Thermochemical Examples of Statistical Pitfalls A Poor, but Real, Linear Plot and a Linear Correlation That is Not J. A. Martinho Simdes and C. Teixeira
Departamento de Engenharia Quimlca, lnstituto Superior Tecnico, 1096 Lisboa Codex, Portugal C. Airoldi and A. P. Chagas lnstituto de Quimlca, Universidade Estadual de Campinas, 13081 Campinas, S. Paulo, Brazil Students and teachers are often confronted with the physical meaning of linear correlations of experimental data. There is a handful of examples in physical chemistry textbooks, such as the integrated Clausius-Clapeyron equation, the van't Hoff equation, the Arrhenius plot, and the Birge-Sponer extrapolation, to name just a few ( I ) . In most cases, a solid theoretical background justifies the linear correlation or even accounts for the exceptions. Some situations, however, are not so well-founded. The famous and widely used Hammett equation (2) is one of them. The relationship is purely empirical and tries to quantify substituent effects on the reactivity of benzene compounds. This goal has been achieved successfully, and the application of the Hammett equation has been extended to many systems.
evaluated as 0.99996, the large value obtained for F (19.3) indicated that the linear relation was not a satisfactory representation of the experimental results, which were, of course, affectedby a rather small standard deviation. The calculations needed to make a proper regression analysis are easily made with any programmable calculator, so there is no good excuse for using correlation coefficients as the sole criterion for goodness of fit. Rather than describing the formulas required for that analysis, which can be found in any statistics textbook (61, the aim of the present paper is to provide a couple of (hopefully)convincing examples of the dangers of assessing linear correlations just by considering that r is either close or far from unity. An illustration in which the correlation passes the F test hut fails the r test is also provided.
Misuse of the Correlation Coefficients
A Poor, but Real, Linear Plot Standard enthalpies of formation of many organic molecules can be accurately estimated through several methods (7-11). The Laidler and the Benson "schemes" are currently the most frequently used. Those prediction methods, most of which rely on the additivity of bond enthalpies or group contributions, would not be available without the existence of much reliable data.
It has been claimed that many of those "extended" uses of the eauation are s~urious.It has even been ~ointedout the logthat a pcrfcct ~ a m m e t plot t is ohtnined hy arithm of the rate constant for the reaction ofdiazodiphenylmethane with substituted benzoic acids against thk logarithm of Sweden's gross national product (3).Although this plot has in turn been refuted (41,the example is useful in stressing that empirical correlations should be analyzed with great caution. The point looks trivial, but it is often forgotten.We believe that the reason is related to the misuse of the so-called correlation coefficients(r's). A rather pedagogic discussion of the use of r values was made by Tiley (51,who stressed that a good criterion for assessing the xoodness of fit of experimental data is to comparethe standard deviation ol'the linear regression o with the standard deviations of the measurements o,.The lower the ratio is. the hetter the linear correlation $11 he. The ratio can also be compared with the tabulated F value for a glven number of degrees of freedom and for a certain probability of rejection, where
In the case of inorganic compounds, the size of the inorganic data hank has been limited by several facton: the large variety of elements and bonds, and the experimental difficultiesin determining enthalpies of formation of many substances. Thus, the development of estimation methods has been hindered. Efforts in this direction have only recently been reported (11-20). One of the methods, which has also been applied to estimate enthalpies of formation of organometallic compounds (12,13), provided the examples of poor, hut real, linear correlations discussed below. Consider a molecule
The misuse of the correlation coefficients was convincingly illustrated by Tiley who calculated both F and r for a linear regression involving 17 data points. Although r was
where M is a metal, and Y and L are organic or inorganic ligands. For a series of molecules of that type, in which the moiety MY,, is the same, and the ligands L belong to a given "family" (e.g., alkyls, alkoxides, thiolates, halides),
Inorganic Compounds
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475
Enthalpy of Formation Data (kJ1mol)for Several Homoleptic Metal Halidesa
dissociation enthalpies follow parallel trends. AH,
Molecule
At+'(@
BeFz BeCIz BeBrz Belz SrFz SrClz SrBr2 Srlz TiFz TiClz TiBrz
