CORRESPONDENCE
Precise Estimation of Reaction O r d e r s n two recent papers (2, 3 ) (the earlier one coauthored by Watson), Kittrell and Mezaki present a method for analyzing isothermal, constant volume kinetic data of irreversible reactions. I n my comments on these two papers, I shall use the nomenclature of Reference 2. Working with a transformed variable ,(A) defined by Equation 1, Kittrell et al. minimize the sum of squares S ( h ) (Equation 2) with respect to the parameters k and n :
I
wherey
j
= =
(1 - x)-I, X = ( n - 1) geometric mean of the N values of y
N and S(X) =
Z
( ~ ( 5) btJ2
i = l where b
=
k C Axo j(1-X)
T h e estimates of n and k thus obtained are claimed to be maximum likelihood estimates. This claim is based on the assumption thaty(&)[hence &))I is a variable which simultaneously achieves (i) (ii) (iii) (iv)
Linearity of the model Equal variance a t all points (homoscedasticity) Xormality of error distribution Independence of observations
Kittrell, Mezaki, and Watson ( 3 ) cite the paper by Box and Cox (1) to justify the choice of the transformation. A careful reading of that paper reveals the following: “We have supposed that after suitable transformation fromy toy(A):(a) the expected values of the transformed observations are described by a model of simple structure; (b) the error variance is constant; and (c) the observations are normally distributed. Then we have shown that the maximized likelihood for X, and also the approximate contribution to the posterior distribution of A, are each proportional to a negative power of the residual sum of squares for the variate &) = y ( A ) / 3 1 i nT.h e ‘over-all’ procedure seeks a set of transformation parameters for which (a), (b), a n d (c) are simultaneously satisfied, and sample information on all three aspects goes into the choice . . . The above procedure depends on spec@ assumptions, but it would be quite wrong f o r fruitful application to regard the assumptions as$nal. The proper attitude of skeptical optimism is accurately expressed by saying that we tentatively 76
entertain the basis for analysis, rather than that we assume it. The checking of the plausibility of the present procedure is discussed in Section 5.” I t is apparent that Kittrell and Mezaki, in their numerical examples, make no attempt to use sample information to check the “plausibility” of the assumptions (as was intended by Box and Cox). Indeed, in the absence of multiple observations a t a given time, it is impossible to conduct either Bartlett’s test or the NeymanPearson test to check assumptions (ii) and (iii), respectively. Assumption (i), on the other hand, is true by definition of ,(A). I t is therefore unjustifiable a priori to regard the estimates of n and k as maximum likelihood estimates. Further, if the choice were given to the experimenter, he would in all probability make the assumptions (ii) to (iv) apply to his obseruations (the Ca4’s)rather than to an [Obviously artificially defined variable such asy(A) or they cannot be true of both the CA’S and they(X)’s.]H e would do this for lack of further statistical information on the errors. T h e criterion that now leads to maximum likelihood estimates is
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COMPARISON OF SUM OF SQUARES CURVES N20 DECOMPOSITION AT 875 O K
00 5
1
I
I\
400005
004
N
2
a
003
-a
w - 002 MEZAKI KITTRELL WATSO~
NON-LINEAR LEAST SQUARES 99% CONFIDENCE INTERVALS
001
0
I0
I
I2
I3
14
REACTION ORDER n
of
I5
i
I6
Figure I . Comparison of sum squares curues f o r N20decomposition at 875 “K. pi represents the partial pressure of N20, which was measured by Pease ( 4 )
We can now subject the data to a nonlinear squares analysis, in the light of Equation 3. For the decomposition of NzO at 875 OK, the results of such an analysis are compared in Table I and Figure 1 with the results obtained by Kittrell, Mezaki, and Watson for the same data points. No significant difference in computing time was noticed.
TABLE I. COMPARISON OF PARAMETER ESTIMATES AND 99% CONFIDENCE INTERVALS FOR N20 DECOMPOSITION AT 875 O K . [Data from Pease ( 4 ) ] Kittrell, Mezaki, Nonlinear least and Watson squares with Ca's k X lo4 (1.45 & 0.38) (1.31 k 0.54) n (1.25 ?- 0.11) (1.23 0.13) Units of k are
(2)
l-nsec-l.
TABLE 11. ANALYSIS OF VARIANCE OF N20 DECOMPOSITION DATA AT 875 "I(' [Data from Pease ( 4 ) ] Mean Squares,
Source of Variation Due to model (2 parameters)
278.86
Nonlinear least squares analysis of observed variable Ravimohan
223.17 278.86 -- 223.17
-=
1
1
Residual Ratio of mean sum of squar,es due t o model to the mean residual a
I t is seen that the results differ not only in the parameter estimates, but more importantly in the confidence limits. T h e confidence limits with the nonlinear analysis, also computed by the procedure of Box and Cox, are gee, to be less optimistic than those of Kittrell and Mezaki. T h e nonlinear analysis performed here does not involve differentiation of the data to obtain reaction rates. Hence the difference in the results cannot be due to computational errors. T h e only explanation is that the basic assumptions are different in the two cases. Kittrell and Mezaki use a least squares criterion on a transformed , claim without further justification that variable z ( ~ )and this leads to maximum likelihood estimates. I t is my contention that in the absence of further statistical information on the errors, the experimenter is better off using a least squares criterion on the observed variable CA and performing a nonlinear analysis as above. T h e only specific complaint which Dr. Kittrell expresses in his reply to my comments (which follow on page 78) seems to be that I did not carry out a residuals test on the data. Table I1 shows the results of such a test, .sing the same example as before-the decomposition of
Degrees of Freedom 1
Least squares analysis of transformed vanable of Kittrell and Mezakt
(3
278.86 1.35
223.17 206.56 -- 9950.0 ___ 0.0235 --
-=
The deviations have been taken from the
1 =
0 value of the observed variable
p,
NzO at 875 O K , with data taken from Pease (4).T h e fit is shown to be much better when the raw untransformed observations are used for the analysis as I suggested above. T h e plausibility of the assumptions regarding z(X) thus remain unverified, even approximately. T h e experimenter can still ask whether, if at all, it helps him to shift the assumptions from the observed variable to an artificially defined variable, z(X). REFERENCES (1) Box, G. E. P., and Cox, D. R., J . Royal Stadis). &., Series 8,26 (2), 21 1 (1966). (2) Kittrell, 3. R,, and Mczaki Rciji Ch. 10 in '!Applied Kinetics and Chemical Go;ring and V. W. Weekman, Eds., American Reaction Engineering " R; Chemical Society Pudlicatrgns, Washington, D.C. (1967). . (3) Kittrell, J. R., Mezaki, Reiji, and Watson, C. C., INn. ENC.CHEM., 58 (5), 51 (1966). (4) Pease, R . k., J . Chcm. Phys., 7 , 769 (1939).
