Nonlinear least squares fitting applied to copolymerization modeling

Abstract

SUMMARYThe choice of the statistical method to determine the reactivity ratios in copolymerization modeling is shown to be very important. Problems in literature, as well as possible pitfalls when using available statistical programs that are in itself correct are pointed out. These problems mainly involve (knowledge of) the error structure, as the error structure determines the weighting scheme of the data points in fitting procedures. A simple, robust, statistically correct non-linear least squares (NLLS) method is reintroduced which is based on the visualization of the sum of squares of residuals in the so-called sum of squares space (SSS). The advantages of this method include the fact that the method is easy to understand and can be implemented in simple computer programs, as well as the fact that the method allows important aspects of the error structure to be incorporated. Furthermore, in the SSS the joint confidence interval (JCI) with exact shape can be constructed. The exact shape of the JCI is not always ellipsoidal and following a normal distribution, depending on the linearity of the fitted equations. This can sometimes lead to wrong conclusions.

Statistical approach to model discrimination for the radical copolymerization of methyl methacrylate and styrene from a posteriori data of composition

“…However, as already concluded [12], from a statistical point of view, the most important is the proper estimation of the relation between relevant errors in variables instead of their absolute values. This agrees with the opinion of van Herk et al [16] on the importance of the error structure in determining the suitability of nonlinear least squares fitting procedures to the modelling of radical copolymerization.…”

Determination of Reactivity Ratios for the System Methyl Methacrylate−n-Butyl Methacrylate

“…The error-in-variables (EVM) method 11 is not used, because the construction of the confidence intervals in the EVM program is based on the 2 distribution, which is not valid for estimated errors, and the available EVM program 11 only gives confidence intervals with approximate shape. 12 The resulting reactivity ratios as well as reactivity ratios reevaluated from the original literature data with the NLLS method are given in Table 3. In Figure 1 the 95% joint confidence intervals for these reactivity ratios are given.…”

Pulsed initiation polymerization as a means of obtaining propagation rate coefficients in free-radical polymerizations. II Review up to 2000

“…Recently [138] it was realized that the wrong statistical procedures to obtain reactivity ratios from pk p P values as a function of feed composition in many cases suggests that both s-values can be obtained from these type of data which turns out not be true. The s-value referring to the addition of the monomer with the highest homopropagation rate coefficient (say s 2 ) usually is poorly determined.…”