-796 f 4 -360 f 11 -229 f 17 -64 f 34 -766 f 4 -473 f 6 -407 f 13 -275 f 6 4 8 8 f 42 -237 f 13 -179 f 21 -20 f 34 -558 f 21 -2~6~ -175 f 42 -67 -67 f 42 154 f 42 213 f 42 274 f 42 83 f 21 211.8~ 301 f 42 591 f 42 -271 .I -92.307' -36.4' 26.48'
Tilz
ZrFz ZrClz ZrBrz Zrlz
TiF TiCl TiBr Til
ZrF ZrCl ZrBr Zrl
= &(M-L)
- mD(GH)+ E ~ ( ~ - ~ ) 2
The usefulness of linear correlations (eq 3) or the parallel trends of D(M-L) and D I L H Jfor estimatine or assessine energetic data has been stressed elsewhe; (13, 14, 2 3 and demonstrated for many compounds. These references also include detailed discussions of the method. Only two important points are noted here. First, eq 3 seems to hold even when the molecules MY.L, are in their standard reference state, that is, their stable physical state at 298 K and 1 atm, which is usually solid. Second, the slope a of the correlation should be equal to the number m of ligands bonded to the metal atom. However, in practice, this is not always observed (13,14). The BeryNium and Strontium Halides The first examples chosen to illustrate the linear correlations are shown in Figure 1,in which literature data for mABeX2, g) and *(SrX2, g) (221, as shown in the table, were plotted against AHF(HX, g). X = F, C1, Br, or I. The least-square fits obtained for the beryllium and the strontium halides are respectively given by eqs 5 and 6.
'
HF HCI HBr HI 'Data from ref. 22,unless indicated otherwise. %el. 23. 'Ref. 24. Uncertainties are smaller than 0.8 Wlmol.
there is a linear correlation between w(MY,L,, g) and (LH, g). In the case of the above families of ligands, LH will of course be identified with alkanes, alcohols, thiols, and hydrogen halides, respectively. In order to appreciate the implications of the linear relationship, it is helpful to express the enthalpy of (hypothetical)reaction 1in terms of the enthalpies of formation of products and reactants, as computed by eq 2.
An interesting point about these two correlations is that the one for BeXz is much better than the one for SrX2,despite the higher error bars assigned to the experimental values of w(BeX2,g), as shown in the table. The correlation coefficients are 0.9999 and 0.9980. The standard deviations of the fits
+ b - y)2 N-2
(ax
If the linear correlation (eq 3 below) is observed and if m =a, then AH1 will be constant for the series of ligands from the same family. q(MY,L,, g) = a q ( L H ,g) + b
(3)
Moreover, when A H l is in turn defined usine bond dissociation enthal~ies. . as in eq 4,-it is concluded that its constancy implies that the M-L mean bond dissocia- Figure 1.Linear correlations of the standard enthalpies of formation of belyllium and strontium tion enthalpies and the L-H bond dihalides against the standard enthalpies of formation of hydrogen halides.
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Journal of Chemical Education
are 5.66 and 16.13kJImol, respectively. Here they's are the experimental values of AH,?(MXz,g), and N is the number of data points. Ifthe correlation coefficients are used to assess the plots, it is concluded that they are rather satisfactory because they both exceed the value 0.950 at the 95% probability level (6). In simple terms, this means that there is only a 5% probability that the correlation is fortuitous. However, according to the F test, the relationship for strontium is pavr. Considering the lowest uncertainty affecting @&SrX2, g), which is 4 kJImoL, F is equal to 16.2. This is much higher than the tabulated value of 3.00 for two degrees of freedom for o and infinite degrees of h e dom for a, at the 5% probability level (6). As above, this means that there is a 5% probability that the F factor of any four pairs of random data exceeds 3.00.If, on the other hand, the highest error in AH,?(SrXz,g), which is 13 kJImol, is taken, then a more favorable conclusion is reached: F = 1.54,which is less than 3.00.However, the average uncertainty in the experimental values supports the first conclusion: that the linear correlation is not genuine. Comments on Data Analysis
The previous analysis deserves a few comments. First, it must be stressed again that good or excellent linear wrrelations analogous to those in Figure 1have been observed for many compounds. This indicates that the method is reliable. When a satisfactory fit is not observed, inconlstency of the experimental data is a very likely explanation. Had we no more examples than those discussed above, the statistical analysis would give us some ground to reject the whole method. For strontium, the correlation may be considered spurious. For beryllium, with such high error bars affecting the experimental values, it could be considered fortuitous (and indeed it may be!). Second, as pointed out by Tiley (5),the use of correlation coefficientsdoes not seem to be a valid miterion forjudging the quality of the plots. Third, and perhaps more relevant, although the usefulness of the F test is apparent, there is a somewhat simpler procedure to assess the linear correla-
