Metodološki zvezki, Vol. 3, No. 1, 2006, 39-48
GD PBIBD(2)s in Incomplete Split-Plot ´ Split-Block Type Experiments
Katarzyna AmbroSy and Iwona Mejza1
Abstract
In this paper we present a method of designing a three-factor experiment with crossed and nested treatment structures. The design considered is called a split-plot ´ split-block design. A kind of design incomplete with respect to all three factors is examined. Additionally, we consider the usefulness of group divisible partially balanced incomplete block designs with two associate classes in planning such experiments. In modeling data obtained from them, we take into account the structure of experimental material and a four-step randomization scheme for the different kind of units. As regards the analysis of the obtained randomization model with seven strata, we adapt an approach typical of multistratum experiments with orthogonal block structure.
1    Introduction
There are many experimental designs that can be considered for use in a three-or-more-factor experiment. For instance, all experimental designs for one- or two-factor experiments can be applied for this purpose. There are also three-or-more-factor designs specifically developed for the type of such experiments used in agricultural research. These are primarily extensions of either a split-plot or a split-block design (Gomez and Gomez, 1984; LeClerg et al., 1962).
The purpose of this paper is to present a method of designing a three-factor experiment with crossed and nested treatment structures. The design considered is called a split-plot ´ split-block (SPSB) design (LeClerg et al., 1962). A kind of such design incomplete with respect to the levels of three factors is examined (AmbroSy and Mejza, 2003). Additionally, we present the usefulness of group divisible partially balanced incomplete block designs with two associate classes (shortly GD PBIBD(2)s, see Clatworthy, 1973) in planning such experiments.
1 Department of Mathematical and Statistical Methods, Agricultural University of Poznañ, Poland; ambrozy@au.poznan.pl; imejza@au.poznan.pl
40                                                                   Katarzyna AmbroSy and Iwona Mejza
The SPSB type design is convenient in field experiments, especially when certain treatments such as types of cultivation, application of irrigation water, varieties etc., may have to be arranged in (crossed and nested) strips across each block.
2    Assumptions and notation
Let us consider a three-factor experiment of SPSB type in which the first factor, say A, has s levels A1, A2, …, As, the second factor, say B, has t levels B 1,
B2,, …, Bt and the third factor, say C, has w levels C 1, C2, …, Cw. Thus the number v (= stw) denotes the number of all treatment combinations in the experiment. The experimental material is assumed to be divided into b blocks each with a row-column structure with kA rows (kA < s) and kB columns of the first order, called I-columns for short (kB <t). So there are kAkB intersection plots of the first order within each block, henceforth called whole plots. Then each I-column has to be split into kC columns of the second order, called II-columns for
short  (kC < w). Then there are  kAkBkC   intersection plots of the second order
within each block, henceforth called small plots. So,  n = bkAkBkC  denotes the
number of subplots, which are required in the experiment. Here the rows correspond to the levels of the factor A, also termed row treatments, the I-columns correspond to the levels of factor B, also I-column treatments, and the II-columns are to accommodate the levels of factor C, termed II-column treatments. The arrangement of the factors in the mixed design is very important from the statistical point of view.
EU s ( II-columns) for the II-column EU s (I-columns) for the I-column treatments           treatments
—       —       —      —     —     —     —
®_____________________________________________________________________j      j      j
®   I                                              I                                              I                                              I     Ii      i      i
Figure 1: Experimental material of a single block in a complete split-plot ´ split-block type experiment with the factors A ( A1 , A2 ), B ( B1 , B2 , B3 ), C ( C1 , C2 , C3 , C4 )
Figure 1 illustrates a sample layout in one block (replication) of a complete SPSB design with two row treatments ( A1, A2 ), three I-column treatments ( B1, B2 , B3 ) and four II-column treatments (C1 , C2 , C3 , C4 ). We consider a situation where
EUs (rows) for the row treatments
GD PBIBD(2)s in Incomplete…
41
the SPSB design is incomplete with respect to three factors and their levels are arranged according to different or identical incidence matrices of some GD PBIBD(2)s.
