Comment on “Calculating the Confidence and Prediction Limits of a

Letter pubs.acs.org/jchemeduc

Comment on “Calculating the Confidence and Prediction Limits of a Rate Constant at a Given Temperature from an Arrhenius Equation Using Excel” Pascal Pernot* Laboratoire de Chimie Physique, UMR8000 CNRS/U-Psud, Bât. 349, Univ. Paris-Sud, 91405 Orsay, France ABSTRACT: In its introductory sections, the title article seems to assert that there is a failure of the method of propagation of uncertainties (PoU), when applied to a linearized Arrhenius model. In consequence, the PoU formula is replaced by an expression of the confidence limits that can be easily implemented in Excel. This letter explicits that the observed “failure” is in fact due to an improper use of PoU, neglecting the essential term involving the parameters covariance, and that the replacement formula is only an alternative expression of the correct PoU formula. KEYWORDS: Second-Year Undergraduate, Physical Chemistry, Computer-Based Learning, Kinetics

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K amounts to s2ln(k) = 0.337201. However, the covariance term is of the same order of magnitude with opposite sign (−0.337061), which results in a strong variance cancellation. The prediction uncertainty resulting from eq 2 is therefore scln(k) = 0.012, much smaller than the value obtained in RH2017 for the incomplete equation (sln(k) = 0.581). It is therefore illegitimate to claim that the PoU method fails in this case because of “enormous” prediction uncertainty. The failure is rather due to an incorrect application of the PoU approach. In a second part, the author turns to an expression for the confidence limits in the prediction by a linear regression model (eq RH2017-3). It is surprising that he does not relate this expression to the PoU method: as shown in the next section, it is in fact a simple reformulation of eq 2.10,13−15 Recalculating the uncertainty provided by eq 2 at 298 K (scln(k)= 0.012) and enlarging it by the Student’s factor for two degrees of freedom (tcrit = 4.3), one recovers indeed the value obtained in RH2017 for the 95% confidence limit: cl(298) = 0.0517.

INTRODUCTION In the section “Identifying the Problem” of the title article1 (referred to as RH2017 in the following), its author seems to imply that there is a failure of error propagation by the method of propagation of uncertainties (PoU), when applied to a linearized Arrhenius model. He then attempts to justify this failure by considerations about the nonvalidity of the Arrhenius law for extrapolation at high temperature, which is a completely unrelated problem, considering that he seeks prediction uncertainty within the calibration temperature range. In fact, for such a linear model, application of the PoU approach is fully legitimate, as documented in the international standard “guide to the expression of uncertainty in measurement”, also known as the GUM, first published in 1993 by the BIPM.2 So, where is the problem?

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BACK TO THE FUNDAMENTALS The problem lies in the fact that eq RH2017-2 (eq 2 of the RH2017 article) s ln(k) =

{

2 s ln( A) +

1 2 s E /R T2 a

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1/2

}

DEMONSTRATIONS For ordinary least-squares (OLS) linear regression, y = a + b*x, the best fit parameters (â, b̂), their uncertainties (ua, ub) and covariance (uab) can be estimated analytically from the data set {xi,yi; i = 1, n} . The corresponding formulas can be found in classical textbooks.16−18 However, the omission of the covariance term formula is not unusual,16,17 which might be a source of confusion when prediction is considered. For the sake of consistency, we report here the main steps in the derivation of eqs RH2017-3/5 by the PoU approach. The link with the Arrhenius equation is done by taking x = −1/T, y = ln k, a = ln A, and b = Ea/R.

(1)

is not the correct expression of the PoU for the parameters of a linear regression. The parameters issued from the least-squares fit are generally correlated, and the proper PoU expression should take the parameters covariance into account,2−5 leading to an additional term in the equation, i.e. c s ln( k) =

{

2 s ln( A) +

1 2 2 s − s ln(A)s Ea / R ρ 2 Ea / R T T

1/2

}

(2)

where ρ = cor[ln(A), Ea/R] is the parameters correlation coefficient. This has been repeatedly demonstrated in the present Journal.6−12 Using the example data in RH2017 and the formulas provided in the next section, one finds that the correlation coefficient between the linear fit parameters is very large (ρ = 0.99985), a very common condition in Arrhenius regression. The variance resulting from eq 1 for a prediction at T = 298.15 © XXXX American Chemical Society and Division of Chemical Education, Inc.

Calibration

1. The best fit parameters can be expressed as Received: April 26, 2017 Revised: June 21, 2017

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DOI: 10.1021/acs.jchemed.7b00251 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education b̂ =

Letter

cl(y) = ±tcrituy

∑i (xi − x ̅ )yi ∑i (xi − x ̅ )

3. To obtain the prediction limits for a new measurement (eq RH2017-5), uy has to be combined quadratically with the expected measurement uncertainty (sreg):

a ̂ = y ̅ − bx̂ ̅ (3)

where x̅ and y ̅ are the arithmetic means of the variables. The latter equation shows clearly that â and b̂ are correlated, unless x̅ is null. This explains why centering x before the regression (the “de Levie approach” in RH2017) cancels the parameters covariance and solves the “problem” of eq 1, in addition to solving possible numerical problems due to a strong correlation.12 2. By assuming that the yi values have an unknown uniform measurement uncertainty, estimated by sreg =

