800
Ind. Eng. Chem. Process Des. Dev. 1985, 24, 890
CORRESPONDENCE
Comments on “Correlation for Vlscoslty Data of Llquld MMures” Sir: Recently, Dizechi and Marschall(1982) proposed an equation (eq 6 of the reference) for correlating ternary viscosity data. Stating that “In eq 6, the last term is so insignificantly small that it can be omitted without loss of accuracy”, the authors recommended the use of the simple eq 11 (nothing but eq 6 without the “last term”). Furthermore, the authors demonstrated the identical results obtained with eq 6 and 11 for six tepary systems in Table I11 of their paper. The objective of this correspondence is to point out that eq 11 (or its generalized version, eq 14) is an exact equation and not a simplified version of eq 6. This is because of the fact that the “last term” is always zero and therefore there should be no surprise in exactly identical results with eq 6 and 11. One can easily prove the vanishing quality of the “last term” of eq 6 if he recogsises the constraint,
Cxi = 1.0 along with the definitions of Cij and Ci.kterms as presented by Dizechi and Marschall(l982). (”he part - C,) of the “last term”, (C1x13+ C2x23+ + 6x1xg3C123 is identically zero.) Equations 11 and 14 are exact; correctly they do not contain the term involving (hN),neither a mixture-property nor a pure-component property. Further work is in progress in this university. Literature Cited DlzecM, M.; Marschall, E. Ind. p g . Chem. Process Des. D e v . 1982, 21,
282.
P. Sabarathinam
Department of Technology Annumalai University Annamalainagar 608 002, India
Received for review January 27, 1984
Comments on “Comparison of Methods for phnllnear Parameter Estlmation” Sir: The recent article by Ricker (1984) on data fitting with errors in all variables contains a misconception that has been around for a t least a decade. When comparing the “classical” method of Deming (1943) with that of Britt and Luecke (1973) and Anderson et al. (1978), the article states: “As noted by Britt and Luecke and others, the resulting estimates [of the Deming method] will be the same as those produced by the method of Anderson et al. when the constraints are linear ...”. A simple example will show that this is not true. Consider the case of three ( x , y ) pairs: (1, l),(1, 2), (2, 2) as s h o p in the accompanying Figure 1. If we want to fit a linear model to these, of the form y = Ax + B, then the constraints are linear in the usual sense of the term. Assuming that the variance of the error in x is the same as that in y, then the maximum likelihood fit is y = x This is the correct result given by the method of Anderson et al. and Britt and Luecke, and it is shown by line 1 in the figure. However, the method of Deming consistivof weighting the points in a way that depends on the slope of the fitting function in their vicinity, and in this case that results in equal weight for all three points. One then does a normal weighted least-squares fit of y against x , which yields y = x / q + 1. This is shown as line 2 in the figure. As one can see, it goes through the third point exactly, and goes half way between the first two points. The total sum of squares of the errors in x and y is 2/5, whereas for the true solution, it is 1/3. Fitting x against y by this method gives a still different line, also with a s u m of squares of 2/5. The reason that Deming’s method does not work is that the product Ax is in a sense nonlinear, since we must solve for both A and each “true” x . This example shows that even for a simple linear model, there is a difference between the old method of Deming
0
2
1
3
X
Figure 1. Example of data fitting. Dotted lines show distancesfrom fits.
+
and the correct method. It is true that the correct method takes considerably more computing, but fitting is usually done only a few times compared to the number of times that the fit will be used, so computing time should not be overly emphasized. If the software is available, and one knows that the “independent” variables are not very accurate, it seems wiser to obtain the correct maximum likelihood fit.
Literature Cited Anderson, T. T.; Abrams, D. S.; Grens, E. A. A I C M J . 1978, 24(1), 20. Brltt, H. I.; Lusdce, R. H. T&1973, 5(2), 233. Demlng. W. E. “StaUstical AdJustmsnt of Data”; Wlley: New York, 1943. Rlcker, N. L. Ind. Eng. Chem. FrocessDes. Dev. 1984, 23, 283.
IMI Institute for Research and Development Haifa 31002, Israel
Eric T. Kvaalen
Received for review December 18,1984 0 1985 American Chemical Society
