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Resolvable Incomplete Split-Plot × Split-Block DesignsI. Mejza a & K. Ambroży a
a Department of Mathematical and Statistical Methods , AgriculturalUniversity, Wojska Polskiego , Poznań, PolandPublished online: 30 Nov 2011.
To cite this article: I. Mejza & K. Ambroży (2007) Resolvable Incomplete Split-Plot × Split-Block Designs,Journal of Statistical Theory and Practice, 1:3-4, 405-416, DOI: 10.1080/15598608.2007.10411849
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Journal of © Grace Scientific Publishing Statistical Theory and Practice Volume 1, Nos. 3 & 4, Dec. 2007
* 1559-8608/07-3&4/$5 + $1pp – see inside front cover
Resolvable Incomplete Split-Plot × Split-Block Designs
I. Mejza, Department of Mathematical and Statistical Methods, Agricultural University, Wojska Polskiego, Poznań, Poland
E-mail: [email protected]
K. Ambroży, Department of Mathematical and Statistical Methods, Agricultural University, Wojska Polskiego, Poznań, Poland
E-mail: [email protected]
Received: Feb. 26, 2007 Revised: Aug. 20, 2007, Sept. 21, 2007. __________________________________________________________________________________________
Abstract
The aim of the paper is to present a randomization model, statistical properties and their consequences for an analysis of some three factor experiments set up in resolvable split-plot × split-block designs. To control several sources of local variation, nested blocking structure is applied. The designs considered are incomplete with respect to all the factors which levels are arranged in resolvable block designs.
AMS Subject Classification: 62K10 and 62K15.
Key-words: General balance, Multistratum experiment, Resolvable block designs, Split-plot × split-block designs, Stratum efficiency factors. __________________________________________________________________________________________
1. Introduction
In practice, particularly in agricultural field experiments, often situations appear in which experimental blocks are grouped into some sets called “superblocks”. Such nested blocking structure allows controlling several sources of local variation. Any additionally added proper system of blocks improves precision of a statistical inference from an experiment (cf. Pearce, 1983). In the case of many systems of blocks, they can be crossed or/and nested. In the paper, one of the most popular blocking systems, from a practical point of view, is discussed. In particular, we consider such a situation with reference to a three-factor experiment set up in a split-plot × split-block (say, SPSB) design (cf. LeClerg et al., 1962). This type of the design is quite often used in field experiments because of a useful arrangement of experimental units. Certain treatments, such as types of cultivation, application of irrigation water, varieties etc., may have to be arranged in (crossed and
© 2007 Grace Scientific Publishing, LLC
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406 I. Mejza & K. Ambroży nested) strips across each block. This arrangement of units in an experiment will be discussed in the present paper. The columns (or the rows) of a split-block design will be split into smaller strips to accommodate a third factor. In this way, the third factor will be in a split-plot relation to the column (or row) treatments. Therefore, the mixed design of the split-plot and split-block designs, which are crossed, will be called a split-plot × split-block (SPSB) design. The complete SPSB design (i.e. when all types of the treatments are assigned to the appropriate for them units within each block) is quite often used in field experiments. Ambroży and Mejza (2003, 2004, 2006) have described statistical properties of incomplete SPSB designs examined under mixed linear model of observations resulting from properly performed randomization. It was assumed that the designs considered possess an orthogonal block structure (cf. Nelder, 1965a), and hence, the overall analysis can be split into so called stratum analyses.
In the ordinary SPSB design, a four-step randomization process according to a stratification of the experimental units leads to seven strata: zero stratum (0), the inter-block stratum (1), the inter-row (within the block) stratum (2), the inter-column I (within the block) stratum (3), the inter-column II (within the column I) stratum (4), the inter-whole plot (within the block) stratum (5), and the inter-subplot (within the whole plot) stratum (6),
The number of the strata will increase when we take into account the nested structure of the blocks. We have, then, additionally one stratum more, the inter-superblock stratum, and other strata, apart from (0), follow it. The resulting design, called the resolvable split-plot × split-block (say, RSPSB) design, is incomplete with respect to the blocks but complete with respect to the superblocks. We assume that the incompleteness is related to all the factors which levels are arranged in the same or different resolvable block subdesigns. It means that in the RSPSB design, the blocks within one superblock have different treatment combinations, and each treatment combination occurs only once inside each superblock.
The aim of the paper is to present a randomization model, statistical properties, and their consequences for an analysis of some three factor experiments set up in RSPSB designs. A concept of resolvability in nested block designs, described by Caliński and Kageyama (2000, 2003), has been adopted for the designs considered.
