Fitting straight lines with replicated observations by linear regression: Part II. Testing for homogeneity of variances

Critical Reviews in Analytical Chemistry

Volumn 34, Issue 3-4, 2004, Pages 133-146

  Sayago, Ana ^a   Asuero, Agustin G ^a  

^a UNIVERSITY OF SEVILLE   (Spain)

Author keywords

Least squares method; Replicated observations; Straight line model; Straight lines; Testing for homogeneity of variances

Indexed keywords

DATA ACQUISITION; TESTING;

STRAIGHT LINE MODEL; TRANSFORMATIONS;

REGRESSION ANALYSIS;

MATHEMATICAL METHOD;

EID: 11144275386     PISSN: 10408347     EISSN: None     Source Type: Journal    
DOI: 10.1080/10408340490888599     Document Type: Review

References (131)

1. A. Sayago, M. Boccio, and A. G. Asuero, Fitting straight lines with replicated observations by linear regression: the least squares postulates. CRC Crit. Rev. Anal. Chem. 34 (2004):39-50.
2. R. de Levie, Curve fitting least squares. CRC Crit. Rev. Anal. Chem. 30 (2000):59-74.
3. P. Valko and S. Vajda, Advanced Scientific Computing in BASIC, with Applications in Chemistry, Biology and Pharmacology (Amsterdam: Elsevier, 1989):139-144.
4. M. Meloun, J. Militki, K. Kupla, and R. G. Brereton, The effect of influential data, model and method on the precision of unvariate calibration. Talanta 57 (2002):721-740.
5. K. A. Connors, Statistical Treatment of Data in Binding Constants: The Measurement of Molecular Complex Stability, Chapter 3. (New York: Wiley, 1987):103-138.
6. M. Davidian and P. D. Haaland, Regression and calibration with non constant error variance. Chem. Intel. Lab. Syst. 9 (1990):231-248.
7. W. E. Deming, Statistical Adjustment of Data (New York: Dover, 1964):14.
8. J. R. J. Belloto and T. D. Sokolovski, Residual analysis in regression. Am. J. Pharm. Educ. 49 (1985):295-303.
9. S. Bolton, Pharmaceutical Statistics: Practical and Clinical Applications, 3rd ed. (New York: Marcel Dekker, 1994).
10. P. P. Mager, Design Statistics in Pharmacochemistry (New York: Wiley, 1991).
11. E. Mullins, Statistics for the Quality Control Laboratory (Cambridge: RSC, 2003).
12. International Conference on Harmonization of Technical Requirements for Registration of Pharmaceuticals for Human Use (ICH), Guidelines for Industry: Validation of Analytical Procedures, Q2A (Geneva: ICH, 1994).
13. K. A. Brownlee, Statistical Methodology in Science and Engineering 3rd ed. (Malabar, Fla.: R. G. Krieger, 1984).
14. J. S. Hunter, Calibration and the straight line: current statistical practices. J. Assoc. Off. Anal. Chem. 64 (1981):574-583.
15. D. L. Mac Taggart and S. O. Farwell, Analytical use of linear regression. Part I. Regression procedures for calibration and quantitation. J. Assoc. Off. Anal. Chem. 75 (1992):594-607.
16. B. Jülicher, P. Gowik, and S. Uhlig, Assessment of detection methods in trace analysis by mean of a statistically based inhouse validation concept. Analyst 123 (1998):173-179.
17. J. Berkson, Are there two regressions? J. Am. Stat. Ass. 45 (1950):164-180.
18. M. Thompson, Regression methods in the comparison of accuracy. Analyst 107 (1982):1169-1180.
19. B. D. Ripley and M. Thompson, Regression techniques for the detection of analytical bias. Analyst 112 (1987):277-383.
20. F. J. Del Rio Bocio, J. Riu, R. Boqué, and F. X. Rius, Limit of detection in linear regression with errors in the concentration. J. Chemometr. 17 (2003):413-421.
21. J. M. Lisy, A. Cholvadova, and J. Kutej, Multiple straight-line least-squares analysis with uncertainties in all variables. Computers Chem. 14 (1990):189-192.
22. R. F. Martin, General Deming regression for estimating systematic bias and its confidence interval in the method-comparison studies. Clin. Chem. 46 (2000):100-104.
23. N. Draper and H. Smith, Applied Regression Analysis, 3rd ed. (New York: Wiley, 1998).
