Step-Wise Group Screening Designs with Unequal APriori Probabilities and Errors in Observations

ةـصلاخ : تلاامتحلاا تاذ ةيجيردتلا ةلبرغلا قرط ءادأب ثحبلا اذه متهي ةعقوتملا براجتلا ددع ةللادب ةيواستملا ريغ ةقباسلا تارارقلل عقوتملا ىصقلأا دحلا ددـعو متي اهيفو ، ةئطاخلا داجيإ ةقيرطلا  ةيجيردتلا ةلبرغلا ميماصتل ىلثملا تلاامتحلال ةقباسلا ريغ علا هاجتا ةفرعم ةلاحل ةيواستملا متي َ اضيأ اهيفو ، أطخلل ةلباقلا تاظحلاملاو ةلتعملا لماو داجيإ داجيإو ةبسانملا فيلاكتلا ةلاد يق يف فيلاكتلا ةلمج لعجت يتلا ةعومجملا مجح ىرغصلا اهتم .


Introduction
T here are investigations where a large number of factors needs to be examined.In such a situation we have to run an experiment to identify the influential factors.The group screening procedure aims at reducing the size of the experiment, thus conserving resources.The method of group testing was first introduced by Dorfman (1943), who proposed that instead of testing each blood sample individually for the presence of a rare disease, blood samples be pooled and analysed together.Watson (1961) considered two stage group screening designs with and without errors in observations and with equal prior probabilities.In the same paper, he laid down the device of using different group sizes when prior probabilities differ.Li (1962) and Patel (1962) generalized Watson's method to more than two stages.Both these authors considered multistage groupscreening designs with equal prior probabilities and without errors in observations.Ottieno and Patel (1984) extended the idea of two stage group screening with unequal prior probabilities to include situations when no prior information is available so that no natural partitioning can be assumed.Odhiambo and Patel (1986) generalized this approach to multi-stage designs.
The group testing procedure first considered by Sterrett (1957) has been extended by Manene (1985), Patel and Manene (1987), Odhiambo and Manene (1987) and Manene (1997) in what they have called step-wise group-screening designs and they have approached the problem from the M.M. MANENE point of view of designs of experiments.They have considered the cases when all factors are defective with equal prior probabilities.
In this paper, we shall extend group screening designs considered by Odhiambo and Manene (1987) to the case when factors are defective with unequal prior probabilities.

Assumptions and design structure
We shall assume that there is a single response variable of interest y, which is related to a set of f factors through the first order linear regression model  (i) The total number of factors, f , can be divided into a fixed number of group-factors in the 'g ' initial step such that where k is the number of factors in the i group factor.The approach here is rather similar to the use of diffuse prior distributions in Bayesian inference.
The step-wise group-screening experiment is performed in steps as follows: in the initial step, the f factors are divided into groups such that the group contains factors ( ) These groups are called group factors.These group-factors are then tested for significance.Those that are declared non-defective are set aside.In step two, we start with any group-factor that is declared defective in the initial step and examine the factors within it one by one till a factor is declared defective.We set aside factors which are declared non-defective, keeping the factor declared defective separate.The remaining factors are then tested in a group.This is done for all group-factors declared defective in the initial step.The test procedure carried out in the initial step and in step two is repeated in subsequent steps successively till the analysis terminates with a group-factor declared non-defective or with a group-factor of size one.
In testing the significance of the group-factors in the initial step, we shall use the orthogonal main effects plans of the type given by Placket and Burman (1946).For testing the significance of individual factors and group-factors in the subsequent steps we shall use non orthogonal designs to simplify computations.

Expected Number of Runs
Suppose that f factors are divided into a fixed number of group-factors in the initial step such that the i group-factor is of size .The group-factors are tested in ' ' g ' ' g th ) ( ) and Var ( )  ( ) Where ( ) .
φ denotes the standard normal distribution function and ( ) Let denote the probability that the i group factor is declared defective in the initial step.Then where i p is the probability that a factor in the group-factor in the initial step is defective.
Define a random variable U such that In the subsequent steps, we shall use non-orthogonal designs.Let ' i p be the probability that a factor chosen at random from the group factor containing th i i δ defective factors that has been declared defective in the initial step is defective.