3     Model of observations
In modeling data obtained from such experiments we take into account the structure of experimental material and a four-step randomization scheme of the different kind of units (Ambrozy and Mejza, 2004a). With respect to the analysis of the obtained randomization model with seven strata, we adapt an approach typical of multistratum experiments with orthogonal block structure (Nelder, 1965a, Nelder, 1965b). In this case, we have a zero stratum (0) generated by the vector of ones, an inter-block stratum (1), an inter-row (within the blocks) stratum (2), an inter-I-column (within the blocks) stratum (3), an inter-II-column stratum (4) (within the I-columns), an inter-whole plot (within the blocks) stratum (5), and an inter-subplot (within the whole plots) stratum (6). The statistical analysis of such a model is connected with the algebraic properties of the stratum information matrices for the treatment combinations. The obtained incomplete SPSB design can thus be characterized with respect to a general balance property and stratum efficiency factors of the design for a set of orthogonal contrasts between treatment combination effects (Houtman and Speed, 1983). These contrasts are connected with comparisons among the main effects of the considered factors and interaction effects. When the SPSB design is complete, then all information about these contrasts is contained in a suitable single stratum. In cases where the design is incomplete, the information is split between two or more strata.
Applying the GD PBIBD(2)s to the construction method for SPSB type designs, we obtain different degrees of balance in the strata with respect to the different contrasts. It can be shown that the potential loss in estimating the treatment contrasts can be limited by the proper choice of experimental design.
4     Incomplete  SPSB  type  designs  generated  by  GD PBIBD(2)s
Let   NA (s x b),   NB  (t x b)   and   NC (w x b)   be  incidence  matrices   of GD
PBIBD(2)s for the row treatments, the I-column treatments and the II-column treatments with respect to the blocks, respectively. In the present paper the construction method for three-factor experiments is based on the Kronecker product of matrices, denoted by ®. Then we have
N1 = NA ® NB ® NC ,                                           (4.1)
42                                                                   Katarzyna Ambroiy and Iwona Mejza
where N1 is the treatment combinations vs. blocks incidence matrix of the SPSB design.
Let C*(* denotes the letters A, B, C in turn) be information matrices of the
subdesigns with the following positive eigenvalues: e( j *) =1-w ( j *) /( r*k*) and their
multiplicities r ( j *) , where r* is the number of replications of the row treatments,
the I-column treatments and the II-column  treatments, respectively, k* is the size
of blocks, w (j    are the eigenvalues of the associate matrices N*N*,   j  = 0, 1, 2,
2                                  2                                          2
and  = r jA) =s-1,    = r ( jB) =t-1 and   =!r jC) =w-1 (Clatworthy, 1973).
To describe properties such as efficiency and balance of the incomplete SPSB design, we will introduce the following abbreviations: let Mf {q,a}  denote the
property that q contrasts among the treatments of factor M (or interaction contrasts) are estimated with efficiency a in the f-th stratum. In other words, we say that the design is Mf{q,a}- balanced. In particular, for a = 1, the design is
