Multiresponse analysis

Multiresponse analysis can provide fuller information, as illustrated in Investigation 11. There is a growing literature on error-in-variables methods. 

Multiresponse analysis or joint analysis uses the concept... 

In this example with only three components, the optimum could have been determined by simply overlaying the individual response contour plots. This approach would be difficult, if not impossible, if the formulation would have many responses or contain four or more components. By contrast, the combination of the desirability function and the Complex algorithm permits an optimization of a multiresponse formulation having many constrained components in addition to providing the basis for sensitivity analysis. 

Figure 5. Multiresponse sensitivity analysis of the overall desirability value showing the optimum and the compositional region complying with all response limits.

Experiments that will be used to estimate the behavior of a system should not be chosen in a whimsical or unplanned way, but rather, should be carefully designed with a view toward achieving a valid approximation to a region of the true response surface [Cochran and Cox (1950), Youden (1951), Wilson (1952), Mandel (1964), Fisher (1971)]. In the next several chapters, many of the important concepts of the design and analysis of experiments are introduced at an elementary level for the single-factor single-response case. In later chapters, these concepts will be generalized to multifactor, multiresponse systems. 

Some problems associated with the analysis of multiresponse data, Technometrics, 15 (1973) 33-51. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

Stamp, J.A. BCinetics and analysis of aspartame decomposition mechanisms in aqueous solutions using multiresponse methods, Ph.D. Thesis, St. Paul, MN University of Minnesota, 432, 1990. 

A sensitivity-analysis has to show finally how the model reacts to changing values of parameters. An efficient biokinetic experimental design has been demonstrated by Johnson and Berthouex (1975a), and multiresponse data are used for parameter estimation by the same authors (1975b). 

Predicted and measured values of Xj,buii( allow to undertake the regression analysis for estimating the kinetic parameters in Table 2. This was performed by employing the pack of routines GREGPAK [8] in multiresponse mode. 

When the v experimental errors are normally distributed with zero mean and those associated with the Mh and kh responses (e.g., in the differential method of kinetic analysis r and r j are statistically correlated, the parameters are estimated from the minimization of the following multiresponse objective criterion ... 

B. Dejaegher and Y. Vander Heyden, Response surface designs part 2. data analysis and multiresponse optimiztion, LC GC Eur., 2009, 22(11), 581-585. 

The regression analysis of experimental data then is performed according to one of the objective functions given in Fig. 15 for non-linear regression in different multiresponse situations. The selection of a specific objective function depends on the information available a priori concerning the different variances as given in the right column of Fig. 15 /29/. Naturally, from the statistical point of view, the different criteria are more efficient if more information about the variances is available. 

There are several techniques for minimization of the sum of squared residuals described by Eq. (7.160). We review some of these methods in this section. The methods developed in this section will enable us to fit models consisting of multiple dependent variables, such as the one described earlier, to multiresponse experimental data, in order to obtain the values of the parameters of the model that minimize the overall weighted) sum of squared residuals. In addition, a thorough statistical analysis of the regression results will enable us to... 

Previously, we have considered only the analysis of single-response data. Here, we discuss multiresponse regression, focusing primarily upon the extension of the least-squares method to the case of multiple, perhaps correlated, responses in each experiment. 

About the most probable estimate of k = 0.0024, this analysis of the multiresponse data of Table 8.3 yields a 95% HPD region for k of... 

Above, we have assumed that we measure the same set of responses in each experiment however, often we estimate parameters from composite data sets that mix different types of data. Here, we treat composite data sets using the sequential learning aspects of Bayesian analysis hence, the routines take the suffix MRSL for multiresponse sequential-learning. 

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