24. J. W. Einax, H. W. Zwanziger, and S. Geiß, Chemometrics Environmental Analysis. (Weinheim: VCH, 1997).
25. International Standard, Air Quality - Assessment of the Accuracy of the Calibration Function of a Measurement Method under Field Conditions ISO 9169 (Geneva: International Organization for Standardization, 1994).
26. International Standard, Air Quality - Assessment of the Accuracy of the Calibration Function of a Measurement Method under Field Conditions ISO/CD 13752 (Geneva: International Organization for Standardization, 1994).
27. J. Keith and G. Duggleby, What happens when the data are fitted to the wrong equation? Biochem. J. 171 (1978):513-517.
28. J. B. Kennedy and A. M. Neville, Basic Statistical Methods for Engineers and Scientists, 2nd ed. (New York: Harper, 1976).
29. G. Kateman, H. C. Smith, and L. Meites, Weighting in the interpretation of data for potentiometric acid-base titration by non linear regression. Anal. Chim. Acta 152 (1983):61-72.
30. A. Olin, Some statistical properties of the results obtained by the standard addition method demonstrated by computer simulation. Chem. Anal. (Warsaw) 40 (1995):939-951.
31. J. S. Garden, D. G. Mitchell, and W. N. Mills, Non constant variance techniques for calibration curve-based analysis. Anal. Chem. 52 (1980):3210-2315.
32. International Standard, Accuracy (Trueness and Precision) of Measurement Methods and Results: Part 2. Basic Method for the Determination of Repeatability of a Standard Measurement Methods, ISO 5725-2 (Geneva: International Organization for Standardization, 1994).
33. D. L. MacTaggart and S. O. Farwell, Analytical use of linear regression. Part II: Statistical error in both variables. J. Ass. Off. Anal. Chem. 75 (1992):608-614.
34. A. G. González and A. G. Asuero, Computational program for validating analytical results. Fresenius Z' Anal. Chem. 346 (1993 ):885-887.
35. J. D. Lee and T. D. Lee, Statistics and Computer Methods in BASIC (New York, Van Nostrand-Reinhold, 1982): 108-109.
36. P. M. Wanek, R. E. Whippe, T. E. Fickies, and P. M. Grant, Inaccuracies in the calculation of standard deviation with electronic calculators. Anal. Chem. 54 (1982):1877-1878.
37. H. E. Solberg, Inaccuracies in computer calculation of standard deviation. Anal. Chem. 55 (1983):1611.
38. B. D. McCullough and B. Wilson, On the accuracy of statistical procedures in Microsoft Excel 97. Comp. Stat. Data Anal. 31 (1999):27-37.
39. M. G. Cox and P. M. Harris, Design and use of reference data sets for testing scientific software. Anal. Chim. Acta 380 (1999):339-351.
40. A. G. Asuero and A. G. González, Some observations of fitting a straight line to data. Microchem. J. 40 (1989):216-225.
41. P. C. Meier and R. E. Zünd, Statistical Methods in Analytical Chemistry, 2nd ed. (New York: Wiley, 2000).
42. A. G. Asuero, G. Gonzalez, F. de Pablos, and J. L. Gomez Ariza, Determination of the optimum working range in spectrophotometric procedures, Talanta 35 (1988):531-537.
43. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge: Cambridge University Press, 1999).
44. J. N. Miller, Basic statistical methods for analytical chemistry. Part 2. Calibration and regression methods. A review. Analyst 116 (1991):3-14.
45. M. Mulholland and D. B. Hibert, Linearity and the limitations of least squares calibration. J. Chromatogr. A762 (1997):73-82.
46. J. N. Miller and J. C. Miller, Statistics and Chemometrics for Analytical Chemistry, 4th ed. (Essex: Ellis Horwood, 2000).
47. L. A. Currie, Recommendations in evaluation of analytical methods including detection and quantification capabilities. Pure Appl. Chem. 67 (1995):1699-1723.
48. S. M. Gort and R. Hoogerbrugge, A user-friendly spreadsheet program for calibration using weighted regression. Chem. Intell. Lab. System, 28 (1995):193-199.
