Electromagnetic Modeling of the Propagation Characteristics of Satellite Communications Through Composite Precipitation Layers Science and Technology, 7 (2002) 323-332 © 2002 Sultan Qaboos University Deletion Designs Michael Kamau Gachii, John Wycliff Odhiambo and Anne Wachira Department of Mathematics, University of Nairobi , P.O.Box 30197, Nairobi, KENYA, Email: gachii@uonbi.ac.ke. التصاميم اإلنشاطية مايكل كامايو جاشي ، جون ويكلف أوديامبو و آن واشيرا يقدم هذا البحث طريقة إلنشاء نوع من التصاميم العاملية ذات التكرارات المحددة المرنة في شكل مجموعات ويعطي :خالصة . تعابير جبرية إليجاد نقص المعلومات في التفاعالت ذات الرتب المنخفضة ABSTRACT: In this paper a method of constructing a class of flexible single replicate factorial designs in blocks is given. Simple expressions for calculating loss of information on low order interactions is presented. KEYWORDS: Asymmetrical Factorial Designs, Generalised Cyclic Designs, Loss of Information. 1. Introduction C onsider a single replicate factorial experiment involving n factors factors F occurring at levels. Let a ,F F iis ...1 2 na ,..., ;1 2 nF a a= denote a treatment combination, where a ( 0 ) is the level of factor F . The number of treatment combinations is then given by i s 1i ia s≤ ≤ − i v 1 n i i = = ∏ (1.1) These treatment combinations will always be lexicographically ordered. That is a treatment combination appears before another treatment combination a if and only if for the first u such that a we have for 1 v a ...1 2 na a a= * * *...1 2 na a a= au ≠ * u * uu aa < ≤ ≤u n. Suppose we wish to construct a v 1 n i i = = s∏ single replicate factorial experiment in blocks. We first construct a single replicate single replicate preliminary block design, say d , using one of the known methods, such that for ...1 2 r× × × ir ≥ n i n r r p s 1, 2,...,i = . We can then select l r ,si i i= − i levels of the i-th factor of d and delete from dp p all treatment combinations where factor F occurs at any of the selected levels. The resulting s sil ...1 2 ns× × × single replicate design is referred to as a j-th order deletion design if levels are deleted from j factors. Bose (1947) laid the foundation of factorial designs. He used finite Euclidean geometry to construct symmetrical factorial designs in blocks. Kishen and Srivastava (1959) extended the method of finite geometries to the construction of balanced confounded asymmetrical factorial designs thereby introducing the idea of deletion. John and Dean (1975) proposed a simple method of confounding based on the properties of generalised cyclic designs from a set of generating 323 M.K. GACHII, J.W. ODHIAMBO and A. WACHIRA treatments or generators and showed that the confounding patterns could easily be determined from these generators. More recently Voss (1986) has constructed nearly orthogonal single replicate factorial designs in blocks. He uses the deletion technique where he deletes from the first factor, without loss of generality, to obtain first order deletion designs. The most recent contribution in this direction is that of Chauhan (1989) who generalized the work by Voss (1986), by constructing efficient single replicate designs using the generalized deletion technique. Starting from an single replicate generalized cyclic design, levels are deleted from the first factors, without loss of generality , to obtain an ( ) deletion design. ns 1m 1 1 m n ms l s −− The overall objective of the present study is to give results for the general order deletion designs of the form s s which are proper, for 1( ) 11 mn m l− − 1l s≤ ≤ − and less than or equal to the number of generators of the preliminary single