i.
A. L. R A V I M O H A N
Division o f Chemical Ehgineeri'ng California Institute o f Technology Pasadena, Calif.
VOL.
61
NO.
5
MAY
1969
77
Replies to Commenfs of
A. L. Ravirnohan
M r . Ravimohan apparently has three primary criticisms of the paper in question ( I ) . H e asserts (a) we did not realize that the Box-Cox transformation does not guarantee maximum likelihood estimates of a reaction order; (b) we did not utilize sample information to test the transformed error variable for normality, independence, and constancy of variance; and (c) in the absence of any numerical information to the contrary, the experimenter should assume that the reactant concentrations possess these three properties rather than any transformation of the concentration. I believe that we belabored point (a) sufficiently in the paper. We stated twice on page 53 and once on page 54 that the transformations only achieve errors with an independent, normal distribution of constant variance to the extent that it is possible. O n page 59 we say, “The method . . . transforms the dependent variable to achieve an error distribution as consistent as possible with the assumptions inherent in a least squares analysis. Hence, zf these assumptions are fulfilled, the maximum likelihood estimates of the reaction order and the transformed rate constant are obtained . . .” A similar statement may be found on page 123 of Ravimohan’s Reference 2. Concerning point (b), we did not formally test the plausibility of these assumptions; M r . Ravimohan pointed out that since insufficient data exist, such tests are impossible. I t might be pointed out that sufficient data are seldom available from kinetic experiments to conduct these particular statistical tests rigorously. M’e did, however, use residuals to approximatell- test these assump-
tions (see our examples). T h e residual tests we used are those covered in Draper and Smith, “Applied Regression Analysis,” pp 86-103, TViley, New York, K.Y. (1 966). These attempt to specify whether the transformed \,ariable or the untransformed variable has constant variance, etc., and are quite distinct from the analysis of variance contained in Mr. Ravimohan’s Table I1 on page 77. Since the residual s u m of squares is being minimized in obtaining Ravimohan’s entry of Table 11, it is not surprising to find a smaller residual mean square is obtained by nonlinear least squares; this is the preferred method, however, only if all data are of equal precision. T h e criticism of point ( c ) is quite subjective. If the experimenter feels that C, has the correct properties, then the unweighted nonlinear least squares analysis is quite appropriate. Similarly, the transformation In C, often has been used to stabilize error variance. (Our procedure approximately reduces to this for a first-order reaction.) However, if no prior knowledge of the error distribution is available. our procedure will be quite valuable in parameter estimation; the point and interval estimates should approach those of the above weighting procedures if those procedures are appropriate.
For the precise determination of reaction orders of power function rate models, Ravimohan recommends the use of nonlinear squares analysis for integral rate equations rather than the use of linear least squares analysis for a transformed variable. T h e reason for this recommendation is that two basic assumptions imposed upon the error variance of the transformed variable (constant error variance at all experimental points and normality of error distribution) are not justified for the transformed variable examplified by Kittrell et al. ( I ) . I n many cases of kinetic study, not enough data have been gathered to obtain the accurate information concerning the behavior of error variance of the dependent variable. With unweighted nonlinear least squares one can approximately maximize the likelihood function. Thus we obtain the most precise estimates of parameters only when we do know the nature of error variance. I n any event, one can check the “plausibility” of these assumptions after a trial fit by using simple tests such as residual plots. If the assumptions are found to be inappropriate, then perhaps a nonlinear model of the original observations should be used. T h e confidence intervals of nonlinear parameter estimates shown in Table I of Ravimohan’s correspondence give some information relative to the allowable range of
parameter estimates. For this case, however, it would be more adequate to show the joint confidence regions for parameter estimates. With limited knowledge on error variance, it would be rather difficult to judge which estimation method is superior. Nevertheless, the use of a transformed variable reported in Reference 1 seems to furnish more accurate estimates of parameters in any situations of error variance of the dependent variable. T h e most important assumption made in these analyses is that the kinetic model is correct. This could lead to more problems in the analysis than any of the other assumptions, especially if the model is not correct.
78
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REFERENCE (1) Kittrell, J. R., hlezaki, R.. and Watson, C C., (1966).
IND. ENG.CHEM.,58 (5), 51
J. R. K I T T R E L L Chevron Research Co. Richmond, Calif.
Acknawledgment
I a m indebted to W.J. Hill for his TTaluable suggestions. REFERENCE (1) Kittrell, J. R . , Mezaki, Reiji, and Watson, C. C., IKD.ENG.CHEY.,58 (5), 51 (1966).
R E IJI MEZAKI Department of Engineering and Applied Science Yale University N e w Haven, Conn.
