A C T A U N I V E R S I T A T I S L O D Z I E N S I S _________ FOLIA OECONOMICA 162, 2002
Bronisław Ceranka*, M ałgorzata Graczyk**
RELATIONS BETWEEN CHEMICAL BALANCE WEIGHING
DESIGNS FOR p = v AND p = v + 1 OBJECTS
ABSTRACT. T h e in c id e n c e m a tric e s o f te rn a ry b a la n c e d b lo c k d e sig n s fo r v tre a tm e n ts h a v e b e e n u sed to c o n s tru c t c h e m ic a l b a la n c e w e ig h in g d e sig n s fo r p = v a n d
/> = v + l o b je c ts w ith u n c o rre la te d e stim a to rs o f w e ig h ts. C o n d itio n s u n d e r w h ich the e x is te n c e o f a c h e m ic a l b a la n c e w e ig h in g d e sig n s w ith u n c o rre la te d e s tim a to rs o f w e ig h ts fo r v o b je c ts im p lie s th e e x is te n c e o f the d e sig n w ith the sa m e re s tric tio n s fo r v + 1 o b je c ts are g iv e n . T h e e x is te n c e o f a ch e m ic a l b a la n c e w e ig h in g d e sig n w ith u n c o rre la te d e s tim a to rs o f w e ig h ts fo r v+1 o b je c ts im p lie s the e x is te n c e o f th e d e sig n w ith th e sa m e re s tric tio n s fo r p < v +1 o b jects.
Key words: c h e m ic a l b a la n c e w e ig h in g d e sig n , te rn a ry b a la n c e d b lo c k d e sig n .
I. INTRODUCTION
In the presented paper we study the problem o f constructing the design matrix X for the chemical balance weighing designs for p = v and p = v + l objects and relations between these designs, when matrix X is based on the incidence matrices o f ternary balanced block designs. The problem is how to choose the matrix X in such a manner that the estimators o f weights are uncorrelated. Several methods o f constructing matrix X are available in the literature. C e r a n k a , К a t u l s k a and M i z e r a (1998), A m b r o ż y and C e r a n k a (1999), C e r a n k a and G r a c z y k (2000) have shown how chemical balance weighing design with uncorrelated estimators o f weights can be constructed from the incidence matrices o f ternary balanced block
Prof., D epartm ent o f M athematical and Statistical Methods, Agricultural University of Poznań.
** Dr, D epartm ent o f M athematical and Statistical Methods, Agricultural University of Poznań.
constructed chemical balance weighing designs with uncorrelated estimators of weights for p = v +1 objects from incidence matrices o f ternary balanced block designs for v treatments. C e r a n k a and K a t u 1 s к a (1991) and (1999) have shown relations between parameters o f chemical balance weighing designs in situation, when matrix X o f chemical balance weighing design was based on the incidence matrices o f balanced incomplete block designs and on balanced bipartite block designs, respectively.
The results o f n weighing operations to determine the individual weights of
p objects with a balance that is corrected for bias will fit into the linear model
y = X w + e,
where у is an n x l random observed vector o f the recorded results o f the weights, X = {xij), i = l , 2 , . . . , n , j = 1 ,2 ,..., p , is an n x p matrix o f known elements with x tJ = - 1 , 1 ,0 if the j -th object is kept on the right pan, left pan, or is not included in the /-th weighing operation, respectively, w is the p x l vector representing the unknown weights o f objects and e is an n x l random vector o f errors such that E (e)= 0 „ and E ( e e ) = o 2I n, where 0„ is the n x l vector with zero elements everywhere, I„ is the n x n identity matrix, "E" stands for the expectation and e is used for transpose o f e. The matrix X is the design matrix and we refer to the chemical balance weighing design X with the со variance matrix a 21„.
If the matrix X X is nonsingular, the least squares estimates o f the true weights are given by
w = (X X )'* X у and the variance-covariance matrix o f w is
Var( w ) = a 2(X X )Л
A ternary balanced block design to be a design consisting o f b blocks, each o f size к , chosen from a set o f size v in such a way that each o f the v elements occurs r times altogether and 0, 1 or 2 times in each block, and each o f the distinct pairs o f elements occurs A times. Any ternary balanced block design is regular, that is, each element occurs singly in p, blocks and is repeated in
p2 designs for p - v objects. C e r a n k a and G r a c z y k (2002) have blocks, where pi and p? are constant for the design. Accordingly we write the parameters o f ternary balanced block design in the form v, b , г, к, A, p |( p 2 .
Let N be the incidence matrix o f ternary balanced block design. It is straightforward to check that:
vr = b k ,
r = p , + 2 p 2 ,
A ( v - l ) = p , ( * - l ) + 2 / > 2( * - 2 ) = r ( * - l ) - 2 p 2, NN = (/>,+ 4 p 2 - X ) l v + X I X = (r + 2p 2 - X ) l v + X l X .
II. MATRIX X BASED ON ONE INCIDENCE MATRIX OF TERNARY BALANCED BLOCK DESIGN
D efinition 2.1. In a chemical balance weighing design the estimators of weights are uncorrelated if matrix X X is diagonal.
Let N denote the incidence matrix o f order v x b o f ternary balanced block design. From this matrix we define matrix X o f chemical balance weighing design in the form:
X = N - M , M v ■N
( 1)
where l a is the a x l vector o f ones. In this design in n = 2b weighings we check weights o f p = v objects.
