Robust Designs And Constrained Optimal Designs In Multiresponse Situations

An experiment in which a number of responses are measured simultaneously for each settingof a group of input variables is called a multiresponse experiment. Multiresponse experimentsare widely used in different areas. In the multiresponse experiment, the response variables shouldbe investigated individually and independently of one another. Interrelationship that may existamong the responses can render such univariate investigation meaningless. The design criterionshould be based on perceiving the responses as a group rather than as individualized entities.As mentioned in Box (1975), a'good'design should have fourteen good properties.In the first part of this thesis, we consider one of the properties: a good design should beinsensitive to wild observations and to violation of the usual normal theory assumption. And inthe second part, we try to deal with several properties and combine them at the same time.In Chapter 1, we try to explore a measure of insensitivity to wild observations based on mul-tiresponse regression model. We consider the design problem for recovering a response surfacewhen the observation of each response has been added to an outlier. A new model-robust opti-mality criterion is introduced. We succeeded in finding the robust design which is insensitive tooutliers on observations. A numerical example for linear and quadratic regression problems isprovided based on the class of central composite designs. Two different factorial portion of thedesign, the resolution II design with I = AB, that is nf = 2 and the 22 design, that is nf = 4are discussed. And based on these two situations, we find the correspondingαvalue so that thenew criterion is as small as possible when the number of center points nc is determined. Youcan clearly see from our figures the change of corresponding criterion values for different numberof center points nc. In our numerical example,it is obvious the value of criterion becomes muchsmaller as the number of corner points nf changes from 2 to 4. That means if more runs are per-missible, it's better to choose 22 design for the factorial portion of the central composite design inour example.In Chapter 2, we attempt to follow another thought suggested by Box and Draper (1975) andBandemer (1980). Looking back to Box and Draper (1975) paper, they commented that"In prac-tice, there are many different considerations that must be taken into account in selecting a design.Thus choosing a design solely on the basis of a single criterion is not recommended. In general,it seems more appropriate to select suitable measures which describe various aspects of a design,and to use them to make sensible subjective compromises."How to combine of different criteriafor experimental designs? Constrained and compound optimal designs represent two well-knownmethods for dealing with multiple objectives in optimal design. Our main concern in that chap- ter is to investigate the combination of several optimality criteria by the technique of constrainedoptimization for multiresponse model. Constraints may be due to some optimality criteria so thatthe designs satisfying the constraints will have at least the minimal quality that an investigatorwishes to maintain. A necessary and sufficient condition similar to Kiefer (Theorem 1, 1974a)is obtained using Fre′chet derivatives. Some examples are given to illustrate some possible appli-cations of the constrained optimality criterion, including Stigler's (1971) C-restricted D-criterionand Lee (1987), combination of A- and D-criteria in multiresponse situations.Finally, some conclusion and future work are given in Chapter 3.