49. M. E. Zorn, R. D. Gibbons, and W. C. Sonzogni, Weighted least squares approach to calculating limits of detection and quantification by modelling variability as a function of concentration. Anal Chem. 69 (1997):3069-3075.
50. R. de Levie, How to Use Excel in Analytical Chemistry and in General Scientific Data Analysis (Cambridge: Cambridge University Press, 2001).
51. E. Desimoni, A. Daghetta, and S. Valsecchi, Evaluation of uncertainty by ordinary linear least squares regression of replicated data. A revised formulation to deal with unbalanced data set. Ann. Chim. (Roma) 88 (1998):601-609.
52. W. Penninckk, C. Hartmann, and D. L. Massart, J. Smeyer-Verbeke, Validation of the calibration procedure in atomic spectrometric methods. J. Anal. At. Spectrosc. 11 (1996):237-246.
53. W. J. Dixon, Processing data outlier. Biometrics 9 (1953):74-89.
54. F. E. Grubbs, Procedures for detecting outlying observations in samples. Technometrics 11 (1969):1-21.
55. F. E. Grubbs and G. Beck, Extension of sample sizes and percentage points for significance tests of outlying observations. Technometrics 14 (1972):847-854.
56. S. Aknazarova and V. Kafarov, Experiment Optimization in Chemistry and Chemical Engineering (Moscow: Mir Publishers, 1982):50-52.
57. L. A. Currie and G. Svehla, Nomenclature for the presentation of results of chemical analysis. Pure Appl. Chem. 66 (1994):595-608.
58. M. J. Cardone, Detection and determination of error in analytical methodology. Part I. In the method verification program. J. Ass. Off. Anal. Chem. 66 (1983):1257-1282.
59. S. N. Deming and S. L. Morgan, Experimental Design: A Chemometric Approach, 2nd ed. (Amsterdam: Elsevier, 1993).
60. G. T. Wernimont, Use of Statistics to Develop and Evaluate Analytical Methods (ed. W. Spendley) (Arlington, VA: AOAC, 1985).
61. G. E. P. Box and N. R. Draper, Empirical-Model-Building and Response Surfaces (New York: Wiley, 1987).
62. R. L. Anderson, Practical Statistics for Analytical Chemistry (New York: Van Nostrand, 1987), 56, 109.
63. J. Cserminski, A. Iwasiewicz, Z. Paszek, and A. Sikorski, Statistical Methods in Applied Chemistry (Amsterdam: Elsevier, 1990), 425.
64. J. Mandel, The Statistical Analysis of Experimental Data (New York: Wiley-Interscience, 1964): 68, 352-353.
65. D. L. Massart, B. G. M. Vandeginste, S. N. Deming, Y. Michotte, and L. Kaufman, Chemometrics: A Textbook (Amsterdam: Elsevier, 1988).
66. J. H. Noggle, Practical Curve Fiting and Data Analysis. Software and Self-Instruction for Scientists and Engineers (Chichester: Ellis Horwood, 1993).
67. D. McCormick and A. Roach, Measurement Statistics and Computation (New York: Wiley, 1987), 303-321.
68. H. Bubert and R. Klockenkämper, Precision-dependent calibration in instrumental analysis. Fresenius Z' Anal. Chem. 316 (1983 ):186-193.
69. L. M. Schwarz, Calibration curve with non uniform variance. Anal. Chem. 51 (1979):723-727.
70. M. Thompson, Variation of precision with concentration in an analytical system. Analyst 112 (1988):1579-1587.
71. K. Baumann and H. Wätzig, Appropriate calibration functions for capillary electrophoresis. II. Heterocedasticity and its consequences. J. Chromatog. A700 (1995):9-20.
72. K. Baumann, Regression and calibration for analytical separation techniques. Part II. Validation, weighted and robust regression. Process and Control Quality 10 (1997):75-112.
73. S. N. Ketkar and T. J. Bzik, Calibration of analytical instruments. Impact of nonconstant variance in calibration data. Anal. Chem. 72 (2000):4762-4765.
74. J. K. G. Watson, Robust weighting in least squares fits. J. Mol. Spectr. 219 (2003):326-328.
75. J. Tellinghuisen, Statistical error calibration in UV-visible spectrophotometry. Appl. Spectrosc. 54 (2000):431-437.
76. J. Tellinghuisen, On the role of statistical weighting in the least squares analysis of UV-visible spectrophotometric data. Appl. Spectrosc. 54 (2000):1208-1213.