M f{q, 1}- orthogonal.
First we turn our attention to three-factor interaction contrasts. From the algebraic properties of the information matrices of the incomplete SPSB design and the subdesigns (Ambrozy and Mejza, 2004b) we have
Theorem 1. Assume that the row treatments, the I-column treatments and the II-column treatments can be grouped into three different associate schemes. Then the
SPSB(GD( ) ,GD(B),GD( )) design with the incidence matrix N1=NAÄNBÄNC   and  the   parameters:   b = bAbBbC,   k = kAkBkC,   v = stw,
r = rArBrC with respect to three-factor interaction contrasts is:
·     (A´B´C)1{ r 1( ) r 1(B) r 1( ) , (1–e1( ))(1-e1( B ))(1–e1( ) )}- balanced, (A´B´C)2{ r 1(A) r 1(B) r 1( ) , e1( ) (1–e1(B) )(1–e1( ) )}- balanced, (A´B´C)3{ r 1( ) r 1(B) r 1( ) , (1–e1( ))e1( B) (1-e1( ) )}- balanced, (A´B´C)4{ r 1(   r 1(   r 1(   , (1–e1(   )e 1    }–, (A´B´C)5{ r 1( ) r 1(B) r 1( ) , e1( ) e1(B) (1–e1( ) )}-, and
(A´B´C)6{ r1(    r1(    r1( C ,e 1 ) e 1 )} - balanced,
·    (A´B´C)1{ r 1( ) r 1(B) r2C ) , (1– e 1 ))(1–e1(B))(1–e2C))}– balanced (A´B´C)2{ r 1( ) r 1(B) r2 ) , e 1 ) (1– e 1( B))(1–e2 ))}– balanced, (A´B´C)3{ r 1( ) r 1(B) r2C ) , (1– e 1 ))e 1B) (1–e2C))}– balanced, (A´B´C)4{ r 1(   r 1(    r2   , (1– e 1   )e2   }–,
GD PBIBD(2)s in Incomplete…
43
(A
(A
(A
(AxBxC)5{ r 1 (AxBxC)6{ r 1(
• (AxBxC)1{ r 1( (AxBxC)2{ r 1( (AxBxC)3{ r 1 (AxBxC)4{ r 1( (AxBxC)5{ r 1( (AxBxC)6{ r 1(
(A
(A
(A
(A
·    (A´B´C)
(A´B´C)2{r1 (A´B´C)3{r1( (A´B´C)4{r1( (A´B´C)5{r1 (A´B´C)6{r1(
(A
(A
(A
(A
(A
•    (AxBxC)1{ r2   r 1(
(A´B´C)2{r2 (A´B´C)3{r2 (A´B´C)4{r2
(A
(A
(A
5{r2(
(A
(A
(AxBxC) (AxBxC)6{ r2 •    (AxBxC)
(A´B´C)2{r2
(A
(A´B´C)3{r2( (A´B´C)4{r2( (A´B´C)5{r2 (A´B´C)6{r2(
(A
(A
(A
(A
•    (AxBxC)1{ r2    r2
(A´B´C)2{r2 (A´B´C)3{r2
(A
(A
(A´B´C)4{
4{r2
(A
r1(B) r2(C) , r1(B) r2(C) ,
e1(A) e1(B) (A)     (C)
(1–e2(C) )}–, and
ee2
e2(C) }– balanced,
(A)
r2(
(B)
r1(C) , (1–e1(A) )(1–e2(B) )(1–e1(C) )}– balanced,
r2(B) r1(C) ,
r2(B) r1(C) ,
r2(B) r1(C) ,
r2(B) r1(C) ,
r2(B) r1(C) ,
(B)
e1(A) (1–e2(B) )(1–e1(C) )}– balanced,
(B)
e1(A) )e2(B) (1–e1(C) )}– balanced,
e1(A) )e1(C) }–,
(1
(1
e1(A) e2(B) (1–e1(C) )}–,
e1(A) e1(C) } – balanced,
1{r1(
(A)
r2
(B) r2(C) ,
(B)
(1–e1(A) )(1–e2(B) )(1–e2(C) )}– balanced,
r2(B) r2(C) ,
r2(B) r2(C) ,
r2(B) r2(C) ,
r2(B) r2(C) ,
r2(B) r2(C) , (A)
(B)
e1(A) (1–e2(B) )(1–e2(C) )}– balanced,
(1 (1
(B)
e1(A) )e2(B) (1–e2(C) )}– balanced,
e1(A) )e2(C) }–,
(A)     (B)
e2(B) (1–e2(C) )}–, and e1(A) e2(C) } – balanced,
(B) r1(C) , (1
(B)
e2(A) )(1–e1(B) )(1–e1(C) )}– balanced,
r1(B) r1(C) ,
r1(B) r1(C) ,
r1(B) r1(C) ,
r1(B) r1(C) ,
r1(B) r1(C) ,
e2(A) (1–e1(B) )(1–e1(C) )}– balanced,
(1 (1
(B)
e2(A) )e1(B) (1–e1(C) )}– balanced,
e2(A) )e1(C) }– ,
e2(
(A)     (B)
2 (A)     (C)
(1–e
(C)
)}–, and
} – balanced,
1{r2(
(A)
r1
(B) r2(C) ,
(B)
(1–e2(A) )(1–e1(B) )(1–e2(C) )}– balanced,
r1(B) r2(C) ,
r1(B) r2(C) ,
r1(B) r2(C) ,
r1(B) r2(C) ,
r1(B) r2(C) , (A)
e2(A) (1–e1(B) )(1–e
(B)
(C)
(1 (1
e2(
)}– balanced e2(A) )e1(B) (1–e2(C) )}– balanced e2(A) )e2(C) }–,
(A)     (B)
2
(1–e2(C) )}–, and
e2(A) e2(C) }– balanced,
(B) r1(C) , (1
(B)
e2(A) )(1–e2(B) )(1–e1(C) )}– balanced,
r2(B) r1(C) , r2(B) r1(C) ,
r2(B) r1(C) , (1
e2(A) (1–e2(B) )(1–e1(C) )}– balanced
(B)
(1–e2(A) )e2(B) (1–e1(C) )}– balanced
e2(A) )e1(C) }–,
44
Katarzyna AmbroSy and Iwona Mejza
(A´B´C)5{
(A)      (B)      (C)        (A)
5{ r 2     r 2     r 1
e 2(A) e 2(B) (
1– e
1 ))}-,
(A´B´C)6{ r 2( ) r 2( B ) r 1( C ) , e 2( ) e 1( C )} – balanced.
