Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers Science and Technology, 8 (2003) 153-165 © 2003 Sultan Qaboos University Step-Wise Group Screening Designs with Unequal A-Priori Probabilities and Errors in Observations M.M. Manene Department of Mathematics, University of Nairobi, P.O. Box 30197, Nairobi, Kenya, Email: mathuon@iconnect.co.ke. السابقة غير المتساوية واألخطاء في المالحظاتاالحتماالتتصاميم الغربلة التدريجية للمجموعات ذات مانين. م.م السابقة غير المتساوية بداللة عدد التجارب المتوقعة يهتم هذا البحث بأداء طرق الغربلة التدريجية ذات االحتماالت : خالصـة السابقة لالحتماالتالمثلى لتصاميم الغربلة التدريجية الطريقة إيجاد الخاطئة ، وفيها يتم وعـدد الحد األقصى المتوقع للقرارات دالة التكاليف المناسبة وإيجاد إيجادوامل المعتلة والمالحظات القابلة للخطأ ، وفيها أيضاَ يتم المتساوية لحالة معرفة اتجاه العغير .متها الصغرىحجم المجموعة التي تجعل جملة التكاليف في قي ABSTRACT: The performance of step-wise group screening with unequal a-priori probabilities in terms of the expected number of runs and the expected maximum number of incorrect decisions is considered. A method of obtaining optimal step-wise designs with unequal a-priori probabilities is presented for the case in which the direction of each defective factor is assumed to be known a - priori and observations are subject to error. An appropriate cost function is introduced and the value of the group size which minimizes the expected total cost is obtained. KEYWORDS: Step-Wise, Designs, Group-Factors, Initial Step, Subsequent Steps, Expected Number of Runs, Incorrect Decisions, Cost Function. 1. 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 group- screening 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 153 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. 2. 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 0 1 f u j u j y x j uβ β ε = = + +∑ (2.1) where is the u response, uy th 0β is a constant term common to every response; is the linear effect of the ( 1j jβ ≥ ) thj factor, is the level of the 1= ±u jx thj factor in the u run, th uε is the u error term. Further, we shall assume that it is possible to partition the th f factors into a fixed number of groups, such that the group contains factors. The factors will be partitioned into groups of unequal sizes by selecting a set of numbers g thi ik { }1 2 .... ; o 1g ip p p p≤ ≤ ≤ < < and identifying as the probability that a factor belonging to the group is defective. Thus ip thi ip s′ and will be variables. This is a generalization of natural partitioning, when ik s′ ip s′ are actual probabilities. Thus, in addition to model (2.1), we shall make the following assumptions; (i) The total number of factors, f , can be divided into a fixed number of group-factors in the ' g ' initial step such that 1 g i i f k = = ∑ where k is the number of factors in the i group factor. i th (ii) The errors are independent normal with means zero and variance 2σ (known). (iii) All the factors in the group-factor have independently the same probability thi ip (variable) of being defective. (iv) A defective factor within the group-factor has a positive effect thi i∆ . 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. g thi ik =1,2,..., .i g 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. 154 STEP-WISE GROUP SCREENIN DESIGNS 3. 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 ' 'gth 1 g i i i k k f =   =    ∑ ( ) ( ) ( ) 4 mod 1, 2, 3, 4 R g g g g h h = + − = + = 4 (3.1) runs. In the subsequent steps, factors within the defective group-factors are tested as explained earlier. Let be the estimate of the main effect of the i group-factor in the initial step, with ˆiG th iδ effective factors each with effect ( )0; 1 ....,i i kδ∆ > = , 2, i i h . Then and Var . ( )i iE G δ= ∆ ( ) 2ˆ /iG gσ= + Define a random variable W by i ˆ /i iW G g hσ= + (3.2) Then ( )i iE W u iδ= and Var ( ) 1iW = , where /i i σu g h= ∆ + (3.3) Consider the hypothesis H u0 : i i 0δ = alternative u 0i iδ = . Let Iiα be the level of significance for testing the i group-factor in the initial step and denote by th ( )Ii,i iuIi δ αΠ the power function of the test. Then ( ) ( )( )Ii Ii Ii, 1i i i iu Zδ α φ α δΠ = − − u (3.4) Where ( ).