Ind. Eng. Chem. Fundam. 1984, 23, 267-268 Garza, G.; Rosales, M. A. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 168. Haynes. H. W.; Sharma, P. N. AIChE J . 1973, 19, 1043. Kocirik, M.; Zikanova, A. Ind. Eng. Chem. Fundam. 1974, 13, 347. Ma, Y. H.; Evans, D.M. AIChE J . 1966, 14, 956. Neogi, P.; Ruckenstein, E. AIChE J. 1980, 26, 787. Ruckenstein.. E.:. Vaidvanathan. A. S.:. Younoouist.. G. R. Chem. E . Sei. 1971. , 26, 1305. Smith, D. M. Ph.D. Thesis, University of New Mexico, Albuquerque, NM, 1982.
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Department of Chemical Engineering Montana State University Bozeman, Montana 5971 7
Douglas M. Smith
I .
Received f o r review May 9,1983 Accepted December 8, 1983
CORRESPONDENCE Comments on “A Maximum-Likelihood Approach to Treating Error Variance Instability in Hougen-Watson Rate Models” Sir: In apaper by Egy (1982), a procedure is described that corrects for nonconstant error variances when fitting linearized forms of Hougen-Watson rate models. The author compared his procedure with what he thought was a technique described by Box and Hill (1974) and concluded that this latter technique was “inappropriate”. The purpose of this correspondence is to point out that the Box-Hill technique was incorrectly stated and improperly applied by the author. Hence, the results attributed to the Box-Hill technique and reported in the Egy paper are not supportable. The problem stems from the fact that the author incorrectly assumes that Box and Hill used the linear model yi(h)= xiB + ui (i = 1,2,..., N observations) where X is a power transformation of the response, yi, say reaction rate, uI’sare independent errors with nonconstant variance V(uJ = u2z:
(Po, ..., PK-J is a vector of coefficients, and xi is the vector of K independent variables for the ith experiment. Here ziis taken as some independent variable (e.g., partial pressure of a reactant), u2 is a constant, and 6 is an unknown power transformation to be estimated from the data. Egy set zi equal to one or another of the independent variables and then estimated 6. For comparison purposes the author claimed that the Box-Hill approach is to set 6 = 0 and estimate A. This is not the Box-Hill model nor the procedure. The correct application of the Box-Hill method is to use the model with coefficients 0
@ =
y I = dxi,o) + ei
where 7 is a linear or nonlinear model function, the e, are
independent errors with nonconstant variance such that where 4 is an unknown weighting parameter to be estimated from the data, and Y iis the calculated value for the response yi. For the particular example discussed (Carr, 1960), yi was taken by Box and Hill to be the inverse of the reaction rate or r-l, which gives a linear form of the Hougen-Watson rate model under investigation. Contrary to what Egy indicated, the Box-Hill procedure does not change the model form from nonlinear to linear, or vice versa, by transforming y to y(A). Rather, the procedure actually finds the weighting parameter 4 (and not A) and then estimates 0 by an appropriately weighted linear or nonlinear least-squares analysis. In summary, we do not disagree with the author’s suggestion of attempting to stabilize variance based on weighting the data with an independent variable. In some examples this might work very well. However, we think it important to point out that the author has misunderstood the Box-Hill approach and we take issue with his conclusion that the latter technique is “inappropriate”. This can discourage engineers from using a useful method for modeling chemical reaction rates.
Literature Cited Box, G. E. P.; Hill, W. J. Technometrics 1974, 76(3), 385-389 Carr, N. L. Ind. Eng. Chem. 1960, 52(5).391-396. Egy, D. J. Ind. Eng. Chem. Fund 1982, 21, 337-339.
Department of Statistics University of Wisconsin Madison, Wisconsin 53706 Buffalo Research Laboratory Allied Corporation Buffalo, New York 14210
G . E. P. Box D. M. Steinberg W. J . Hill*
Response to Comments on “A Maximum-Likelihood Approach to Treating Error Variance Instability in Hougen-Watson Rate Models” Sir: In their correspondence referencing my 1982 paper, Mssrs. Box, Steinberg, and Hill take exception to some conclusions which I drew regarding a weighting scheme developed by Box and Hill (1974). The Box and Hill scheme to stabilize the error variances in Hougen-Watson
rate models transformed in some fashion for estimation purposes is, I stated, “inappropriate” for reasons which I attempted to demonstrate in my paper and which I shall attempt to defend here. I would like to apologize to Mssrs. Box and I-iill for what
0196-4313/84/1023-0267$01.50/00 1984 American Chemical Society
