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相關係數(r)與決定係數(R2)之評論

 

中興大學 生物系統工程研究室  陳加忠

 
 

利用R2(Coefficient of determination )以評斷模式與數據的符合姓,這幾乎是對於統計學外行人的慣用方式。自1976年至今,已有許多不同領域之學者提出警語,但是並未被學術界加以重視。R2稱為決定係數,r稱為相關係數(Correlation coefficient)兩者常被混用,其實其意義並不相同。”r”代表兩個變數之間的相關程度,”R2代表一個回歸模式對於反應值(yresponse value)所能解釋之比例。許多檢定單位仍然以R2r為校正方程式的判別標準。近日一篇評論(review)對此加以整理,有關的評論與其原始文獻依出版年代排序如下。在此將原論文之內容重新排序整理,作為研究人員之參考,也由此可知,學問真理推行之不易。

 

Title: Evaluation of analytical calibration based on least-squares linear regression for instrumental techniques: A tutorial review

Author: Francisco Raposo

Trends in Analytical Chemistry

http://dx.doi.org/doi: 10.1016/j.trac.2015.12.006

 

 

Table 1. Literature relating to r/R2 as misleading linearity criterion

 

Reference

Authors

Data

Coefficients

[1]

Davis & Pryor 

1976

r

 

 

 

 

Statement:

r, although widely used as a measure of GOF, does not accurately reflect the deviations of points from the line”

 

Reference

Authors

Data

Coefficients

[2]

Hunter

1981

r&R2

 

 

 

 

Statement:

“In fitting functional models values of r and R2 close to ±1 do provide an aura of respectability, but not much else”

 

Reference

Authors

Data

Coefficients

[3]

Van Arendonk et al.

1981

r

 

 

 

 

Statement:

“ A practice that should be discouraged is the use of r as a means of evaluating goodness of fit of linear models”

 

Reference

Authors

Data

Coefficients

[4]

Mitchell & Garden

1982

r

 

 

 

 

Statement:

“The r value does not indicate whether the chosen mathematical model adequately fits the data”

 

Reference

Authors

Data

Coefficients

[5]

Analytical Methods Committee

1988

r

 

 

 

 

Statement:

“A large value of r does not indicate a linear relationship between two measurements”

“The r does not indicate linearity or the lack thereof”

 

Reference

Authors

Data

Coefficients

[6]

Sahai and Singh

1989

R2

 

 

 

 

Statement:

“A large value of R2 does not insure a good fit neither the model predict well”

 

Reference

Authors

Data

Coefficients

[7]

Thompson

1990

r

 

 

 

 

Statement:

r is often misapplied to calibration data in an attempt to support the presumption of linearity”

r≈1 does not necessarily imply an underlying linear relationship”

 

Reference

Authors

Data

Coefficients

[8]

Miller

1991

r

 

 

 

 

Statement:

“ The magnitude of r, considered alone, is a poor guide of linearity”

 

Reference

Authors

Data

Coefficients

[9]

Karnes and March

1991

r

 

 

 

 

Statement:

r is a poor indicator of how well a linear regression equation fits the linear model”

r is of little value in documenting adherence to the linear model”

 

Reference

Authors

Data

Coefficients

[10]

Miller

1991

r

 

 

 

 

Statement:

“A high value of r is thus seen to be no guarantee at all that a straight line rather than a curve, is appropriate for a given calibration plot”

 

Reference

Authors

Data

Coefficients

[11]

Cassidy & Janosky

1992

r& R2

 

 

 

 

Statement:

“Values of r and R2 tell us whether there is a reasonable probability that x and y are directly related. They are not intended to measure the degree of linearity of the line of best fit. Consequently, neither r nor R2 should be used the linearity of a calibration curve”

 

Reference

Authors

Data

Coefficients

[12]

MacTaggart & Farwell

1992

r

 

 

 

 

Statement:

r gives only a relative idea of the linearity inherent in a particular data set”

 

Reference

Authors

Data

Coefficients

[13]

Analytical Methods Committee

1994

r

 

 

 

 

Statement:

“Hence, r is misleading in the context of testing for linearity”

“It is better used for correlation, not for quantify linearity”

 

Reference

Authors

Data

Coefficients

[14]

Mulholland & Hibbert

1997

r& R2

 

 

 

 

Statement:

“Many analysts depend entirely on the use of R2 (or r) value between 0.999 and 1.000 as an acceptability criterion. This is well known to be inadequate”

r does not give any indication of the errors associated with an individual measurement”

 

Reference

Authors

Data

Coefficients

[15]

Van Loco et al.