·    (A´B´C)1{ r2 ) r 2( B ) r 2 ) , (1-e2 ))(1–e2( B))(1–e2 ))}- balanced,
(A ´ B´C)2{
(A)      (B)      (C)        (A)
2{ r2    r2    r2
e 2(A) (
1–e2( B))(1–e2C))}– balanced, (A´B´C)3{ r2 ) r 2( B ) r 2( ) , (1– e 2( ))e 2( B )(1– e 2( C ))}– balanced,
(A ´ B´C)4{r2     r2     r2     , (1– e 2(      )e2     }–,
5{r2   ) r 2( B ) r2   ) , e 2(   ) e 2( B )(1– e 2( C ))}–,    and
2 (C)
(A ´ B´C)5
(A ´ B´C)6{ r2 ) r2( B) r2C), e2 ) e 2( C )} – balanced.
Table 1: Stratum efficiency factors of the incomplete SPSB design with respect to some types of orthogonal contrasts (T]i (*) = 1 - e i (*), where i = 1, 2 and (*) denotes the letters (A),
(B), (C) in turn.).
Type of contrast	Number of contrasts	Stratum (1)	Stratum (2)	Stratum (3)	Stratum (4)	Stratum (5)	Stratum (6)
A	r 1(A) r(A)	h 1(A) h(A)	e 1(A) e(A)				
B	r(B)	h 1(B)		e 1(B)			
	r(B)	h (B) 2		e 2(B)			
C	r 1( C )	h 1( C )			(C) e 1		
	r 2( C )	h 2( C )			e(C)		
	r(A)   r(B)	h 1(A) h 1(B)	e 1(A) h 1(B)	h 1(A) e 1(B)		(A)     (B) e 1     e 1	
A ´ B	r(A)   r(B)	h 1( A ) h 2( B )	e 1(A) h(B)  2	h 1(A) e(B)		e(A)  e(B)	
	r(A)   r(B)	h(A) h 1(B)	e(A) h 1(B)	h(A) e 1(B)		e(A)  e(B)	
	r(A)   r(B)	h 2( A ) h 2( B )	e(A) h(B)	h(A) e(B)		e(A)  e(B)	
	(A)     (C) r 1     r 1	h 1( A ) h 1( C )	e 1(A) h 1(C)		h 1(A) e 1( C)		(A)     (C) e 1     e 1
A´C	r(A)   r(C)	h(A) h(C) 1      2	e 1(A) h(C)  2		h 1(A) e(C)		e(A)  e(C)
	r(A)   r(C)	h 2( A ) h 1( C )	e(A) h 1(C)		h(A)   C^) 2   e 1		(A)     (C) e2     e 1
	r(A)   r(C)	h(A) h(C)	e(A) h(C)		h(A) e(C)		e(A)  e(C)
	r(B)   r(C)	h(B) h(C) 1      1		e 1(B) h 1(C)	e 1(C)		
B´C	r 1( B ) r 2( C )	h 1( B ) h 2( C )		e 1(B) h(C)  2	e2		
	r(B)   r(C)	h(B) h 1(C)		e 2(B) h 1(C)	(C) e 1		
	r 2( B ) r 2( C )	h 2( B ) h 2( C )		e 2(B) h(C)	e2		
GD PBIBD(2)s in Incomplete…
45
Conclusion 1. In the incomplete SPSB design generated by three GD designs, information about the basic interaction contrasts of type AxBxC is contained in all strata if the GD designs are of R types. When the generating designs are of type SR or S, then some of the contrasts are estimable in one stratum only.