φ denotes the standard normal distribution function and ( )IiZ α satisfies ( )( )Ii Ii1 Zα φ α= − (3.5) Thus if i0 for 0i ,δ µ= = then ( )Ii Ii Ii0,α αΠ = and if 0iδ ≠ and 1 /σ∆ is large, then (Ii i iu )Ii,δ αΠ tends to 1. Let denote the probability that the i group factor is declared defective in the initial step. Then * IiΠ th ( ) (*Ii Ii Ii 0 1 i ii i ki ki i i i i i k p p u δδ δ ), ,δ α δ − =   Π = − Π    ∑ (3.6) where ip is the probability that a factor in the group-factor in the initial step is defective. Define a random variable U such that thi i ( ) * Ii1 with probability 0 otherwise 1, 2,...i U i g  Π =  = (3.7) Then ( ) *IiiE U = Π (3.8) 155 M.M. MANENE In the subsequent steps, we shall use non-orthogonal designs. Let 'ip be the probability that a factor chosen at random from the group factor containing thi iδ defective factors that has been declared defective in the initial step is defective. Then ( )1* * 1 1 ' / , / 1 i i i i i k i k i i Ii i i Ii i i Ii i Ii Ii i k p p p q pδ δ δ δ µ α δ − − + = −  = Π Π = Π Π −  ∑ (3.9) where ( ) (1 Ii 1 1 1 1 i i ii i k ki Ii i i i i Ii i k p p δδ δ ),δ µ α δ −−+ = −  Π = − Π −  ∑ (3.10) Let siα be the probability of declaring a non-defective factor from the group-factor in the initial step as defective and thi siγ be the probability of declaring a defective factor as defective in the subsequent steps. Further let iβ + 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. thi Then ( ) ( ) ** * Ii Ii ' 1 ' / / si i si ii i si si si Ii Ii p p p β γ α γ α α β + + + = + −  = − Π + Π Π =  iΠ (3.11) where ( ) * * Ii Iii si si sii pβ γ α α += − Π + Π  (3.12) Let *siα 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 th * siα will take different values at different steps. However for simplicity in algebra, we shall assume *siα to be of uniform value, say *iα . Denote by ( )*jkp j the probability that exactly j factors from the group-factor in the initial step that has been declared defective in the subsequent steps. thi Then ( ) ( ) ( ) ( ) * * * * * * 1 1 1 1 ; 0 1 1 ; 1, 2,......., i i j i k i Ii k k j i ii i Ii j p j k j k j β β β −     − − − =   Π    =    − = Π   (3.13) Let 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), ( )* ik E R thi j 156 STEP-WISE GROUP SCREENIN DESIGNS ( ) ( ) ( ) ( ) ( )( ) ( ) 2 * * i for =0 2 1 1 1 1 1 1 1 for =1,2,.....,k 1 i i ii i k j i i i i i i i i i i k j j k jjk kj j j E R j j j k k j j k k k j k j j k k α ξ     1 j + −  = + + − + − − +  + + − + + − −       − −−  − (3.14) where *iα is as already defined and 0iξ = if * 0iα = and 1 otherwise. Let siR be the number of runs required to analyse the group-factor once it has been declared defective in the initial step. Then thi ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * 1 * * * 1* * * * 1* * * * ** 1 1 1 2 1 1 1 1 1 1 1 1 1 1 i i i i i i i k si k j k j k i i ii i Ii Ii k i i i i Ii i k k i ii i i i Ii i E R E R p j k k k k k β ξ β ξ β β β β α β β β β β = + + =    = − − − − − +    Π Π     + + + − − −   Π       + − − − − − − +  Π    ∑ *2 *2 i i iβ (3.15) Using (3.13) and (3.14). If sR is the number of runs required to analyse all the group factors declared defective in the initial step, then ( 1 g )s i s i iR U E R = = ∑ (3.16) 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 g thi ( )1, 2,....,ik i g= ( ) ( ) ( ) ( ) ( ) ( ) 