Bayesian Analysis of Multi-Stratum Designs and Probability-based Optimal Designs with Separation.

Abstract

Industrial experimental design is an important area under design of experiments and factorial design hold a rm place in industrial experiments. The generalization of factorial designs results in split-plot type designs when complete randomization of runs is not possible. More speci cally, hard-to-set factors lead naturally to split-plot type designs and mixed models. Mixed models are used to analyze multi-stratum designs as each stratum may have a random e ect on the responses. The study of random e ects in mixed models might be di cult using likelihood methods because of small number of groups or whole plots in multi-stratum and split-plot designs. Also, zero estimates of variance components could be due to estimating multiple variance components in a hierarchical model. Therefore, likelihood-based inference is often unreliable with the variance components being particularly di cult to estimate for small samples. A Bayesian method considering some noninformative or weakly informative priors for variance components could be a useful tool to solve the problem. Fuel economy experiments, conducted by Shell Global Solutions UK, fall under small sample trap during variance components estimation. Using SAS procedure MIXED, experimenters estimated the variance components to be zero which were unrealistic. Also, the experimenters were unsure about the parameter estimates obtained by likelihood method from linear mixed models. Therefore, we looked for an alternative to compare and found the Bayesian platform to be appropriate. Bayesian methods assuming some non-informative and weakly informative priors enable us to compare the parameter estimates and the variance components. Pro le likelihood and bootstrap based methods veri ed that Bayesian point and interval estimates are not absurd. Also, simulation studies have assessed the quality of likelihood and Bayesian estimates in this study. A polypropylene experiment was conducted by four Belgian automobile industries to look for economical plasma treatments which lead to a good adhesion to various coatings. The e ects of several additives were also studied in addition to the plasma treatments. The likelihood-based estimates were not reliable completely due to the existence of moderate number of whole-plots. Also, some of the variance components due to batch were zero for some coatings. Assuming noninformative priors for xed e ects and some weakly informative priors for variance components we have obtained more sensible estimates of variance components which were inestimable or poorly estimated by the likelihood-based method using SAS procedure GLIMMIX. In this study, the Bayesian methods appeared to give comparable results with classical methods. One response variable in the polypropylene experiment was categorical which was converted to binary to see the e ects of additives on the outcome of interest. Unfortunately for binary responses we failed to obtain estimates of the logistic parameters for some of the coatings as the system did not converge. One of the reasons for this was due to having the separation problem in the data. When one or more explanatory variables completely separate the responses, the problem is known as separation. This problem causes the non-existence of likelihood estimates of logistic regression parameters. We have done some novel methodological works on the separation issue to minimize the problem in the light of optimal design techniques. Though the information based D-optimality criterion is widely used in practice, it fails to handle the separation problem appropriately. We have proposed new probability-based optimality criteria to handle the separation problem at the design stage of a study. Our proposed criteria Ps- and DPs- might be worthwhile to take into account reduction of the separation problem. However, Ps-criterion alone is not suitable to deal with separation problem as it produces worse designs in terms of precision of the parameter estimates, i.e. with respect to D-optimality. On the other hand the compound DPs-criterion makes a balance between D- and Ps-optimality and produces better designs. To make designs less sensitive to parameter misspeci cation, pseudo-Bayesian design criterion DPSB- has been proposed. Simulation studies have veri ed that Bayesian designs perform better than non-Bayesian designs by providing less bias, less median squared errors and above all less probability of separation. Thus, newly devised Bayesian and non-Bayesian design criteria could be useful in practice to control separation problem at the design stage of a study.