2002

r

 

 

 

 

Statement:

r is not useful indicator of linearity in the calibration model, even for r>0.997”

“r is not suitable for assessing the linearity of calibration curves”

 

Reference

Authors

Data

Coefficients

[16]

De Levie

2003

r

 

 

 

 

Statement:

r can easily be misinterpreted by chemists as a measure of GOF which it is not”

 

Reference

Authors

Data

Coefficients

[17]

Huber

2004

r

 

 

 

 

Statement:

r describes the quality of the fit only poorly and its linearity not at all”

“If r is used for testing the quality of the fit with subsequent proof of linearity, it is severely biased”

 

Reference

Authors

Data

Coefficients

[18]

Kiser & Dolan

2004

R2

 

 

 

 

Statement:

“Even if the standard curve has R2>0.9990, the fit will not necessarily be very good”

R2 is a poor measure of the curve fit quality”

 

Reference

Authors

Data

Coefficients

[19]

Emer

2005

r

 

 

 

 

Statement:

r is neither a proof or linearity, nor a suitable quantitative parameter”

 

Reference

Authors

Data

Coefficients

[20]

Hibbert

2005

r

 

 

 

 

Statement:

r is not the statistic of choice to determine the extent of linearity”

Reference

Authors

Data

Coefficients

[21]

De Souza & Junqueira

2006

r &R2

 

 

 

 

Statement:

“the improper recommendation to establish linearity that is most frequently written into protocols and papers is the use of r or R2

 

Reference

Authors

Data

Coefficients

[22]

Asuero et al.

2006

r

 

 

 

 

Statement:

r close to unity does not necessarily indicate a linear calibration function”

“Analyst should avoid being misled by r”

“It is surprising that r had been used so frequently to assess the linearity of calibration graphs”

“In short, r value is in reality not a measure of model adequacy”

 

Reference

Authors

Data

Coefficients

[23]

Lee et al.

2006

R2

 

 

 

 

Statement:

R2 is not useful for evaluating the quality of a calibration curve model because it does not penalize model complexity and consequently encourages overfitting”

 

Reference

Authors

Data

Coefficients

[24]

Sonnergaard

2006

r

 

 

 

 

Statement:

r is often misused as a universal parameter expressing the quality in linear regression analysis”

 

 

Reference

Authors

Data

Coefficients

[25]

Singtoroj et al.

2006

R2

 

 

 

 

Statement:

R2 alone is not adequate to demonstrate linearity since values above 0.999 can be achieved even when the data shows signs of curvature”

 

Reference

Authors

Data

Coefficients

[26]

Analytical Methods Committee

2006

r

 

 

 

 

Statement:

“Given the importance of linear calibration, it is strange that most analytical chemists are willing to use r as an indicator of linearity”

r in the context of linearity testing is potentially misleading, and should be avoided.

 

Reference

Authors

Data

Coefficients

[27]

Araujo

2009

r

 

 

 

 

Statement:

“It is extremely important to emphasize that an r-test to check the linearity does not exist. We cannot say that r=0.999 is more linear that r= 0.997”

 

Reference

Authors

Data

Coefficients

[28]

Komsta

2012

r &R2

 

 

 

 

Statement:

r and R2 are completely unrelated to several phenomena that can occur during calibration. Very high values can be obtained for curves with significant curvi-linearity”

 

Reference

Authors

Data

Coefficients

[29]

Rozet et al.

2013

R2

 

 

 

 

Statement:

R2 do not allow to properly select an adequate response function for the calibration curve”

 

References

[1] W. H. Davis Jr. and W. A. Pryor, “Measures of goodness of fit in linear free energy relationships,” J. Chem. Educ. 53 (1976) 285–287.

[2] J.S. Hunter, “Calibration and the straight line: current statistical practices”, J. Assoc. Anal. Chem. 64 (1981) 574-583.

[3] M. D. Van Arendonk, R. K. Skogerboe, and C. L. Grant, “Correlation coefficients for evaluation of analytical calibration curves,” Analytical Chemistry 53 (19781) 2349–2350.

[4] D. G. Mitchell and J. S. Garden, “Measuring and maximizing precision in analyses based on use of calibration graphs,” Talanta 29 (1982) 921–929.

