 20 'Y20) are plotted in Figure 3.1. The relationship between X and Y appears to be approximately linear.
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Table 3.1 Acid Content Data
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123 109 62 104 57 37 44 100 16 28 138 105 159 75 88 164 169 167 149 167
Source: Daniel and Wood (I980, p. 46).
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Plot of the acid content data.
Figure 3.2 The residual point (x" Y,).
is the vertical distance from the estimated regression line to the data
Let us try the linear regression model given in equation (1.1). For this example, the value of p is 1. That is, we have only one explanatory variable, in which case the model is called a simple linear regression model. Strictly following equation (1.1), one would write Y = {3o + {31 X I + e, but to avoid unnecessary subscripts, let us write the model as