77. N. V. Nagaraja, J. K. Paliwal, and R. C. Gupta, Choosing the calibration model in assay validation. J. Pharm. Biom. Anal. 20 (1999 ):433-438.
78. Y. V. Heyden, The ruggedness of analytical methods. Analysis 22(5) (1994):M27-M29.
79. S. Dowy and S. Wearden, Statistics for Research, 2nd ed. (New York: Wiley, 1991), 348-356.
80. E. Desimoni, S. Mannino, and B. Brunetti, On the assessment of compliance with legal limits. Part 1: Signal concentration domains. Accred. Qual. Assur. 6 (2001):452-458.
81. J.-C. Lee, and M. H. Ramsey, Modelling measurement uncertainty as a function of concentration: an example from a contaminated land investigation. Analyst 126 (2001):1784-1791.
82. R. L. Watters, R. J. Carroll, and C. H. Spiegelman, Error modelling and confidence interval estimation for inductively coupled plasma calibration curves. Anal. Chem. 59 (1987):1639-1643.
83. L.-J. Hwang, Impact of variance function estimation in regression and calibration. Meth. Enzymol. 240 (1994):150-170.
84. E. D. Prudnikov and Y. S. Shapling, Random errors in analytical methods. Analyst 109 (1984):305-307.
85. J. H. Ottoway, Normalization in the fitting data by iterative methods. Application to tracer kinetics and enzyme kinetics. Biochem. J. 134 (1973):729-736.
86. R. D. Cook and S. Weisberg, Diagnostics for heteroscedasticity in regression. Biometrika 70 (1983):1-10.
87. J. P. Franke, R. A. de Zeeuw, and R. Hakkert, Evaluation and optimization of the standard addition method for absorption spectrometry and anodic stripping voltammetry. Anal. Chem. 50 (1978):1374-1380
88. International Standard, Water Quality - Calibration and Evaluation of Analytical Methods and Estimation of the Performance Characteristics: Part 1. Statistical Evaluation of the Linear Calibration Function, ISO, 8466-1 (Geneva: International Organization for Standardization, 1990).
89. International Standard, Water Quality - Calibration and Evaluation of Analytical Methods and Estimation of the Performance Characteristics: Part 2. Calibration Strategy for Non Linear Second Order Function, ISO, 8466-2 (Geneva: International Organization for Standardization, 1990).
90. H. Scheffe, The Analysis of Variance (New York: Wiley, 1959).
91. E. Desimoni, A program for the weighted least squares regression of unbalanced response arrays. Analyst 124 (1999):1191-1196.
92. M. S. Bartlett, Some examples of statistical methods of research in agriculture and applied biology. J. Roy. Statist. Soc. 4(2) (1937, suppl.):137-170.
93. M. S. Bartlett, Properties of sufficiency and statistical tests, Proc. Roy. Soc. Ser. A160 (1937):268-282.
94. M. S. Bartlett, A note on the multiplying factors for various approximations. J. Roy. Statist. Soc. Ser. B16 (1954):296-298.
95. W. G. Cochran, Some difficulties in the statistical analysis of replicated experiments, Empire Journal of Experimental Agriculture 6 (22) (1938):157-175.
96. W. G. Cochran, The distribution of the largest of a set of estimated variances as a fraction of the total. Ann. Eugenics 11 (1941):47-53.
97. W. G. Cochran, Some consequences when the assumptions for the analysis of the variance are not satisfied. Biometrics 3 (1947):22-38.
98. W. G. Cochran, The combination of estimates from different experiments, Biometrics 10 (1954):101-129.
99. I. W. Burr and L. A. Foster, A Test of Equality of Variances, Department of Statistics Mimeo Series No. 282 (Lafayette, Ind.: Purdue University, 1972).
100. E. Carbonell, J. B. Dewis, R. Calvo, F. Gonzalez, and W. Pruñozosa, Regresión lineal, un enfoque conceptual y práctico (Madrid: INIA, Ministerio de Agricultura, Pesca y Alimentación 1983), 97-99, 175-176.
101. H. O. Hartley, The maximum F-ratio as a short-cut test for heterogeneity of variance. Biometrika 37 (1950):308-312.
102. K. E. Peace, Biopharmaceutical Statistics for Drug Development (New York: Marcel Dekker, 1988).
