Multiple responses

RSM has the ability to model as many responses as one wishes to measure. For example, one may not only be interested in optimum yield, but also the level of a difficult to remove impurity. Both the yield and impurity levels could be modeled using data from the same set of experiments. Decisions could then be made between the cost to remove an impurity and changes in yield. 

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Buzzi-Ferraris, G., P. Forzatti, G. Emig and H. Hofmann, "Sequential Experimental Design for Model Discrimination in the Case of Multiple Responses", Chem. Eng. Sci., 39(1), 81-85 (1984). 

This parameter-estimation technique has also been extended to the multiple-response case (D3). Just as was seen in the multiple-response... 

Fig. 2.1 Lead discovery requires an integrated approach to address a multiple response surface optimization problem properties related to pharmacokinetics, toxicology and pharmacodynamics require optimization for novel chemotypes. The increased difficulty of finding such optima is suggested by darker backgrounds. Modified from [4] with permission.

However, there are prices to pay for the advantages above. Most empirical modeling techniques need to be fed large amounts of good data. Furthermore, empirical models can only be safely applied to conditions that were represented in the data used to build the model (i.e., extrapolation of such models is very dangerous). In addition, the availability of multiple response variables for building a model results in the temptation to overfit models, in order to obtain artificially optimistic results. Finally, multivariate models are usually much more difficult to explain to others, especially those not well versed in math and statistics. 

The PLS technique gives a stepwise solution for the regression model, which converges to the least squares solution. The final model is the sum of a series of submodels. It can handle multiple response variables, highly correlated predictor variables grouped into several blocks and underdetermined systems, where the number of samples is less than the number of predictor variables. Our model (not including the error terms) is ... 

T. Smith, C. Oji, D. Boning, and J. Chung, Bias and Variance in Multiple Response Surface Modeling, Third International Workshop on Statistical Metrology, Honolulu, HI, June 1998. 

Multivariate A multivariate measurement is defined as one in which multiple measurements are made on a sample of interest. That is, more than one variable or response is measured for each sample. Using a sensor array to obtain multiple responses on a vapor sample is a multivariate measurement. 

Response Matrix (R) Multh-ariate calibration and pattern-recognition techniques make use of multiple responses that are represented by the matrix R, The rows of R contain the measurement vectors for individual samples. This matrix has the dimensions nsamp X nvars. 

In many applications, one response from an instrument is related to the concentration of a single chemical component. This is referred to as univariate calibration because only one instrument response is used per sample. Multivariate calibration is the process of relating multiple responses from an instrument to a property or properties of a sample. The samples could be, for example, a mixture of chemical components in a process stream, and the goal is to predict the concentration levels of the different chemical components in the stream from infrared measurements. The methods are quite powerful, but as Dr. Einstein noted, the application of mathematics to reality is not without its limitations. It is, therefore, the obligation of the analyst to use them in a responsible manner. 

Lack of fit of the obtained model has to be statistically checked, so that, if needed, the polynomial degree may be augmented. Knowing the mathematical model of the research subject for several responses is a prerequisite in solving optimization with multiple responses. The computation of this is solved geometrically or by use of computers and the method of linear algebra. 

In the introduction, we have already classified the optimization problems as deterministic and stochastic. It is evident that deterministic problems are based on functional models or models that disregard experiment error. Problems where one cannot neglect experiment error are stochastic ones and, as established, they are the subject of this book. Besides, optimization problems are by the number of factors divided into one-dimensional and more-dimensional. The Optimization problem grows with dimension. The problem becomes even more complicated if optimization is not done by one but by more responses simultaneously-multiple response processes. 

Dose-Response Assessment for Chemicals That Cause Deterministic Effects. For hazardous chemicals that cause deterministic effects and exhibit a threshold in the dose-response relationship, the purpose of the dose-response assessment is to identify the dose of a substance below which it is not likely that there will be an adverse response in humans. Establishing dose-response relationships for chemicals that cause deterministic effects has proved to be complex because (1) multiple responses are possible, (2) the dose-response assessment is usually based on data from animal studies, (3) thousands of such chemicals exist, and (4) the availability and quality of data are highly variable. As a consequence, the scientific community has needed to devise and adhere to a number of methods to quantify the most important (low or safe dose) part of the dose-response relationship. 

With cell retention accomplished by 3D flow control, calcium mobilization in a single yeast cell has been studied over a long period of time, and in the manner of multiple stimuli and multiple responses. Loading of the Ca2+- sensitive dye Fluo-4 in the cell was accomplished either (1) after cell wall removal or (2)... 

Chowdhury and Fard (2001) presented a method for estimating dispersion effects from robust design experiments with right censored data. Kim and Lin (2002) proposed a method to determine optimal design factor settings that take account of both location and dispersion effects when there are multiple responses. They based their approach on response surface models for location and dispersion of each response variable. 

Kim, K. J. and Lin, D. K. J. (2006). Optimization of multiple responses considering both location and dispersion effects. European Journal of Operational Research, 169,133-145. 

Further research is needed to derive the overall probability of correctly classifying the individual factors as important or unimportant in our sequential procedure, which tests each factor group individually see Lewis and Dean (2001, pages 663-664), Nelson (2003), Westfall et al. (1998), and Wan et al. (2004). Also, the extension of sequential bifurcation from single to multiple responses is an important practical and theoretical problem that requires further work. 

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