replicate generalized cyclic design. The efficient proper single replicate designs of the form 1m ( ) 1ns l s −− given by Chauhan (1989) thus become special cases of the results obtained in this study. The method proposed by John and Dean (1975) is used to construct the preliminary single replicate factorial design, which is always symmetric. That is, factor occurs at siF i s= levels for all 1, 2,...,i n= . Conditions are given which guarantee the existence of either proper or improper deletion designs. Simple formulas for calculating the loss of information, due to confounding with blocks, on main effects and two factor interactions are given. A simple method of choosing a fraction for estimating main effects and low order interactions is also given. 2. Notations We shall first assume the fixed effects linear model ah a h ahy µ τ β ε= + + + (2.1) where denotes the observed yield from treatment combination a in the h-th block; ahy aτ denotes the effect of the treatment combination a ; hβ denotes the effect of the h-th block and ahε are uncorrelated random errors with mean zero and variance 2σ . Let ( )ahy y= and ( )aτ τ= denote v 1× vectors of observations and treatment effects respectively, each lexicographically ordered by a. That is the i-th row corresponds to the i-th treatment in the above arrangement of the v treatment combinations. We shall denote the incidence matrix, the intrablock matrix, the diagonal matrix of block sizes and the number of blocks, respectively, by N, A, K and b . The i-th row of the incidence matrix N corresponds to the i-th lexicographically ordered treatment combination . The qa 1× vectors of ones and of zeros will be denoted by 1 and , respectively. A generalized interaction will be denoted by a where q 0q x ...1 2 nx x x x= such that 1x j = if jF is present in the interaction and otherwise. A v contrast vector will be denoted by c where 0jx = 1× x c c (2.2) ........1 2x xx 1 2c c= ⊗ ⊗ ⊗ n x n with being an vector of ones if ixic lis × 0,jx = otherwise it is an s li × contrast vector. The minimum variance unbiased estimator of the generalized interaction a is represented by cx x τ′ , where iτ , the i-th element of τ , is the estimate of the fixed effect of the i-th treatment combination. We shall denote the set of n factors { by , ,..., }1 2 nF F F { }1, 2,..., n . Then for a non-empty subset will denote the vector (factorial) space of contrast vectors { , ,..., } {1, 2,.., }, ( , ,..., )1 2 r 1 2 ri i i n V i i i⊂ 324 DELETION DESIGNS xc corresponding to the estimator of the generalized interaction xα , where { } { , ,..., }1 2 rj i i i⊂ and jx 0= otherwise. All the notations corresponding to the preliminary design will carry the subscript p while those corresponding to the deletion design will carry no subscript. m n 1 = sumg 2 3 ,..., mg g g ( -1 m 1m d P a j s≤ / 2 / 2jl a s= = 1j 1 / 2j s s s a − 1, ..., s a = − + ( ) -ssI J ( ) sI F 3. Some Properties of Deletion Designs We start by giving results useful in constructing deletion designs which can be used to estimate the main effects and also the results are useful in calculating loss of information due to confounding in blocks. Theorem 1: If is a generalized cyclic design generated by generators such that of the first rows of an identity matrix of order and are the last rows of an identity matrix of order , then there exists a proper deletion design of order , provided . ns ( -n m + 1 ≤ )1 ) j , n m m Let be the jD ir s× matrix obtained from an js s j× identity matrix by deleting the t-th row if the t-th level is deleted from factor in the preliminary design d to obtain the deletion design . In our