1 n−2

2 pl(y) = ±tcrit uy2 + sreg

CONCLUSION The importance of parameters covariance in Arrhenius fitting has been stated repeatedly in the past decades, and the method of uncertainty propagation has been standardized more than 25 years ago.2 Calculating error limits for rate constants derived from a fitted Arrhenius expression should not be problematic any more, and 21st century students should not be left with the impression that uncertainty propagation is unable to deal with such simple linear problems. They should also be aware that taking into account the covariance of a model’s parameters is an essential feature of reliable predictions by this model.

n

∑ (yi − a ̂ − bx̂ i)2 i=1

(4)

ua2 2 uab

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1 ∑i (xi − x ̅ )2

=

2 sreg

=

2 ⎜1 sreg ⎜

=

2 sreg

(8)

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and that the xi values are not uncertain (OLS hypotheses), one can apply PoU to the above expressions and obtain ub2

(7)

2

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

⎛

⎞ x̅ 2 ⎟ + 2⎟ ∑i (xi − x ̅ ) ⎠ ⎝n

ORCID

−x ̅ ∑i (xi − x ̅ )2

The author declares no competing financial interest.

Pascal Pernot: 0000-0001-8586-6222 Notes

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ACKNOWLEDGMENTS The author is grateful to a reviewer for mentioning a sign error in the manuscript.

(5)

The latter expression is obtained by developing the covariance and observing that cov(y,̅ b̂)=0: 2 = cov(a ̂, b)̂ uab

REFERENCES

(1) Hites, R. A. Calculating the Confidence and Prediction Limits of a Rate Constant at a Given Temperature from an Arrhenius Equation Using Excel. J. Chem. Educ. 2017, 94, 398−400. (2) BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML. Evaluation of Measurement DataGuide to the Expression of Uncertainty in Measurement (GUM); Technical Report 100:2008; Joint Committee for Guides in Metrology: Sévres, France, 2008. (3) Heberger, K.; Kemeny, S.; Vidoczy, T. On The Errors Of Arrhenius Parameters And Estimated Rate-Constant Values. Int. J. Chem. Kinet. 1987, 19, 171−181. (4) Hébrard, E.; Pernot, P.; Dobrijevic, M.; Carrasco, N.; Bergeat, A.; Hickson, K. M.; Canosa, A.; Le Picard, S. D.; Sims, I. R. How measurements of rate coefficients at low temperature increase the predictivity of photochemical models of Titan’s atmosphere. J. Phys. Chem. A 2009, 113, 11227−11237. (5) Turányi, T.; Nagy, T.; Zsély, I. G.; Cserháti, M.; Varga, T.; Szabó, B. T.; Sedyó, I.; Kiss, P. T.; Zempléni, A.; Curran, H. J. Determination of rate parameters based on both direct and indirect measurements. Int. J. Chem. Kinet. 2012, 44, 284−302. (6) Wentworth, W. E. Rigorous least squares adjustment: application to some non-linear equations, I. J. Chem. Educ. 1965, 42, 96−103. (7) Kim, H. Computer programming in physical chemistry laboratory: least-squares analysis. J. Chem. Educ. 1970, 47, 120−122. (8) Sands, D. E. Correlation and covariance. J. Chem. Educ. 1977, 54, 90−94. (9) Meyer, E. F. A note on covariance in propagation of uncertainty. J. Chem. Educ. 1997, 74, 1339−1340. (10) Bruce, G. R.; Gill, P. S. Estimates of precision in a standard additions analysis. J. Chem. Educ. 1999, 76, 805−807.

= cov(y ̅ , b)̂ − x ̅ cov(b ̂, b)̂ = −xu̅ b2 We note that implementation in Excel of the formula for u2ab does not seem to be an obstacle to the application of the PoU approach through eq 2. Prediction

1. The standard uncertainty of a model prediction is obtained by inserting the expressions derived above into the PoU expression 2 uy2 = ua2 + x 2ub2 + 2xuab

⎛ ⎞ 2xx ̅ x̅ 2 x2 2 ⎜1 ⎟ = sreg + − ⎜n + 2 2 2⎟ ∑i (xi − x ̅ ) ∑i (xi − x ̅ ) ∑i (xi − x ̅ ) ⎠ ⎝ ⎛ (x − x ̅ )2 ⎞ 2 ⎜1 ⎟⎟ = sreg ⎜n + ∑i (xi − x ̅ )2 ⎠ ⎝

(6)

which is the expression reported in eq RH2017-3, up to an enlargement factor. 2. The confidence limits for a model prediction (eq RH2017−3) is obtained by multiplying the standard uncertainty uy by the adequate Student’s factor (tcrit), B

DOI: 10.1021/acs.jchemed.7b00251 J. Chem. Educ. XXXX, XXX, XXX−XXX

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(11) Salter, C. Error analysis using the variance-covariance matrix. J. Chem. Educ. 2000, 77, 1239−1243. (12) De Levie, R. Collinearity in least-squares analysis. J. Chem. Educ. 2012, 89, 68−78. (13) Taylor, J. R. Simple examples of correlations in error propagation. Am. J. Phys. 1985, 53, 663−667. (14) Danzer, K.; Currie, L. A. Guidelines for Calibration in Analytical Chemistry. Pure Appl. Chem. 1998, 70, 993−1014. (15) Danzer, K. Analytical ChemistryTheoretical and Metrological Fundamentals; Springer: Berlin, 2007; pp 123−175. (16) Taylor, J. R. Introduction To Error Analysis, 2nd ed.; University Science Books: Sausalito, CA, 1997. (17) Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1992. (18) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1993.

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