利用R^{2}(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。R^{2}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，”R^{2}”
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R^{2}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 leastsquares 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/R^{2} 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&R^{2} 




Statement:
“In fitting functional models values of r and R^{2}
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 
R^{2} 




Statement:
“A large value of R^{2} 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& R^{2} 




Statement:
“Values of r and R^{2} 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 R^{2} 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 
R^{2} 




Statement:
“Even if the standard curve has R^{2}>0.9990, the
fit will not necessarily be very good”
“R^{2} 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 &R^{2 } 




Statement:
“the improper recommendation to establish linearity that is most
frequently written into protocols and papers is the use of r
or R^{2}” 
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 
R^{2 } 




Statement:
“R^{2} 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 
R^{2} 




Statement:
“R^{2} 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 rtest 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 &R^{2 } 




Statement:
“r and R^{2 }are completely unrelated to
several phenomena that can occur during calibration. Very high
values can be obtained for curves with significant curvilinearity”

Reference 
Authors 
Data 
Coefficients 
[29] 
Rozet et al. 
2013 
R^{2 } 




Statement:
“R^{2} do not allow to properly select an
adequate response function for the calibration curve”

References
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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) 574583.
[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) 59.
[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?,”
LCGC 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) 594608.
[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,”
LCGC North Am. 22 (2004) 112–117.
[19] J. Ermer and H. J. Ploss, “Validation in pharmaceutical analysis:
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criteria and for verifying and improving the quality of analytical
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[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)
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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,
“Fitforpurpose 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.
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[29]
E.
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