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:
tion: to compare the standard deviations assigned toa and b with the uncertainties of experimental data. The Zirconium and Titanium Halides
Thus, the quality of the strontium halide plot inFigure 1 indicates some inconsistency of one or more experimental values. Other cases exist for which the fits are even worse. In Figure 2 for example, Wf(ZrI, g) is clearly discrepant with the remaining values. The best thing to do is probably to disregard it. Even so, the least-squares fit
AHJ?(ZrX,g)= (0.88i 0.17)@m
g) + (316f 28)
(7)
is rather unsatisfactory on the basis of the calculated correlation coefficient, r = 0.982,as compared with the value tabulated for one degree of freedom and a 95% probability level, r = 0.997 (6). Interestingly, when the F test is applied, the large uncertainty intervals that affect the experimental data lead to
which is smaller than the tabulated value at the 5% confidence level, which is about 3.9.In other words, while the correlation fails the r test, it looks fairly good when the F test is applied, which is the opposite conclusion to that drawn for SrXz. &, and ZrXz, also The correlations involving TiX, T shown in Figure 2 for comparison are represented by eqs 8-10.
All three pass both the r test and the F test. The values of r are 0.9986, 0.9949, and 0.9986,respectively. All are higher than the tabulated value for two degrees offreedom and a 95% probability, 0.950. The values for o are 9.74, 35.44, and 13.42,respectively. These yield the following values: F = 0.05 for TiX, F E 1.7 for TiX2, and F z 0.2 for ZrXz. As referred above, the tabulated value for F is 3.00 (6). Despite these statistically well-behaved correlationsobserved for TiX,T&, and ZrX2,the one for TiXzis rather unsatisfactorj from a thennochemical point of view because it indicates a significant inconsistency of data. This is not at all surprising, given the large erron assigned to the experimental values of the standard enthalpies of formation.
In summary, the examples involving the compounds SrX2,ZrX, and Ti& cover three different possibilities of statistical -400 analysis. For the strontium halides, the correlation should be rejected on the grounds of a poor value of F. For the zir- 6 ~ 0 1Z r X ~ conium halides, the F test is good, but the rejection is advised by the correlation coTiX2 efficient test. Finally, for the titanium ha-800 lides, both the F test and the r test recom-300 -250 200 -150 -100 50 0 50 mend acceptance of the correlation. However, probably the best criteria of ~~~,[~~.gl/[k~/moll analysis of those experimental values, Figure 2. Linear correlations of the standard enthalpies of formation of Several titanium and which were supposed to follow the linear zirconium halides against the standard enthalpies of formation of hydrogen halides. correlation, is made simply by looking at
,/
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477
the standard deviations of a and b. As stated previously, these quantities are good measurements of the internal consistency of data. In the three families above, the inconsistency is apparent, despite other favorable statistical criteria. A Linear Correlation That is Not Figure 3 shows a plot of the enthalpies of formation of gaseous hydrogen halides against hydrogen-halide bond dissociation enthalpies. The least-squares fit corresponds to eq 11,for which the correlation coefficient is 0.9944. q ( H X , g) = -1.107D(H-X)
+ 367.5 (11)
These are the parameters usually derived with simple calculators. According to the r test, a fairly honest straight line was obtained. Indeed, the tabulated value for r a t the 99% probability level is only 0.990 (6).A straightforward analysis of the plot in Figure 3 demonstrates, however, that a linear fit is impossible. The definition of bond dissociation enthalpy, eqs 12 and 13,shows that the above relationship is not linear Figure 3. A linear correlation that is not: standard enthalpies of formation of hydrogen halides because q ( X , g) is, of course, not con- against hydrogen halide bond dissociation enthalpies. stant for X = F, Cl, Br, and I. Literature Cited 1. A t h s , P W.% p i d Chemishy, 4th e l . ; W d Univnaitypkss: O d d , 1990 2. Johnmn, C.D.The Hommlf Equation; Cambridge University Preaa: Camb"dge,
.l..W .. " ~
Thus, the attempt to derive eq 11 was naive, although the correlation coefficient seemed to indicate otherwise. This type of problem, which may occnr whenever a proper (and simple!) statistical analysis is not made (a physical analysis is not always possib~),can be settled by deiiving the standard deviations of the slope and the intercept. The obtained values, 0.083 and 35.6,respectively, show immediately that the correlation is not genuine because the errors affectingthe experimental values of W(HX,g) and D(H-X) are fractions of 1kJImol. In other words. the standard deviation of the fit (o= 16.6)leads to huge values ofF.