1** * 1 * * * 1 1 * ** * 1 1 1 ** 1 1 2 1 1 1 2 1 1 1 1 i i i g k i i i i g g i i i ii i ii i g g gk i ii i i i i g k i i i i E R h f g k k k k k k α β β ξ α β 2 Ii i iβ α α ξ α β + = = = = = = = β   = + + − − − −     − + − − + Π      − − − − + −   − − ∑ ∑ ∑ ∑ ∑ ∑ ∑ Where * *, iiβ α and iξ are as defined earlier. Proof : In the initial step we require 1 R g h= + runs ( )1,2,3,4h = . The number of runs 157 M.M. MANENE required in the subsequent steps is ( ) 1 g s i s i iR U E R = = ∑ with ( ) ( ) 1 g s i i E R E U E R = si   =     ∑ (3.17) The theorem then follows on using (3.8) and (3.15) in (3.17), simplifying and noting that ( ) ( )I sE R R E R= + . Corollary 3.1 : For large values of i s σ ′∆ and small values of ,ip s , the expected number of runs in a step-wise group screening design with group-factors in the initial step, the group- factor being of size is approximately equal to g thi ( 1, 2,......,ik i g= ) ( ) ( ) ( )( ) ( ) g * 1 g g 2 1 1 1 2 1 1- 1 1 1 2 1 i si i i i i i i i Ii i Ii i i i g h k p k k k k k p ξ α α α α = = =  − + + − − + − +    + + − ∑ ∑ ∑ Proof : If i s σ ′∆ are large, then Π ≈ ( )* 1 1 ,ikIi Ii i Iiqα− − Π ≈ 1 ( ) ( )*1 and 1 1 1 iksi i si i si Ii ip qγ β α α α ≈ ≈ − + − −  Further if ip s′ are small, then ( ) ( )1 1 1ikIi i Ii i i Iiq k pα α α− − ≈ − + up to order ip . The corollary then follows on using these approximations in the expression for given in theorem 3.1. ( )RE 4. 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 ( ) ik E j thi ( ) * 1 ik i i i Ii E j k k iβ β + + = Π (4.1) where * */ Iii iβ β + = Π Let d be the number of factors declared defective in the subsequent steps. Then 158 STEP-WISE GROUP SCREENIN DESIGNS g 1 i ii i d k Uβ + = = ∑ (4.2) where U is as defined in (3.7). i Let ( )0ip be the probability that a factor chosen at random from the initial step group- factor declared non-defective, is defective. Then thi ( ) ( ) ( )0 *1 / 1i i Ii Iip p += −Π −Π (4.3) Let ip + be the probability that a factor from the i group factor is non-defective given that it is declared defective. Then th ( )1 /i si i ip pα β ++ ′= − (4.4) Theorem 4.1 : Let RM 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 g th ik ( )1, 2,.....,ip i g= . Then ( ) g i=1 R i i IE M k p i siγ += Π∑ Proof : The expected total number of factors declared defective in the subsequent steps is given by ( ) ( ) g g * 1 1 i i ii i i E d k E U kβ + = = Iiiβ + = =∑ ∑ Π (4.5) The probability that a factor which is declared defective from the group-factor, is defective is given by 1 thi ip +− . Therefore ( ) ( ) ( ) ( ) g 1 g * 1 g * 1 1 1 R i i ii i i sii i i i Ii si i E M k p E U k P k p β β α γ + + = + = = = −  ′ Ii = − − Π = Π ∑ ∑ ∑  i i (4.6) as required. Let denote the expected number of defective factors declared non-defective in the initial step. II Then ( ) ( ) ( ) g 0 1 g 1 1 1 I i i i i Ii i I E U k p k p = + =   = −    = −Π ∑ ∑ (4.7) Theorem 4.2 : In a step-wise group screening design with errors in observations and unequal a- priori probabilities, the expected number of defective factors declared non-defective in the subsequent steps is given by ( ) g 1 1 .s i i Ii si i I k p γ = = Π −∑ 159 M.M. MANENE Proof : The expected total number of defective factors in all the group-factors in the initial step is equal to . Therefore g 1 g i i i k p = ∑ ( ) ( ) ( ) 0 1 1 1 1 1 g g s i i i i i R i i g i i Ii si i I E k p U k p M k p γ = = + =   = − − −    = Π − ∑ ∑ ∑ (4.8) This completes the proof. Theorem 4.3 : Let uM be the number of non-defective factors declared defective in the subsequent steps. Then ( ) ( ) g * 1 u i si Ii i i E M k pα Ii + = = Π − Π∑ Proof : The total number of factors declared defective in the subsequent steps is g 1 i ii i k Uβ + = ∑ . Thus g 1 u i ii i iM k U pβ + + = = ∑ which implies that ( ) ( ) g 1 g * 1 u i i ii i i si Ii i Ii i E M E k U p k p β α + + = + =   =     = Π − Π ∑ ∑ (4.9) 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 g thi ik ip ( )1, 2i g= ,....., . Then ( ) g g g * 1 1 1 i i Ii si i si Ii i Ii i i i I kp k p k pγ α+ + = = = = − Π + Π − Π∑ ∑ ∑ Proof : The expected total number of incorrect decisions is given by ( ) ( ) ( ) ( ( ) g g g * 1 1 1 g g g * 1 1 1 1 1 I s u i i Ii i i Ii si i si Ii i Ii i i i i i i i Ii si i si Ii i Ii i i i I I I E M k p k p k p k p k p k p γ α γ α + + = = = + + = = = = + + = −Π + Π − + Π − = − Π + Π − Π ∑ ∑ ∑ ∑ ∑ ∑ )+Π (4.10) using (4.7), (4.8) and (4.9). This completes the proof. Corollary 4.1: 160 STEP-WISE GROUP SCREENIN DESIGNS ( ) ( ) ( )( ) ( ){ } g g 1 1 g 1 , 1 ,i i i i i i si Ii Ii Ii i i k k i si Ii i i Ii i Ii Ii i Ii Ii Ii i Max I k p k p k q q k p γ φ α α α φ α φ α = = = = − Π + + − Π − Π ∑ ∑ ∑ , Proof : ( ) ( ) ( ) g g g * 1 1 1 g 1 i i i i Ii si i si Ii i Ii i i i i i R u i I k p k p k p k p E M E M γ α+ + = = = = = − Π + Π − = − + ∑ ∑ ∑ ∑ Π (4.11) Hence I will take its maximum value when ( )RE M is minimum and is maximum. But takes its maximum value when ( uE M ) )(E M R Iii+Π is replaced by ( ),Ii Ii Iiφ αΠ . That is ( ) ( g 1 ,R i si i Ii Ii i Min E M k p )Iiγ φ α = = Π∑ (4.12) ( uE M ) will take a minimum value when *IiiΠ is replaced by its maximum value and Ii+Π is replaced by its minimum value. That is when *IiΠ is replaced by ( ) ( ){ }1 ,i Iiki ik kIi i i Ii i Iq qα φ+ − Π α ), and is replaced by Ii +Π ( ), .Ii Ii Iiφ αΠ Thus ( ) ( ) ( ) ({ } g 1 1 ,i ik ku i si Ii i i Ii i Ii Ii i Ii Ii Ii i Max E M k q q k pα α φ α φ α = = + − Π − Π∑ (4.13) The result follows on using (4.12) and (4.13) in (4.11). Corollary 4.2 : For large i σ ∆ and small 'ip s ( ) g 1 1i si Ii i Ii i i i Max I k p p kα α α = ≈ − + −  ∑ Proof : For large i σ ∆ and small ( )' , , 1, 1 and 1iki Ii i Ii Ii si i i ip s s q kφ α γΠ ≈ ≈ ≈ − p . The result follows immediately on replacing these values by their approximations in the expression for Max I given in corollary 4.1. 5. 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 1 2 ( )1 2 MaxC c E R c I= + 161 M.M. MANENE We shall only discuss the case when 'i s σ ∆ are large and 'ip s are small. For simplicity we shall only consider the special case when ( )* *, 1, 2,...,si s and i gα α α= = = =Ii Iα α iα . Theorem 5.1 : If *, andIi I si s I * ,α α α α α α= = = then for large 'i s σ ∆ and small 'ip s , the value of which minimizes the expected total cost is given by ik 1 1 / g g i i i i i i i k f G H G G G H = =   = − +    ∑ ∑ i i (5.1) where ( ) ( )( ){ } ( )* 1 2 1 2 1 1 1 2 1 i I s i s I G c p c pα α α α α i − = − − − + + − and ( )( ) ( ) (*1 2 1 1 3 ; 1, 2,..., g . 2i I s i s I i H c p c p iα α α α α = + − − − − =    ) Proof : The problem is to minimize ( )1 2C c E R c Max I= + subject to ( ) ( ) g 1 i ii 0 ; 1, 2,..., i i i k f k i = = > = ∑ g That is to minimize ( ) ( ) ( )( ) ( ) ( ) g g 2 1 1 1 g g * * 1 1 g g 2 2 1 1 1 1 1 1 1 3 1 2 2 1 I s i I i i 2 i i s i i s i i i i s I s i i s I i i i i C c h g f p k p k p k p c f k p k p α α α α α α α α α α α = = = = = =  = + + − − + −   + − − − −     + − + −    ∑ ∑ ∑ ∑ ∑ ∑ (5.2) subject to ( ) ( ) g 1 i ii 0 ; 1, 2,..., i i i k f k i = = > = ∑ g using corollaries (3.1) and (4.2) noting that * *i 1, , andIi I si s iξ α α α α α α= = = = . Using the method of Lagrange multipliers, let 162 STEP-WISE GROUP SCREENIN DESIGNS ( ) ( ) g 1 2 g 1 2 1 , ,....., , Max i i F k k k c E R c I k fλ λ =   = + +     ∑ − where λ is the Lagrange multiplier. Assuming continuous variation in , the critical value of is obtained from the equations ik ik / 0 ; 1, 2,..., , and / 0iF k i g F λ∂ ∂ = = ∂ ∂ = (5.3) The theorem follows immediately on solving equations (5.3). 