study, . We now state the following result. jF p , s s j= = 1, 2,..., j n Lemma 1: Suppose levels are deleted in descending order from factor F and let j be an s s× permutation matrix with 1 in the -th column of the 0-th row. Then for aja j we have ( ) ( ) ' ' , if 0 if 0 and 1, 2,..., 1 2 , if 1, 2,..., and , if 1, 2,..., 1 0, if , 1,...and 0 or if j j a j j js l s l j j j j j j j s l s l a l D P D l s l l a a s l a l a a a l s a s a ≠− −  =  − = = −  = − =  − − = + + −  = − − + ≠ and for a s we have / 2 j > ( ) ( ) ' ' , if 0 - 2 if 1, 2,..., , if ..., 0, if , 1, 1 j ja j js 1 s 1 j j j j s l s l l a l D P D l a l l s a l a a s − − =   =  − = − −  = + − Next let c be the contrast vector represented by one of the columns of the matrix j ( ) s which span the space of dimensional contrast vectors, where s is an s dimensional identity matrix and that is an s'( ) ( ) 1 1 ,s s sJ = ( ) s× matrix of 1 . Also, let the rows and the columns of the matrix ' s ( ) ( ) s ssl J− be numbered as 0,1, 2,..., 1s − . Note if levels were deleted from factor l jj in the preliminary factorial design, then the contrast vector c is one of the columns of the matrix ( ) ( ) s l ( )ls s lJ− −− 1,I − where l s 2,..., 1.= − 325 M.K. GACHII, J.W. ODHIAMBO and A. WACHIRA Lemma 2: ( ) ( ) ( ) ' ' 1 , if 0 , if 0 and 0 1 if 0 and 1, 2,..., 1 j j a j j j j j j s s a c D P D c s l a s l s l a l s  − =  = − = ≠  − − − = = − Next for a and we have the following results. 0j ≠ 1, 2,..., 1l = s − a Lemma 3: If a s and ( provided a s/ 2j ≤ 1, 2,..., jl = / 2j ≠ when / 2l s= ) or if and then / 2ja s> 1, 2,...,l = js a− , for or and 2 , for , otherwise j 1 j 2 j 1 a j j j j j j 1 2 l c c c c D P D C s l c s 2Ω Ω Ω Ω Ω − ∈ ∈ ∉ ′ ′ = − ∈ ∩  − Ω∩ l= ) where of the matrix {the columns 1, 2,..,1 ja iΩ = − ( ) (( ) s l ss l I J l− −− − } = {the columns , 1 2Ω , 0, 1, 2, ......, 1ja i i l+ = − ,1,0s l− − ,...,2 −= li of the matrix ( ) ( ) ( )s l J s ls l I − −− − }. Lemma 4: If a sj ≤ / 2 and l a a s aj j j= + + − −1 2, ,..., 1 then 1, for or and , for , otherw ise j j j 2 j 1 a j j j j j j j 1 2 j a c c c c D P D C s l a c l s a Ω Ω Ω Ω Ω  − 2∈ ∈ ∉  ′ ′ = − − ∈ ∩  − − ∩ Ω where of the matrix {the columns 0,1,2,...,1 jaΩ = ( ) ( ) ( )s l ss l I J l− −− − }, {the columns s-l-2 jaΩ = of the matrix ( ),i i+ = 0,1, 2,.., 1ja − ( ) ( )s ls l I J s l− −−− }. Lemma 5: If a s and l s or if / 2j < , 1,..., 1j ja s a s= − − + − / 2 and / 2ja s l s= = then 0.= a j Cj j j jc D P D′ ′ Lemma 6: If a s and then / 2j > 1, 2,..., 1j jl s a s a a= − + − + −j , for and , for 2 , otherw ise j j 1 j 2 j 1 a j j j j j j j 1 2 j a s c or c c c D P D C a l c a l s 2Ω Ω Ω Ω Ω  − ∈ ∈ ∉  ′ ′ = − ∈ ∩  − − Ω∩ where of the matrix {the columns , 1, 2,...,1 ja i i lΩ = − = ( ) ( ) ( )s l ss l I J l− −− − }, {the columns s- , 0,1, 2,.., 12 ja l i i s aΩ = − − = j− − of the matrix ( ) ( ) ( )s l ss l I J l− −− − }. Lemma 7: If a s and then c D/ 2j > , 1,..., 1j jl a a s= + − 0. a j j j j jP D C′ ′ = 4. Loss of Information on Main Effects Dean (1978) showed that for a given vector c , the loss of information x , 0 1x xϕ ϕ≤ ≤ , due to confounding in blocks is given by 326 DELETION DESIGNS 1x x c x x x c NK N c c ϕ ′ − ′ ′ = (4.1) where is the incidence matrix and is a diagonal matrix of block sizes. N K We shall use the notation ( ) a j a s l j j sj d 1 D P D 1 l− −′ ′= where ( a j a s l j j sj d 1 D P D 1−′ ′= )l− d is as given in lemma 1 and d d .......a a a a a a1 2 k 1 2 kd= × × × We shall also write d c* a j a j j jj D P D c j′ ′= (4.2) where c D is as given in lemmas 2, 3, 4, 5, 6, and 7. a j j j j jP D c′ ′ 1− ms We consider deletion designs of the form blocks of size