CONSTRUCTION OF CONNECTED OPTIMAL DESIGNS IN DIFFERENT REPLICATES

By

AIYELABEGAN. Adijat Bukola

BEING A PAPER PRESENTED AT THE INTERNATIONAL CONFERENCE ON ADVANCE PURE AND APPLIED

SCIENCES (ICAPAS) 2014 AT KUALA LUMPUR, MALAYSIA

BETWEEN 3RD – 4TH NOVEMBER 2014

1.0 INTRODUCTION

In order to compare two designs d1 and d2 , we consider their efficiency factors E1 and E2 respectively and choose the design with the higher efficiency factor. In particular, if the efficiency factor of one of those designs achieves the upper bound for that class of designs, we could consider that design to be optimal in some sense. Such consideration has led to the development of the notion of optimal design and to various optimality criteria.

In this paper, we propose an algorithm that is able to quickly construct a good design of experiments in different replications with different number of treatments .The proposed method is also useful when we have large information matrix of designs, the proposed method is also able to compute if not within seconds the eigenvalues of the information matrix and also quickly determine A-, D-and E-optimality criteria as well as the trace of the information matrix of the designs. This algorithm enables us to search for connected optimal designs within the shortest possible time by comparing the traces of information matrix of various classes of design and determine the criterion values for each optimality criteria. In this work we only determine A-,D -and E-optimality criteria. Further research is still going on other types of optimality criteria.

PROPERTIES OF MINIMALLY CONNECTED DESIGNSHere we present three Lemmas given by Bapat and Dey (1991).Lemma 2.1. Any connected design is necessarily binary.Let the blocks of be numbered as ,,,……, in such a manner that , where is the empty set. This is clearly possible because design is assumed to be connected. Define …. and let be the complement of with respect to the set of treatment symbols, . Let

here j=2,3,…..b; and denotes the number of elements in a set. It can be seen that is the number of treatments in block not occurring in the previous blocks.

Lemma 2.1. Under (2.0) for any design , Corollary 2.0 Under (2.0) for any connected design , no pair of blocks have more than one treatment in common, that is, no pair of treatments occurs in more than one block.Lemma 2.2 Under (2.1) for any connected design , there is only one unbiased estimate for any treatment contrast thus the ordinary least squares estimate (OLSE) and the general least squares estimate (GLSE) for are the same.Definition 2.0 A chain in a block design is a sequence of experimental units such that two consecutive units either share the same treatment or the same block, but not both.Corollary 2.1 Under (2.1) any two blocks are connected by exactly one chain.

OPTIMALITY CRITERIASeveral optimality criteria have been considered for studying optimal designs. These criteria can be expressed conveniently in terms of the eigenvalues of C*d or, equivalently, the nonzero eigenvalues of Cd say μd1 μd2 ……… μd, t-1.

Suppose ήd is the BLUE of ή using a design d with . It is reasonable to define an optimality criterion as a meaningful function of

INFORMATION FUNCTION

An information function or optimality criterion is then a real-valued function ø that has the following properties Function ø is a monotonic function; that is, an information matrix C* is at least as good as another information matrix D* if ø (C*) ø (D*);Function ø is a concave function, that is ø [(1-α) C*+αD*]=(1-α)ø(C*)+αø(D*)] for αE (0,1) Function ø is positively homogenous , that is, ø (δC*)= δø(C*)Consider (2) says that information cannot be increased by interpolation. And (3) says that even if we define the information matrix to be directly proportionality to the number of observation, n, and inversely proportional to σe

2, that is, the information matrix of the form (n/ σe

2) C*, we need to consider only C*

Table 1. Compute the criterion values and the trace values of the information matrix of the design t=6, , ,

=2 =3 =4 =5

A-optimality 0.90908 0.7843137 0.02950 0.173566

D-optimality 0 0 0.12465 0.0309335

E-optimality 1 1 0.9140 0.9604705

Trace() 2 4 4.78 3.2019

Table 2. Compute the criterion values and the trace values of the information matrix of the design t=9 , ,

=2 =3 =4 =5

A-optimality 0.851063 0.776665 0.034561 0.035671

D-optimality 0 0 0.198754 0.004211

E-optimality 0.8333 0.69995 0.79445 0.976012

Trace() 3 3.68 4 4.25

Table 3.0 shows the result of for average criterion

0.01768 0.13272 7.445x

3.0854 1.6781 23

0.04492 0.04418 0.89723

In general it is found that with similar number of replications this algorithm always achieves better designs with respect to the criterion. This is also confirmed statistically by the s ins, which are all smaller than 0.001%. Since the comparison focuses on how efficient the algorithm is compared to others by using the same amount of reasonable numbers of replications which are considered as small relative to the size of the designs.

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