6. 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 Table 1 below. The corresponding values of and Max ( )C )(C ) (E R I are also given. Table 1 : 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. )C * *, and , for 100Ii I i si s fα α α α α α= = = = ( )C 1 ( ) * 2 1a 3, 13, 0.05 , : 3 : 5 , 0.035I s ih g c c p pα α α= = = = = = ≤ = . i Ip Ik 1 2 3 4 5 6 7 8 9 10 11 12 13 0.008 0.009 0.010 0.013 0.015 0.017 0.020 0.022 0.025 0.027 0.030 0.033 0.035 17.088 15.024 13.372 9.942 8.418 7.252 5.940 5.265 4.454 4.014 3.463 3.013 2.755 Total 100.000 ( ) 28.577, 0.770, 29.039.E R Max I Min C= = = ( ) 26.739E R = The corresponding value of min when incorrect observations are not considered. ( ) * 2 1b 4, 20, 0.05, : 3 : 5 , 0.100I s ih g c c p pα α α= = = = = = ≤ = . 163 M.M. MANENE i Ip Ik 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.040 0.045 0.050 0.053 0.055 0.060 0.062 0.065 0.070 0.075 0.078 0.080 0.082 0.085 0.087 0.090 0.092 0.095 0.098 0.100 9.565 8.337 7.354 6.853 6.550 5.880 5.642 5.313 4.827 4.403 4.179 4.037 3.902 3.712 3.592 3.423 3.316 3.164 3.020 2.931 Total 100.000 ( ) 51.422, 1.545, 52.349E R Max I Min C= = = ( ) 49.219E R = . 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 ( )E R I is used, the number of runs increases. It should be noted that these tables are just an illustration. The values of 'ip s used are not unique; neither is the ratio c 2 1: .c References DORFMAN, R. 1943. The de1ection of defective members of large populations, Annals of Mathematical Statistics, 14: 438 -440. LI, C.H. 1962. A sequential method for screening experimental variables. Jour. Arner Statistical Assoc. 57: 455-477. MANENE, M.M. 1985. Further investigations of group screening designs: Step-wise designs; Ph.D. thesis submitted to University of Nairobi. MANENE, M.M. 1997. On Two Type Step-wise group screening designs, Proc of fifth scientific con£. Of Eastern. Central and Southern Africa Network of I. B.S., 57 –62. ODHIAMBO, J.W. and MANENE, M.M. 1987. Step-wise Group Screening Designs with Errors in Observations. Comm. In Statistics Theor. Meth. 16(1 0): 3095-3115. ODHIAMBO, J.W. and PATEL, M.S. 1986. On Multistage Group Screening Designs. Comm. in Statistics Theor. Meth. 15(5): 1627-1645. OTTIENO, J.A.M. and PATEL M.S. 1984. Two Stage Group Screening Designs with Unequal A- priori Probabilities. Comm in Statistics Theor. Meth. 13(6): 761-779. PATEL, M.S. 1962. Group Screening with more than Two Stages; Technometrics, 4: 209-217. 164 STEP-WISE GROUP SCREENIN DESIGNS PATEL, M.S. and MANENE, M.M. 1987. Step-Wise Group Screening with Equal Prior Probabilities and no errors in Observations, Comm. In Statistics Simula_and Comguta, 16(3): 817-833. PLACKET, R.L. and BURMAN, J.P 1946. The Design of Optimum Multifactor Experiments. Biometrika, 33: 305-325. STERRET, A. 1957. On the Detection of Defective Members of Large Populations, Annals of Mathematical Statistics. 28: 1033-1036. WATSON, G.S. 1961. A study of the Group Screening Method, Technometrics, 3: 371-388. Received 30 December 2002 Accepted 20 November 2003 165 Step-Wise Group Screening Designs with Unequal A-Priori Probabilities and Errors in Observations Department of Mathematics, University of Nairobi, P.O. Box 30197, Nairobi, Kenya, Email: mathuon@iconnect.co.ke. ÊÕÇãíã ÇáÛÑÈáÉ ÇáÊÏÑíÌíÉ ááãÌãæ� ã.ã. ãÇäíä 3. Expected Number of Runs 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 Proof : In the initial step we require � runs �. The number of runs required in the subsequent steps is Calculation of the expected number of incorrect decisions Proof : The total number of factors declared defective in the subsequent steps is �. Proof : and � Proof : The problem is to minimize Total Total References