Then
( ) α be the probability of declaring a non-defective factor from the group-factor in the initial step as defective and th i si γ be the probability of declaring a defective factor as defective in the subsequent steps.Further let i β + be the probability that a factor chosen at random from the group-factor in the initial step is declared defective in the subsequent steps if the group-factor was declared defective in the initial step.
where ( ) Let * si α be the probability of declaring a non-defective factor from the i group-factor in the initial step group-factors is declared defective at any step but on testing individual factors within it, no factor is declared defective due to errors in observations.Obviously be the expected number of runs required to declare exactly factors defective from the initial step group-factor which has been declared defective.Then following Odhiambo and Manene (1987), ( ) where * i α is as already defined and Let si R be the number of runs required to analyse the group-factor once it has been declared defective in the initial step.Then Using (3.13) and (3.14).If s R is the number of runs required to analyse all the group factors declared defective in the initial step, then Theorem 3.1: The expected total number of runs in a step-wise group screening design with (fixed) group-factors in the initial step such that the group-factor is of size is given by , i i β α and i ξ are as defined earlier.
Proof : In the initial step we require 1 R g h = + runs ( ) . The number of runs M.M. MANENE required in the subsequent steps is ( ) The theorem then follows on using (3.8) and (3.15) in (3.17), simplifying and noting that ( ) ( ) Corollary ( 1, 2,......, up to order i p .The corollary then follows on using these approximations in the expression for given in theorem 3.1.

Calculation of the expected number of incorrect decisions
We shall consider the same cases of incorrect decisions as were considered by Odhiambo and Manene (1987) i.e. (i) declaring defective factors as non-defective in the initial step (ii) declaring defective factors as non-defective in subsequent steps (iii) declaring non-defective factors as defective in the subsequent steps Let denote the expected number of factors declared defective from the group-factor that is declared defective in the initial step.Then where U is as defined in (3.7).i Let ( ) 0 i p be the probability that a factor chosen at random from the initial step group-factor declared non-defective, is defective.
Let i p + be the probability that a factor from the i group factor is non-defective given that it is declared defective.Then .Then ( ) The expected total number of factors declared defective in the subsequent steps is given by ( ) ( ) The probability that a factor which is declared defective from the group-factor, is defective is given by 1 as required.
Let denote the expected number of defective factors declared non-defective in the initial step.
Theorem 4.2 : In a step-wise group screening design with errors in observations and unequal apriori probabilities, the expected number of defective factors declared non-defective in the subsequent steps is given by ( ) The expected total number of defective factors in all the group-factors in the initial step is equal to .Therefore This completes the proof.
Theorem 4.3 : Let u M be the number of non-defective factors declared defective in the subsequent steps.Then ( ) ( ) The total number of factors declared defective in the subsequent steps is Theorem 4.4 : Let I be the expected total number of incorrect decisions in a step-wise group screening design with group-factors in the initial step such that the group-factor of size contains factors with a-priori probability of being defective The expected total number of incorrect decisions is given by ( using (4.7), (4.8) and (4.9).This completes the proof.
Corollary 4.1: Hence I will take its maximum value when ( ) is minimum and is maximum.But takes its maximum value when ( u E M )will take a minimum value when , and is replaced by The result follows on using (4.12) and (4.13) in (4.11).
Corollary 4.2 : For large i σ ∆ and small ' i p s ( ) The result follows immediately on replacing these values by their approximations in the expression for Max I given in corollary 4.1.

Optimum sizes of initial group-factors in relation to total cost
We define the expected total cost ( ) C as a linear function of the expected number of runs and the expected number of incorrect decisions and obtain the sizes of the group-factors so that the expected total cost is minimum.
Let c be the cost of inspection per run and c be the loss for each incorrect decision made.Then the expected total cost is given by Using the method of Lagrange multipliers, let ( ) where λ is the Lagrange multiplier.Assuming continuous variation in , the critical value of is obtained from the equations i k i k / 0 ; 1, 2,..., , and / 0 The theorem follows immediately on solving equations (5.3).

Examples of screening plans
The screening efficiency of step-wise group screening design with unequal group sizes can be measured in terms of the minimum expected total cost.A small value of indicates better performance on the average.Examples of group screening plans which minimize the expected total cost are given in   .The corresponding value of min when incorrect observations are not considered.
From the two tables it is easily seen that when a cost function involving both and Max ( ) I is used, the number of runs increases.It should be noted that these tables are just an illustration.The values of ' i p s used are not unique; neither is the ratio c 2 1 : .c the u run, α be the level of significance for testing the i group-factor in the initial step and denote by th th* si α will take different values at different steps.However for simplicity in algebra, we shall assume * si α to be of uniform value, that exactly j factors from the group-factor in the initial step that has been declared defective in the subsequent steps.

M
be the number of defective factors declared defective in a step-wise group screening design with initial group-factors, the factors in the i group-factor of size being defective with a priori probability expected total cost is given by

Table 1 :
Table 1 below.The corresponding values of and Max Optimum group-sizes obtained by minimizing expected total cost ( , when and for selected unequal apriori probabilities.The minimum given is a relative figure using c (the cost of observing a run) as the unit.