103. L. Sach, Applied Statistics: A Handbook of Techniques, 2nd ed. (New York: Springer-Verlag, 1982).
104. S. D. Roof, An IBM 704 Computer Program for computing Bartlett's test of homogeneity of variance. Biometrics 18 (1962):251-252.
105. J. Philippe, Les Methodes Statistiques en Pharmacie et en Chimie (applications a la recherche, a la production et au controle) (Paris: Masson, 1967), 125-126.
106. D. J. Finney, Statistical Methods in Biological Assay, 3rd ed. (London: Griffin, 1978), 55-57.
107. S. Ebel and M. Lorz, Weyandt-Spangenberg, Optimierte kalibrierung. Fresenius Z. Anal. Chem. 335 (1989):960-965.
108. S. C. Gad, Statistics and Experimental Design for Toxicologists, 3th ed. (Boca Raton, Fla.: CRC Press, 1999), 42-44.
109. H. O. Hartley, Testing the homogeneity of a set of variances. Biometrika 31 (1940):249-255.
110. G. E. P. Box, Non-normality and tests on variances. Biometrika 40 (1953):318-335.
111. K. Danzer and L. A. Currie, IUPAC Guidelines for calibration in analytical chemistry. Part1. Fundamentals and single component calibration. Pure Appl. Chem. 70 (1998):993-1014.
112. M. Meloun and J. Militki, Computer-assisted data treatment in analytical chemistry. IV. Classical estimates of parameters of location, scale and shape. Chem. Papers 49 (1995):68-73.
113. W. R. Ott, Environmental Statistics and Data Analysis (Boca Raton, Fla.: CRC Press, 1995).
114. V. P. Spiridonov and A. A. Lopatkin, Tratamiento Matemático de Datos Físico-Químicos (Moscow: Mi, 1973), 99-101, 111-112.
115. R. Caulcutt and R. Boddy, Statistics for Analytical Chemistry, (London: Chapman and Hall, 1983), 70-71.
116. C. I. Bliss, W. G. Cochran, and J. W. Tukey, A rejection criterion based upon the range. Biometrika 43 (1956):418-422.
117. V. L. Anderson and R. A. McLean, Design of Experiments: A Realistic Approach (New York: Marcel Dekker, 1974).
118. J. Neter and W. Wasserman, Applied Linear Models (Homewood, Ill.: R. D. Irving, 1974).
119. F. J. Rohlf and R. R. Soakal, Statistical Tables, 3rd ed. (New York: W.H. Freeman, 1995).
120. H. Levene, Robust tests for equality of variances, in Contribution to Probability and Statistics, Essays in Honor of Harold Hotelling; ed. I. Olkin et al. (Stanford: Stanford University Press, 1960), 278-292.
121. B. Schultz, On Levene's test and other statisticas of variation. Evolutionary Theory 6 (1983):197-203.
122. J. R. Green and D. Margerison, Statistical Treatment of Experimental Data (Amsterdam: Elsevier, 1977).
123. M. B. Brown and A. B. Forsythe, The small sample behaviour of some statistics which test the equality of several means. Technometrics 16 (1974): 129-132.
124. R. G. O'Brien, Robust techniques for testing heterogeneity of variance effects in factorial designs. Psychometrika 43 (1978):327-342.
125. W. J. Conover, M. E. Johnson, and M. M. Johnson, A comparative study of tests for homogeneity of variances with applications to the outer continental shelf bidding data. Tecnometrics 23 (1981):351-361.
126. R. J. Carroll and H. Schneider, A note on Levene's tests for equality of variances. Statistics and Probability Letters 3 (1985): 191-194.
127. T.-S. Lim and W. Y. Loh, A comparison of tests of equality of variances. Comp. Stat. Data Anal. 22 (1996):287-301.
128. H. O. Hartley, The use of range in analysis of variance. Biometrika 37 (1950):271-280.
129. E. S. Pearson and H. E. Hartley, Biometrika Tables for Statiscians 8th ed. (Cambridge: Cambridge University Press, 1966), Table 316.
130. K. G. Owens, C. F. Bauer, and C. L. Grant, Effects of analytical calibration models on detection limit estimates, in Detection Limits in Analytical Chemistry, Importance, Theory and Practice, L. A. Currie, ed. ACS Symposium Series 361 (Washington, D.C.: ACS, 1988). chap. 10, 194-207,
131. R. T. Leslie and B. M. Brown, Use of range in testing heteroscedasticity of variance. Biometrika 53 (1966):221-227.