相關係數(r)與決定係數(R2)之評論

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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(y，response value)所能解釋之比例。許多檢定單位仍然以R²或r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 R²are 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.