Optimal hierarchial factorial designs: the multiple design multiresponse case with cost constraint

In designing multivariate experiments, it will often be the case that different responses have different design matrices. This most often occurs when certain responses are not influenced by various factors. If not all responses are measured on each observational unit, this gives rise to the More General Linear Multiresponse (MGLM) design. For a factorial experiment, denote by V a variance-covariance matrix of estimable functions of the parameters of any effects one wishes to study. Optimal designs are found that minimize the trace of V when the size of the design is restricted by a total cost constraint, thus minimizing the average variance of each estimable function. It is shown that Hierarchical MGLM designs (HMGLM), a subset of the MGLM designs, need only be considered. In a HMGLM design a hierarchy of the responses V₁,...,V_p’ can be found such that if i < j (i,j = 1,...,p'), V_i should be measured on at least as many experimental units as V_j is measured and V_j is measured only on units where V_i is also measured. Given the costs and 'a priori' variance estimates, optimal designs for 2^v factorial experiments are found where k_i effects are under study for V_i. The procedure is then extended to include p^v factorial experiments. We consider next the minimization of the determinant of V as a criterion for optimality. This criterion results in the confidence ellipsoids for the estimable functions to be of minimum volume. Due to the difficulty in defining the off-diagonal covariance matrices of V for the general class of MGLM designs, certain well-defined subclasses were considered where the covariance matrices of these designs could be found in general. First a rather natural subclass of MGLM designs, called Restricted MGLM designs (RMGLM), was investigated. HMGLM designs are ·a subclass of RMGLM designs, as are Multiple Design Multiresponse (MDM) designs which assume that all responses are measured on each experimental unit. The general situation, assuming p' responses, was investigated first. Due to difficulty in finding the determinant of the matrix V, a general solution for the optimal RMGLM design was found for only certain specific situations. In an attempt to ease the difficulty in determining the general form of det(V), the two-response case was considered. The optimal RMGLM design was then determined for more general situations. Finally, the complement subclass (CRMGLM designs) of the RMGLM designs in the class of MGLM designs was investigated for two-response situations. The optimal MGLM design can then be determined by comparing the optimal RMGLM and CRMGLM designs. For most situations, the optimal CRMGLM design can be found, but for those situations where it cannot be found, the optimal RMGLM design (a HMGLM design) can still be determined, giving a design at least as.good as, and often better than, the generally accepted MDM design.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Statistics

Department dc:contributor.department

Statistics

Grantor dc:publisher

Virginia Polytechnic Institute and State University

Year dc:date.issued

1973

Author dc:creator

• Givens, Samuel V.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10919/74683

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/74683