derived from an s generalized cyclic design d with 1 1( ) with n m m n ms s l b sλ− −− = p 1(1/ ) ( ) n m mk s s lλ −= n (1/ ) and m np pk s bλ λ −= = where n is the number of factors, is the order of the deletion design constructed, is the number of generators and with 1m ib m 1 1/ m i λ = = ∏ ( ), ; 1, 2,..., ,i iCF s g i m= =b H as given by John and Dean (1975). We give here two results on loss of information on main effects. Theorem 2: Loss of information due to confounding in blocks on any factor whose levels were not deleted from d to obtain d , is given by 1 ( 1, 2,..., )jF j n m= − p 1 2 1 * . 1 22 1 ... .. ... ..... ...... ( 1)( ) 1 2 n aa a jj 1 j 1 na a a 1 2 j j 1 n x n m mm w da a a a a a a a s s l s λ ϕ − + − + + − ∑ ∑ ∑ = − − d where 1, ... is in the initial block of ... 0, otherwise 1 2 n p 1 2 n if a a a d w a a a = =   a Proof: The contrast vector c is as given in (2.2) c is an s unit vector for is an x xt t 1× , 1, 2,..., ; xt1 tt j t n m c≠ = − ( ) 1s l− × unit vector for t n 1, 2, ...,m n1 1n m= − + − + and j x jc is any of the columns of the matrix ( ) ( )s sJsI − . Without loss of generality, let c be the i-th column of the matrix x j j ( ) ( )s ssI − J . Then we have { ( 1) ( 1) } ( )n m j mx x j 1 2 j 1 j 1 1 1c c is s s s i s s s l− −′ − − − = + − + − − −  = (4.3) ( 1)( ) m n m1s s l s −− − 1 327 M.K. GACHII, J.W. ODHIAMBO and A. WACHIRA But (( ) 1 n mm m m s1 1 K s s l λ ) I λ− −−= − (4.4) From John and Dean (1975), and Chauhan (1989) we have, ' 1 1 1 1 1 11 2 1 2 . ... ...... x ax a xx x n n1 n n n nna a an c N N c w c D P D c c D P D ca a a ′′′ xn′ ′= ⊗ ⊗∑ ∑ ∑ = (4.5) 1 2 * 1 2 1 2 1 1 . ... ... ... ... n a jn j j na a a w da a a a a a a a− +∑ ∑ ∑ d Therefore using (4.2), (4.3), (4.4) and (4.5) in (4.1) we obtain Theorem 2. Theorem 3: Loss of information due to confounding in blocks on any factor whose levels were deleted from d to obtain d , is given by 1 1 ( 1, 2,..., )jF j n m n m n= − + − + p 1 * 1 2 1 1 2 1 1 2 22 1 ... ... ... ... .... ( 1)( ) a jj 1 j n j 1 j na a an x n m mm w da a a a a a a a a a s l s l s λ ϕ − + − + + − ∑ ∑ ∑ = − − − d where 1 2 1 2 1, if ... is in the initial block of w =... 0, otherwise n p n a a a a d a a a =   Proof: The contrast vector c is as given in (2.2), c is an sx txt 1× unit vector for t n ; is an ( unit vector for 11, 2,...,= − m xt tc ) 1s l− × 1 1, and 1, 2,..., j x jt n m n m n c≠ = − + − +t j is any of the columns of the matrix ( ) (( ) )s l J− sI− − l−s l . Without loss of generality, let jx jc be the i-th column. Then we have 1 1 1 1 1 12 1 { ( ) ( 1) ( ) ( 1) ( ) }( ) n m j n m 1 x x j n mn m n m j n m n j is s l c c s l s s l s l i s s l s l − − + − ′ − + −− − − + −  − + =   − − − + − − − − −  1 − = ( (4.6) )( )s l s l sm n m− − − −1 1 1 But we know that * 1 1 ' 1 2 1 2 1 2 . ... ... ... ... a jn j - j+ n x x a a an c NN c w d da a a a a a a a ′ = ∑ ∑ ∑ cf.(4.4). Therefore using (4.4), (4.5) and (4.6) in (4.1) Theorem 3 follows. 5. Confounding in Deletion Designs The following results in confounding in generalized cyclic designs are due to John and Dean (1975). The number of degrees of freedom confounded with blocks for any given interaction, α x is given by Y k (5.1) 1 2 11 2 1/ ... ( )... j jn n n xx a a a a j w za a a = = ∑ ∑ ∑ ∏ 328 DELETION DESIGNS where 1, if 0 and 1 1, if 0 and 1 1, if 0 j j j x j jj j s a x z aa x  x − = =  = − ≠ =  = and 1 2 1 2 1, ... is in the initial block of ... 