Multiple response ridge analysis

Response surface methodology (RSM) is a collection of some statistical and mathematical tools used widely in the area of quality engineering, manufacturing and pharmaceutical industry, etc. It covers various issues in data exploration, experiment design, model building and process optimization.; In RSM, one important tool is ridge analysis. Researchers can use ridge path to obtain a graphical estimate of the optimal operating conditions when designing a new product or improving an existing one. Since the true model of the response is usually unknown and has to be estimated, the resulting ridge path is just a statistical estimate of the true ridge path with some random errors. To address the issue of sampling errors, researchers can construct the corresponding simultaneous confidence band to visually check how accurate the estimate is. However, the current discussion on this problem in the statistical literature is limited to the case of a single response variable and little is known for the multiple response case.; The motivation of this thesis is to extend ridge analysis to multiple response surfaces, where several response variables need to be optimized simultaneously. This is a more frequently encountered case than single response in practice. The organization of the thesis is as follows. We first extend the definition of ridge path to the settings of multiple responses by a desirability function approach. Numerical examples in different situations are given to illustrate the usefulness of this approach. Then, for the situation of small samples, a computationally complex min-max method is used to construct a simultaneous confidence band for the ridge path. An easy-to-compute large sample simultaneous confidence band is also developed based on delta method.