| dc.description.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. | en_US |