0, otherwise n p n if a a a a d wa a a = =   If the number of degrees of freedom in (5.1) is zero the interaction is unconfounded with blocks and if it is the interaction is totally confounded with blocks. Consider any interactions between the n factors, say the interactions of the factors Let ( 1) n x j j 1 s = −∏ ., ,...,i i i1 2 rF F F (5.2) '(( ))ijA a= where a is from the i-th generator ij 1` 2 1 2... , 1, 2,..., and , ,..., .i i i ing a a a i m j i i ir= = = Consider all the f f× sub matrices contained in the 1j -th, 2j -th, .... fj -th rows of A and let ...1 2 fj j j h be the absolute values of their determinants ( , ).f r f m≤ ≤ Define fH as follows . (5.3) 1 2 1 21 2 1, if =0 ( \ { , ,..., } { , ,..., }) if 0< <... 0, if f f j j j f r f H HCF h j j j i i i f f m = ⊂ >      m = In our case the treatment combinations in the initial block are of the form 1 1 2 2 ... ( 0,1,..., 1; 1, 2,..., )m m i iu g u g u g u k i m+ + + = − = The number of treatments in the initial block with -th, -th, ..., -th factors all zero is given by where 1i 2i ri 1 2... m r i i ir s w− 1 -1 1 - 1 2 1 22 1 2 ( , / ) if and 0 s ( \ { , ,..., } { , ,..., }) if . .. ...r r f f r f = r g i i i g rj j g HCF s H H r m H w HCF w j j j i i i g r mj ≤ ≠∏ = ⊂ < ≤ 1 2 1 2 1 2 (w \ { , ,.., } { , ,.., }, if (5.4) .. r g m rj j m s HCF j j j i i i r mj −       ⊂ >  where g is such that and 1 20 and ... 0.g g gH H H+ +≠ = = 1g = if 1 0H = . Let Y be denoted by x ..1 2 hj j j Y where x has the 1j -th, 2j -th, ..., hj -th digits unity and the remainder zero. Then it can be shown that for the interactions of the factors the number of degrees of freedom confounded with blocks is given by 1 2 ,i iF F ,..., r iF (5.5) 1 ... ... ... 1 2 1 21 2 1 1 21 ( \ { , ,.., } { , ,.., }) 1 r r g r i i i i i j j j g r g Y w Y j j j i i i − = = − ⊂∑ − 329 M.K. GACHII, J.W. ODHIAMBO and A. WACHIRA We now give the following results. Theorem 4: For the generalized cyclic designs of theorem 1, all the main effects are estimable with full efficiency. Proof: For the main effect of factor , 1, 2,...,jF j n= we have, using (5.3), and w1 1H = 1j = by (5.4). Thus, using (5.5), Y 0j = . Theorem 5: All the r-factor interactions among any number of the first ( )1n m− + factors, and all the r-factor interactions among at least two of the first ( )1n m− + factors, and any number of the last factors, are partially confounded with blocks provided ( 1m − ) r m≤ . Proof: For the r-factor interaction, r we have from equation (5.4) 2, 3,.., 1,n m= − + )r 1 ...1 2 r r - j j jw s= and thus equation (5.5) yields ...1 2 0 ( r r j j jY s≠ < − Thus all the interactions between at least two of the first ( )1n m− + factors are partially confounded with blocks. For the submatrices of the matrix corresponding to the r-th factor interactions between at least two of the first 3, 4,..., ,r n= r r× A ( )1n m− + factors and any number of the last ( )1m − factors are singular. Therefore using equation (5.4) we obtain 1 2 ...2 r r j j jw s −= and using equation (5.5) we get ...1 2 0 ( r r j j jY s )r≠ < − We can therefore conclude that all these factor interactions are partially confounded with blocks. Hence the theorem. r Confounding in deletion designs has been studied by Chauhan (1989). Theorem 6 below is due to her. Let α x be a given interaction. Then the factors or simply {1 2, ,..., nF F F }1, 2,..., n can be partitioned into three mutually exclusive and exshaustive sets and1 2 3Ω , Ω Ω as follows: 1Ω contains the factors whose levels were not deleted from d to obtain d , that is {p }1, 2,..., n m− 1 ; contains the factors whose levels were deleted from d to obtain d and these factors are not in the factorial space V , that is the factors 2Ω p x { }1,n m 2,...,1 1n m 1n m a− + − + − + ; 3Ω contains the factors whose levels were deleted from d to obtain d and these factors are in the factorial space , that is factors{ p xV }1 11,n m a n m− + + − ( ),..., rj 1 j 2,..., ;n a+ ... jx= = = 0= 1 r = ,1, 2,..., ma+ 1 2 j jx x . We shall write the factorial space as V j j if and all other xV 1 2, x ‘s are zero, where 1 2 , } {{ , ... r 1, 2,..., }j j j n 1 2, ,..., {ri i i⊆ . Let { , then we have the following theorem. 