Conclusion 2. From the above theorem we can easily obtain information about the remaining basic contrasts. For instance, if we want information about s -1 (=
(A)
r 1 that
with
+
1    =
) contrasts among the main effects of the factor A we have to assume
,(B)=    (B) ;2         fc0
(= 0) with
r 1
(B)
r 2( B )
r
(B)
(= 1) and
(C)        (C)
=
(C)
(= 0)
r 1( ) = r 2 ) = r 0 ) ( = 1). If we are interested in the interaction contrasts of type
BxC, then in the theorem we have to assume that
(A)
(A)
(A)
(= 0) with
r 1
(A)
r(2A)
r
(A)
(= 1), only. We present the conclusions about these contrasts we
present in Table 1.
5
Example
In this example we illustrate how orthogonal contrasts can be estimated in a (4x6x4) factor experiment. Assume that four row treatments can be grouped into an associate scheme with two groups (m( A )   = 2) and two treatments within
each group (p( A ) = 2), so s = m( A )p( A) = 4. Similarly, six I-column treatments and four II-column treatments can be grouped in two different associates schemes, therefore t = m( B )p( B ) = 6, w = m( C )p( C ) = 4. Then the experiment is set up in an incomplete SPSB(GD( ) , GD(B),  GD( )) design, in which the  GD( ) design
and GD( ) design are of type SR and GD(B) is of type S (plans: SR1 and S1, respectively, cf. Clatworthy, 1973). Thus the parameters of the generating designs are as follows:
GD(   :   ba = 4,   ka = 2,   s = m ( ) p (   = 4,   ra = 2,    -^        0,
(A)   (A)
(A)
A(
(A)
1,
Ž
(B)
GD(   :   bB = 3, kB = 4,  t = m ( )p(   = 6, rB = 2, /^     = 2,
GD(   :   bC = 4,   kC = 2,   w = m ( )p(    = 4,   rC = 2,    /^ ) = 0,     /l^    = 1.
Then the incidence matrix with respect to the blocks of the SPSB design is
(5.1)
N1 =
		r1    0    1^		
^10    10^		110		r1  0   1   0N
10    0    1	®	0    11	®	10    0    1
0    10    1		10    1		0    10    1
v0    1    1    0j		110 v0    1    1)		v0    1    1    0y
(5.2)
.
46                                                                   Katarzyna Ambroiy and Iwona Mejza
The parameters of the SPSB(SR1, S1, SR1) design are equal to:    b = 48, k = 16, v = 96, r = 8. The eigenvalues of the matrices Ca, Cb and CC are: e 1(     = e1( C =
1/2   with   multiplicities   r 1(     = r 1( C       2,   e 2(        e2            with   multiplicities
r2        r2         ,    1           with multiplicity r 1(    = 3, and e2( B = 3/4 with r (2    = 2.