[5] Analytical Methods Committee, “Uses (Proper and improper) of correlation coefficients,” Analyst 113 (1988) 1469–1471.

[6] H. Sahai, R.P. Singh, “The use of R2 as a measure of goodness of fit: an overview”, Va J Sci 40 (1989) 5-9.

[7] M. Thompson, “Statistics. Abuse of statistics software packages,” Anal. Proc. 27 (1990) 142–144.

[8] J.N. Miller, “Is it a straight line?,” Spectrosc. Int. 3 (1991) 41–43.

[9] H. T. Karnes and C. March, “Calibration and validation of linearity in chromatographic biopharmaceutical analysis,” J. Pharm. Biomed. Anal. 9 (1991) 911–918.

[10] J. N. Miller, “Basic statistical methods for analytical chemistry. Part 2. Calibration and regression methods. A review”, Analyst 116 (1991) 3–14.

[11] R. Cassidy; M. Janoski, “Is your calibration curve linear?,” LC-GC 10 (1992) 692–695.

[12] D.L. MacTaggart, S.O. Farwell, Analytical use of linear regression. Part I: regression procedures for calibration and quantitation, J. of AOAC Int 75 (1992) 594-608.

[13] Analytical Methods Committee, “Is my calibration linear?”, Analyst 119 (1994) 2363–2366.

[14] M. Mulholland and D. B. Hibbert, “Linearity and the limitations of least squares calibration,” J. Chromatogr. A 762 (1997) 73–82.

[15] J. Van Loco, M. Elskens, C. Croux, and H. Beernaert, “Linearity of calibration curves: Use and misuse of the correlation coefficient,” Accredit. Qual. Assur. 7 (2002) 281–285.

[16] R. De Levie, “Two linear correlation coefficients,” J. Chem. Educ. 80 (2003) 1030–1032.

[17] W. Huber, “On the use of the correlation coefficient r for testing the linearity of calibration functions,” Accredit. Qual. Assur. 9 (2004) 726.

[18] M. M. Kiser and J. W. Dolan, “Selecting the best curve fit,” LC-GC North Am. 22 (2004) 112–117.

[19] J. Ermer and H. J. Ploss, “Validation in pharmaceutical analysis: Part II: Central importance of precision to establish acceptance criteria and for verifying and improving the quality of analytical data,” J. Pharm. Biomed. Anal. 37 (2005) 859–870.

[20] D. B. Hibbert, “Further comments on the (miss-)use of r for testing the linearity of calibration functions,” Accredit. Qual. Assur. 10 (2005) 300–301.

[21] S. V. C. De Souza and R. G. Junqueira, “A procedure to assess linearity by ordinary least squares method”, Anal. Chim. Acta 552 (2005) 23–35.

[22] A. G. Asuero, A. Sayago, and A. G. González, “The correlation coefficient: An overview,” Crit. Rev. Anal. Chem. 36 (2007) 41–59.

[23] J. W. Lee, V. Devanarayan, Y. C. Barrett, R. Weiner, J. Allinson, S. Fountain, S. Keller, I. Weinryb, M. Green, L. Duan, J. a. Rogers, R. Millham, P. J. O’Brien, J. Sailstad, M. Khan, C. Ray, and J. a. Wagner, “Fit-for-purpose method development and validation for successful biomarker measurement,” Pharm. Res. 23 (2006) 312–328.

[24] J.M. Sonnergaard, “On the misinterpretation of the correlation coefficient in pharmaceutical sciences,” Int. J. Pharm. 321 (2006) 12–17.

[25] T. Singtoroj, J. Tarning, a. Annerberg, M. Ashton, Y. Bergqvist, N. J. White, N. Lindegardh, and N. P. J. Day, “A new approach to evaluate regression models during validation of bioanalytical assays,” J. Pharm. Biomed. Anal. 41 (2006) 219–227.

[26] “AMC Technical Brief 3,” Committee, Anal. Methods 3(2006) 1–2.

[27] P. Araujo, “Key aspects of analytical method validation and linearity evaluation”, J. Chromatogr. B 877 (2009) 2224–2234.

[28] L. Komsta, “Chemometric and statistical evaluation of calibration curves in pharmaceutical analysis - a short review on trends and recommendations,” J. AOAC Int. 95 (2012) 669–672.

[29] E. Rozet, E. Ziemons, R. D. Marini, and P. Hubert, “Usefulness of information criteria for the selection of calibration curves”, Anal. Chem. 85 (2013) 6327– 6335.