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Least squares
International Tables for Crystallography (2006). Vol. C, ch. 8.1, pp. 678-688 [ doi:10.1107/97809553602060000609 ]
... of definitions and fundamental operations in linear algebra (Stewart, 1973; Prince, 1994) and of basic definitions and concepts in mathematical statistics ... vector v is transformed by Hv into the vector , where e represents a vector with e1 = 1, and ei = 0 for ... the estimated standard deviation or standard uncertainty (Schwarzenbach, Abrahams, Flack, Prince & Wilson, 1995) of . This criterion states that the ...
Influential data points
International Tables for Crystallography (2006). Vol. C, Section 8.5.3, p. 708 [ doi:10.1107/97809553602060000613 ]
... to that for (8.4.4.7) , it can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that Note that, if , all elements of become ... . Theory of optimal experiments, translated by W. J. Studden & E. M. Klimko. New York: Academic Press. Prince, E. & Nicholson, W. L. (1985). Influence of individual ...
Lack of fit
International Tables for Crystallography (2006). Vol. C, Section 8.5.2, pp. 707-708 [ doi:10.1107/97809553602060000613 ]
... systematic error to be characterized. References Abrahams, S. C. & Keve, E. T. (1971). Normal probability plot analysis of error in ... diagnostics. New York: John Wiley & Sons. Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics. New York: John Wiley & Sons. ...
Accuracy
International Tables for Crystallography (2006). Vol. C, Section 8.5.1, p. 707 [ doi:10.1107/97809553602060000613 ]
... nature's `correct' value lies, as well as its width (Prince, 1994). In performing a refinement, we have assumed implicitly ... expression of uncertainty in measurement. Geneva: International Organization for Standardization. Prince, E. (1994). Mathematical techniques in crystallography and materials science, ...
Detection and treatment of systematic error
International Tables for Crystallography (2006). Vol. C, ch. 8.5, pp. 707-709 [ doi:10.1107/97809553602060000613 ]
... nature's `correct' value lies, as well as its width (Prince, 1994). In performing a refinement, we have assumed implicitly ... to that for (8.4.4.7) , it can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that Note that, if , all elements of become ... require great patience and perseverance. References Abrahams, S. C. & Keve, E. T. (1971). Normal probability plot analysis of error ...
Influence of individual data points
International Tables for Crystallography (2006). Vol. C, Section 8.4.4, pp. 705-706 [ doi:10.1107/97809553602060000612 ]
... not all carry the same amount of information (Shoemaker, 1968; Prince & Nicholson, 1985). For an obvious example, consider a space ... observations in their leverage. It can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that the observations with the greatest leverage also ... . Theory of optimal experiments, translated by W. J. Studden & E. M. Klimko. New York: Academic Press. Prince, E. & Nicholson, ...
Comparison of different models
International Tables for Crystallography (2006). Vol. C, Section 8.4.3, pp. 704-705 [ doi:10.1107/97809553602060000612 ]
... models, was introduced by Williams & Kloot (1953; also Himmelblau, 1970; Prince, 1982). Consider a set of observations, , and two models ... the program from which this table was generated appears in Prince (1994). Table 8.4.3.1| | Values of t for which the ... . Process analysis by statistical methods. New York: John Wiley. Prince, E. (1982). Comparison of the fits of two ...
The F distribution
International Tables for Crystallography (2006). Vol. C, Section 8.4.2, pp. 703-704 [ doi:10.1107/97809553602060000612 ]
... the program from which the table was generated appears in Prince (1994). Table 8.4.2.1| | Values of the F ratio for ... estimation, hypothesis testing and least squares. New York: Ronald Press. Prince, E. (1994). Mathematical techniques in crystallography and materials science, ...
The [chi]2 distribution
International Tables for Crystallography (2006). Vol. C, Section 8.4.1, pp. 702-703 [ doi:10.1107/97809553602060000612 ]
... The beta function is defined by It can be shown (Prince, 1994) that . Making the substitution , (8.4.1.12) becomes By a ... Filliben, unpublished). Fortran code for this program appears in Prince (1994). Table 8.4.1.1| | Values of [chi]2/[nu] for ... H. (1981). Applied regression analysis. New York: John Wiley. Prince, E. (1994). Mathematical techniques in crystallography and materials ...
Statistical significance tests
International Tables for Crystallography (2006). Vol. C, ch. 8.4, pp. 702-706 [ doi:10.1107/97809553602060000612 ]
... The beta function is defined by It can be shown (Prince, 1994) that . Making the substitution , (8.4.1.12) becomes By a ... Filliben, unpublished). Fortran code for this program appears in Prince (1994). Table 8.4.1.1| | Values of [chi]2/[nu] for ... the program from which the table was generated appears in Prince (1994). Table 8.4.2.1| | Values of the F ratio ...