We can see, that matrix X is the nonsingular matrix if and only if v Ф к . L em m a 2.1. In the chemical balance weighing design with matrix X given by (1) the estimated weights are uncorrelated if and only if
Ь + Л - 2 г = 0.
N ow we consider the design matrix X in the following form:
X = N “ M v l b
M l - N ' 1, (3)
In this design in n = 2b weighing operations we check weights o f p = v +1 objects.
We can see, that matrix X is the nonsingular matrix if and only if v Ф к . Lem m a 2.2. In the chemical balance weighing design with matrix X given by (3) the estimated weights are uncorrelated if and only if (2) holds.
Proof o f this Lemma was given by Ceranka and Graczyk (2000). From Lemma 2.1 and Lemma 2.2 we have the following Theorem.
T heorem 2.1. In a chemical balance weighing design with matrix X given by (1) the estimated weights are uncorrelated if and only if in a chemical balance weighing design with matrix X given by (3) the estimated weights are uncorrelated.
T heorem 2.2. If in a chemical balance weighing design with matrix X given by (3) the estimated weights are uncorrelated, then any p < v +1 columns o f this matrix constitute a chemical balance weighing design for p objects in 2b weighings, in that the estimated weights are uncorrelated.
Proof. In a chemical balance weighing design with matrix X given by (3) the estimated weights are uncorrelated if and only if matrix X X is diagonal matrix. This means that in a chemical balance weighing design with matrix X the estimated weights are uncorrelated if and only matrix X is a (2b ) x (v + 1) matrix o f such elements -1 ,1 and 0, columns o f this matrix are orthogonal, which yields the assertion o f the theorem.
III. MATRIX X BASED ON TWO INCIDENCE MATRICES OF TERNARY BALANCED BLOCK DESIGNS
Let N. denote the incidence matrices o f order v x b t o f ternary balanced block designs with the parameters: v, bit kt , A,-, p u , p 2i, for / = 1,2. From
these matrices we define matrix X o f chemical balance weighing design in the form:
(4)
In this design in п = Ь{ +Ь2 weighing operations we check weights o f p = v objects. We can see, that matrix X is the nonsingular matrix if and only if v Ф k { or v Ф k 2 .
L em m a 3.1. In the chemical balance weighing design with matrix X given by (4) the estimated weights are uncorrelated if and only if
Proof o f this Lemma was given by C e r a n k a and G r a c z y k (2000). N ow we consider the design matrix X in the following form:
In this design in n = b, + b2 weighing operations we check weights of
p = v + 1 objects.
We can see, that matrix X is the nonsingular matrix if and only if v Ф k2 .
L em m a 3.2. In the chemical balance weighing design with matrix X given by (6) the estimated weights are uncorrelated if and only if (5) holds and
Proof o f this Lemma was given by C e r a n k a and G r a c z y k (2002). From Lemma 3.1 and Lemma 3.2 we have the following Theorem:
T heorem 3.1. If in a chemical balance weighing design with matrix X given by (4) the estimated weights are uncorrelated and condition (7) is satisfied, then in a chemical balance weighing design with matrix X given by (6) the estimated weights are uncorrelated.
b\ + b2 + A, + Я2 - 2 + r2) = 0. (5)
T heorem 3.2. If in a chemical balance weighing design with matrix X given by (6) the estimated weights are uncorrelated, then any p < v + 1 columns o f this matrix constitute a chemical balance weighing design for p objects in + b 2
weighings, in that the estimated weights are uncorrelated.
Proof. In a chemical balance weighing design with matrix X given by (6) the estimated weights are uncorrelated if and only if matrix X ’X is the diagonal matrix o f order v + 1 . When conditions (5) and (7) hold we obtain theorem.
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B r o n is ła w C era n ka , M a łg o r z a ta G ra c zy k
RELACJE POMIĘDZY CHEMICZNYMI UKŁADAMI WAGOWYMI DLA p = v ORAZ p = v + 1 OBIEKTÓW
W p ra c y z a jm u je m y s ię c h e m ic z n y m i u k ła d a m i w a g o w y m i o m a c ie rz y u k ła d u sk o n s tru o w a n e j z m a c ie rz y in c y n d e n c ji tró jk o w y c h u k ła d ó w z ró w n o w a ż o n y c h o b lo k a c h n ie k o m p le tn y c h . R o z w a ż a m y z a le ż n o śc i p o m ię d z y p a ra m e tra m i ty ch u k ła d ó w d la p = v i /; = v +1 o b ie k tó w . P rz e d s ta w ia m y w a ru n k i, p rz y k tó ry c h istn ie n ie c h e m ic z n e g o u k ła d u w a g o w e g o d la p = v o b ie k tó w o n ie s k o re lo w a n y c h e s ty m a to ra c h w a g o b ie k tó w im p lik u je is tn ie n ie c h e m ic z n e g o u k ła d u w a g o w e g o d la p = v + 1
o b ie k tó w , w k tó ry m e sty m a to ry w a g o b ie k tó w s ą n ie s k o re lo w a n e . Z k o le i istn ie n ie c h e m ic z n e g o u k ła d u w a g o w e g o d la p = v +1 o b ie k tó w o n ie s k o re lo w a n y c h e sty m a to ra c h w a g o b ie k tó w im p lik u je istn ie n ie c h e m ic z n e g o u k ła d u w a g o w e g o d la d o w o ln e g o p < v +1 o n ie s k o re lo w a n y c h e s ty m a to ra c h w a g o b ie k tó w .