By Theorem 1 above, the incomplete SPSB(SR1, S1, SR1) design is:
A1{2, 1/2}- balanced and    A2{2, 1/2}- balanced,    A 2{1, 1}- orthogonal,
B1{2, 1/4}- balanced   and   B3{2, 3/4}- balanced,    B3{3, 1}- orthogonal,
C1{2, 1/2}- balanced and   C4{2, 1/2}- balanced,    C4{1, 1}- orthogonal,
(A ´ B)1{4, 1/8}-,    (A ´ B)2{4, 1/8}-,      (A ´ B)3{4, 3/8}-,      and      (A´B)5{4, 3/8}-
balanced,
(A ´ B)3{6, 1/2}-,    and     (A ´ B)5{6, 1/2}- balanced,
(A´B)2{2, 1/4}- ,   and     (A ´ B)5{2, 3/4}- balanced,
(A ´ B)5{3, 1}- orthogonal;
(A´C)1{4, 1/4}-,   (A´C)2{4, 1/4}-,      (A´C)4{4, 1/4}-,      and    (A´C)6{4,  1/4}-
balanced,
(A´C)2{2, 1/2}-,    and     (A´C)6{2, 1/2}- balanced,
(A´C)4{2, 1/2}- ,   and     (A´C)6{2, 1/2}- balanced,
(A´C)6{1, 1}- orthogonal;
(B´C)1{4, 1/8}-,    (B´C)3{4, 3/8}-,    and    (B´C)4{4, 1/2}- balanced, (B´C)3{6, 1/2}-,    and     (B´C)4{6, 1/2}- balanced, (B´C)4{5, 1}- orthogonal;
(A ´ B´C)3{12, 1/4}-, (A ´ B´C)4{12, 1/4}-,   (A ´ B´C)5{12, 1/4}-, and (A ´ B´C)6{12, 1/4}- balanced;
(A ´ B´C)4{10, 1/2}-,   and (A´B´C)6{ 10, 1/2}- balanced;
(A ´ B´C)1{8, 1/16}-,                (A ´ B´C)2{8, 1/16}-,                (A ´ B´C)3{8, 3/16}-,
(A ´ B´C)4{8, 1/4}-,   (A ´ B´C)5{8, 3/16}-, and (A´B´C)6{8, 1/4}- balanced;
(A ´ B´C)5{6, 1/2}-,   and (A´B´C)6{6, 1/2}- balanced;
(A´B´C)6{5, 1}- orthogonal;
GD PBIBD(2)s in Incomplete…
47
(A´B´C)2{4, 1/8}–,   (A´B´C)5{4, 3/8}–,   and (A´B´C)6{4, 1/2}– balanced.
Summing up, it can be noticed that in the generated SPSB design, some of the basic contrasts are estimated with full efficiency equal to 1. This results from the generating designs of types SR and S. These contrasts are estimable in one stratum only, as in a complete SPSB design. Information about other basic contrasts can be split into two or more strata. Statistical inferences (estimates and tests) about them can be obtained using the information from one stratum separately or by performing for them the combined estimation and testing based on information from all the strata in which they are estimable (Caliñski and Kageyama, 2000).
Additional remarks. No standard software package known to the authors can be used to implement the designs under consideration. At the moment this paper is only theoretical. The authors would be very grateful for suggestions concerning the possible use of software for planning the generated designs and statistical inferences.
References
[1] AmbroSy, K. and Mejza, I. (2003): Some split-plot ´ split-block designs. Colloquium Biometryczne, 33, 83-96.
[2] AmbroSy, K. and Mejza, I. (2004a): Split-plot ´ split-block type three factors designs. Proc. of the 19th International Workshop on Statistical Modelling, 291-295.
[3] AmbroSy, K. and Mejza, I. (2004b): Incomplete split-plot´split-block designs based on Kronecker type products. Colloquium Biometryczne, 34, 27-38.
[4] Caliñski, T. and Kageyama, S. (2000): Block designs: A randomization approach, Vol. I: Analysis. Lecture Notes in Statistics, 150, Springer-Verlag, New York.
[5] Clatworthy, W.H. (1973): Tables of Two Associate Class Partially Balanced Designs. NBS Applied Math. Ser. 63. Washington, D.C, USA.
[6] Gomez, K.A. and Gomez, A.A. (1984): Statistical Procedures for Agricultural Research. Wiley, New York.
[7] Gupta, S.C. (1985): On Kronecker block designs for factorial experiments. J. Statist. Plann. Infer., 52, 359-374.
[8] Houtman, A.M. and Speed, T.P. (1983): Balance in designed experiments with orthogonal block structure. Ann. Statist., 11, 1069-1085.
[9] LeClerg, E.L., Leonard, W.H., and Clark, A.G. (1962): Field Plot Technique. Minneapolis: Burgess.
48                                                                   Katarzyna AmbroSy and Iwona Mejza
[10] Nelder, J.A. (1965a): The analysis of randomized experiments with orthogonal block structure. 1. Block structure and the null analysis of variance. Proc. of the Royal Soc. of Lond. Ser. A, 283, 147-162.
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