International Tables for Crystallography (2006). Vol. C, ch. 8.1, pp. 678-688 [ doi:10.1107/97809553602060000609 ]
... of definitions and fundamental operations in linear algebra (Stewart, 1973; Prince, 1994) and of basic definitions and concepts in mathematical statistics ... vector v is transformed by Hv into the vector , where e represents a vector with e1 = 1, and ei = 0 for ... the estimated standard deviation or standard uncertainty (Schwarzenbach, Abrahams, Flack, Prince & Wilson, 1995) of . This criterion states that the ...
Influential data points
International Tables for Crystallography (2006). Vol. C, Section 8.5.3, p. 708 [ doi:10.1107/97809553602060000613 ]
... to that for (8.4.4.7) , it can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that Note that, if , all elements of become ... . Theory of optimal experiments, translated by W. J. Studden & E. M. Klimko. New York: Academic Press. Prince, E. & Nicholson, W. L. (1985). Influence of individual ...
Lack of fit
International Tables for Crystallography (2006). Vol. C, Section 8.5.2, pp. 707-708 [ doi:10.1107/97809553602060000613 ]
... systematic error to be characterized. References Abrahams, S. C. & Keve, E. T. (1971). Normal probability plot analysis of error in ... diagnostics. New York: John Wiley & Sons. Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics. New York: John Wiley & Sons. ...
Accuracy
International Tables for Crystallography (2006). Vol. C, Section 8.5.1, p. 707 [ doi:10.1107/97809553602060000613 ]
... nature's `correct' value lies, as well as its width (Prince, 1994). In performing a refinement, we have assumed implicitly ... expression of uncertainty in measurement. Geneva: International Organization for Standardization. Prince, E. (1994). Mathematical techniques in crystallography and materials science, ...
Detection and treatment of systematic error
International Tables for Crystallography (2006). Vol. C, ch. 8.5, pp. 707-709 [ doi:10.1107/97809553602060000613 ]
... nature's `correct' value lies, as well as its width (Prince, 1994). In performing a refinement, we have assumed implicitly ... to that for (8.4.4.7) , it can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that Note that, if , all elements of become ... require great patience and perseverance. References Abrahams, S. C. & Keve, E. T. (1971). Normal probability plot analysis of error ...
Influence of individual data points
International Tables for Crystallography (2006). Vol. C, Section 8.4.4, pp. 705-706 [ doi:10.1107/97809553602060000612 ]
... not all carry the same amount of information (Shoemaker, 1968; Prince & Nicholson, 1985). For an obvious example, consider a space ... observations in their leverage. It can be shown (Fedorov, 1972; Prince & Nicholson, 1985) that the observations with the greatest leverage also ... . Theory of optimal experiments, translated by W. J. Studden & E. M. Klimko. New York: Academic Press. Prince, E. & Nicholson, ...
Comparison of different models
International Tables for Crystallography (2006). Vol. C, Section 8.4.3, pp. 704-705 [ doi:10.1107/97809553602060000612 ]
... models, was introduced by Williams & Kloot (1953; also Himmelblau, 1970; Prince, 1982). Consider a set of observations, , and two models ... the program from which this table was generated appears in Prince (1994). Table 8.4.3.1| | Values of t for which the ... . Process analysis by statistical methods. New York: John Wiley. Prince, E. (1982). Comparison of the fits of two ...
The F distribution
International Tables for Crystallography (2006). Vol. C, Section 8.4.2, pp. 703-704 [ doi:10.1107/97809553602060000612 ]
... the program from which the table was generated appears in Prince (1994). Table 8.4.2.1| | Values of the F ratio for ... estimation, hypothesis testing and least squares. New York: Ronald Press. Prince, E. (1994). Mathematical techniques in crystallography and materials science, ...
The [chi]2 distribution
International Tables for Crystallography (2006). Vol. C, Section 8.4.1, pp. 702-703 [ doi:10.1107/97809553602060000612 ]
... The beta function is defined by It can be shown (Prince, 1994) that . Making the substitution , (8.4.1.12) becomes By a ... Filliben, unpublished). Fortran code for this program appears in Prince (1994). Table 8.4.1.1| | Values of [chi]2/[nu] for ... H. (1981). Applied regression analysis. New York: John Wiley. Prince, E. (1994). Mathematical techniques in crystallography and materials ...
Statistical significance tests
International Tables for Crystallography (2006). Vol. C, ch. 8.4, pp. 702-706 [ doi:10.1107/97809553602060000612 ]
... The beta function is defined by It can be shown (Prince, 1994) that . Making the substitution , (8.4.1.12) becomes By a ... Filliben, unpublished). Fortran code for this program appears in Prince (1994). Table 8.4.1.1| | Values of [chi]2/[nu] for ... the program from which the table was generated appears in Prince (1994). Table 8.4.2.1| | Values of the F ratio ...
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