1}m} 1, 2,..., n⊂ − Theorem 6: (Chauhan (1989)). Let the contrast vector 1 2 1( , ,..., , 1, x rc V i i i n m a∈ − + + 1n m− + a + 2,..., )n and let 330 DELETION DESIGNS c (5.6) 'x xp D c= then where 1 2 1 1( , ,..., , , 1, 2,..., ) x p p rc V i i i g n m a n m a∈ ⊕ − + + − + + n )2(g Ω∈ Ρ 1 2 ...D ; that is g belongs to the power set of denotes the direct sum, where and 2Ω ⊕ nD D D= ⊗ ⊗ ⊗ again where is as defined in lemma 1. jD We now state the following results: Theorem 7: For the deletion designs of order m derived from generalized cyclic designs of theorem 1, all the main effects of the first factors are partially confounded with blocks, while all the main effects of the last m factors are fully estimable. ( n m− ) ) This makes it possible to proof the following theorem. Theorem 8: If then for the deletion designs of the form 1,m = 1 (ns s l− − the main effects of the first factors are partially confounded with blocks and the loss of information on factor F is given by ( 1n − 1, 2,...,j = ) j ( )1n − ϕ x l s s = − −( )(1 l) provided . / 2l s< Proof: From theorem 7 we know that the main effects of the first ( )1n − factors are partially confounded with blocks. ( ) ( ) 1 2 1 2 1 1 1 2 * ... ... ...... 1 n j- j+ n n a a a a a a a a a a a a x 2 n m 2 w d d s s l s ϕ + − = − − ∑ ∑ ∑ j 2 (5.7) using theorem 2. But by lemma 1, if 0 ≠ l / then (5.8) , 0 2 , , n n a n n n n n s l if a d s l if l a or a l s s l a if a l or if a l s − =  = − ≤ + ≤  − − < + > Therefore due to the nature of the treatment combinations in the initial block, lemma 2 and equations (5.7), (5.8) yield: 2 2 2 1 2 1 ( 1)( ) 2 ( ) ( 2( 1) 1) ( 1)( ) n l 1 n n n n a x n s s s s l s s s l a s s s l s s l s ϕ − − − − = − − − − − − − − − −∑ = − − l s l = − as required. 6. Concluding Remarks These designs give us a simpler method of constructing asymmetrical factorial designs in incomplete blocks. We note that confounding patterns are easily determined from the information gathered from the preliminary factorial designs. Expressions for loss of information in terms of the number of levels, s, of the factors in the preliminary design and the number of levels, l, deleted from ‘j’ factors have been derived. 331 M.K. GACHII, J.W. ODHIAMBO and A. WACHIRA References BOSE, R.C. 1947. Mathematical theory of the asymmetrical factorial design, Sankya, 8: 107-166. CHAUHAN, C.K. 1989. Construction of efficient single replicate designs, preprint. GACHII, K.M. 1993. On the construction of deletion designs, Ph.D thesis, University of Nairobi. GACHII, K.M. and ODHIAMBO, J.W. 1998. Asymmetric single replicate designs, South African Statistical Journal, 32:1-18. JOHN, J.A. and DEAN, A.M. 1975. Single replicate factorial experiments in generalized cyclic designs: I symmetrical arrangements, Journal of the Royal Statistical Society, B, 37: 63-71. VOSS, D.T. 1986. First order deletion designs and the construction of efficient nearly orthogonal factorial designs in small blocks, Journal of the American Statistical Association, 81: 813-818. Received 9 November 2001 Accepted 28 October 2002 332 Deletion Designs Michael Kamau Gachii, John Wycliff Odhiambo and Anne Wachira Department of Mathematics, University of Nairobi , P.O.Box 30197, Nairobi, KENYA, Email: gachii@uonbi.ac.ke. ãÇíßá ßÇãÇíæ ÌÇÔí ¡ Ìæä æíßáÝ ÃæÏ 2. Notations (2.2)