3. Butler, A R. Chem. Betoin 1989,25,997. 4. Wells, T N. C. Chem. Bnfoin 1990.26.25. 5. Tiley, P F Chem.B~ifoinIsBS,21,162. 6. Bevmgton, P. R. Doto Reduction and Error Analysis for the P h y s i d Seicncea; MeGraw-Hill: New Yo& 1969. 7. Car,J. D.;Pileher,G.ThermochPmistryofOwnicond OrganomlallicCompounda: Academic Presp: London and New Ymk, 1970. 8. B e m n . S. W ThemchemleolIlmotic8; Wiley: NnvYmL. 1976. 9. Pedley, J. B.: Naylor, R. 0 ; Kirby, S. P. Thermochemiml Dvto ofOrgonic Campounds; Chapman and Hall: landan and New York, 1986. 10. Molecular SfrwlumnndEmrgetics; L i e h a n , J. F;Greenbelg,A.,Eda.;VCH: New York, 1981:Vol. 2. 11. sanderson, R. T. chemlml Bond6 ond Bond E n e m ; Academic Press: New Yorh 1976.Sandemn,R. T P h r Couaisme;Academic Ress: New York, 1983.Sanderson,R. T J A m . Chem. Soe. 1975.97,1367.Sandemn,R.T J. Org. Cham. 1982. "9 ,s,c -.,
"W".
Conclusion When there is strong experimental and theoretical evidence for a linear relat~&ship,the r test and theFtest are less useful methods for assessing experimental data than a simple consideration of the standard deviations of the slope and the intercept of the least-squares fit. These quantities are essential for assessing the quality of the correlation, and their calculation should always be made. The goodness of fit should never rely exclusively on the value of the correlation coefficient. Acknowledgment We thank FundagHo de Ampam a Pesquisa do Estado de S. Paulo (FAPESP,Brazil) and Junta Nacional de Investigagio Cientifica e Tecnol6gica (JNICT, Portugal) for pmviding the funds for collaboration between our groups.
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Journal of Chemical Education
12. mas, A. R.; Ma-0 sima~aJ. A,; eanomptd. Chem. 1987.335.71.
l'eueim, C.: Aimldi, C.; Chagaa, A. P J
Or
17. 0hashi.H. Thwmochim. Acfo 1988,130,115. 18. Ohashi, H.Thermochim.Ado 1988,132,187. 19. Kiselev, Yu. M.; Popv, A. I.:Kopelw, N . S.; Spitsyn, V. I. L h h l h o d . Nauk SSSR
----,-IPR .-, .A-. ldlE
20. Kim,K-Y; Johnmn,C. E. J. C h . T&rmodyn. 198,13,13. 21. Bnmdza, H.E.;Fong, L. K; Patiello, R.A.;Tam, W.; Bereaw, J. E. J Am. Cham.Soc. 1987,109,1444. 22. Chase. Jx. M. W ;naviee, C. A,; Downey, Jr,J. R.; Frurip, D. J.:~ ~ D a n a lR. d ,A,; Syverud, A. N. J Phvs C h . R& Data I W ,14, supplement no. 1. 23. Efimov, M. E.; Pmkwenko, I. V.; Modvedev, V. A,: Tsirelnlkou, V. I.; Berezovskii, G. A.:Paukov,I.E. J Chem Thormodyn. 1988,21,677. 24. Waenan,D.D.;Evana,WH.;ParLer,V.B.;Sehumm,R.H.;Hal~al~al,I.;Bsiley,S.M.; Chumey, K. L.:NuttaU,R.L.J Phys. Cham. Ref h l o 1982,11,supplementno.2.