2. Material structure
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 B1, B2, …, Bt, and the third factor, say C, has w levels C1, C2, …, Cw. Thus, the number v (= stw) denotes the number of all treatment combinations in the experiment. To control several sources of local variation, an experimental material is assumed to be divided into R superblocks consisting of b blocks, each block has a row–column structure with k1 (≤ s) rows (horizontal strips) and k2 (≤ t) columns (vertical strips I) of the first order, shortly, columns I. So, there are k1k2 intersection plots of the first order, within each block, below called whole plots. Then, each column I has to be split into k3 (≤ w) columns (vertical strips II) of the second order, shortly, columns II. Therefore, there are k1k2k3 intersection plots of the second order, within each block, below called subplots. In this case, the rows correspond to the levels of the factor A, termed also as row treatments, the columns I correspond to the levels of the
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Resolvable Incomplete Split-Plot x Split-Block Designs 407 factor B, called also column I treatments, and the columns II are to accommodate the levels of the factor C termed as column II treatments.
Figure 1 illustrates a sample layout in one block inside each superblock (replication) of a complete SPSB design with four row treatments ( 1A , 2A , 3A , 4A ), four column I treatments ( , , , ), and four column II treatments ( , ,
, ). In the paper, we consider a situation where the RSPSB design is incomplete with respect to three factors and their levels are arranged according to different or identical incidence matrices of resolvable incomplete block designs (see example in section 5).
1B 2B 3B 4B 1C 2C
3C 4C
The arrangement of levels of the factors in the SPSB design is very important from the statistical point of view. It affects a precision of estimation of treatment combination effects. Let us note that by randomization we map out randomly the starting (master, theoretical) design onto experimental units (cf. e.g. Kachlicka and Mejza, 1990, Mejza and Mejza, 1994, Caliński and Kageyama, 2000). Hence, the desirable statistical properties should be included in proper choice of the theoretical design.
Figure 1. An experimental material of a single block inside each superblock in a complete split-plot × split-block type experiment
EUs (I-columns) for the column I treatments
EUs (II-columns) for the column II
treatments
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ → → →
EUs (rows) for
the row treatments
→
3. Linear model and its analysis
A randomization model of observations considered here has a form and properties strictly connected with the performed randomization process in the experiment. It is based on an extension of that developed by Ambroży and Mejza (2003, 2004, 2006) for SPSB designs. The problem of randomization in similar structure of an experimental material for one or two factors was earlier considered by e.g. Mejza and Mejza (1984, 1994), Kachlicka and Mejza (1990, 1996).
In this paper, we generalize the previous considerations to three factorial designs with the structure of experimental units as described in section 2. Used here a randomization scheme consists of five randomization steps performed independently, i.e. by randomly permuting the superblocks within the total area, by randomly permuting blocks within the superblocks, then the rows within the blocks, the columns I within the blocks, and the columns II within the columns I in blocks. Further, assuming the usual unit-treatment
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408 I. Mejza & K. Ambroży additivity and uncorrelation of the technical errors, with zero expectation and a constant variance 2
eσ , the model of observations can be written as 7
1f f
f =
′= +∑Δ τ D η e'y + (3.1)
where y is an n × 1 vector of observations, 1 2 3n Rbk k k= , (n × v) is a known design matrix for v treatment combinations, (n × R),
'Δ
1′D 2′D (n × Rb), 3′D (n × Rbk1), (n × Rbk2), (n × Rbk2k3), (n × Rbk1k2),
4′D
5′D 6′D 7′D (n × n) are design matrices for superblocks, blocks (within the superblocks), rows (within the blocks), columns I (within the blocks), columns II (within the columns I), whole plots (within the blocks) and subplots (within the whole plots) respectively, (v × 1) is the vector of fixed treatment combination effects,
(R × 1), (Rb × 1), (Rbk1 × 1), (Rbk2 × 1), (Rbk2k3 × 1), (Rbk1k2 × 1), (Rbk1k2k3 × 1), and e (n × 1) are random effect vectors of the superblocks, the blocks,
the rows, the columns I, the columns II, the whole plots, the subplots and technical errors, respectively.
τ1η 2η 3η 4η 5η 6η
7η
The design matrices ′Δ and f′D (f = 1, 2,…, 7) can be expressed by, respectively:
1 2[ : : : ]R′ ′=Δ Δ Δ Δ , 1 2 31 R bk k k′ = ⊗D I 1 ,
1 2 32 Rb k k k′ = ⊗D I 1 ,
1 2 33 ,Rbk k k′ = ⊗D I 1 1 2 34 ,Rb k k k′ = ⊗ ⊗ ⊗D I 1 I 1 (3.2)
1 25 3Rb k k k′ = ⊗ ⊗D I 1 I , 1 2 36 Rbk k k′ = ⊗D I 1 , 7 n′ =D I
where xI is the identity matrix of order x, x1 is the x-dimensional vector of ones,
x x x′=J 1 1 , and ⊗ denotes Kronecker product of matrices.
Let (f = 1, 2,..., 7) denote, respectively, the variances of the effects of the superblocks, the blocks, the rows, the columns I, the columns II, the whole plots, the subplots. Then, under accepted assumptions, we can write the first two moments of distributions of the random variables (f = 1, 2,...,7), i.e.
2σ f
fη
E ( )f =η 0 and Cov( )f f=η V
where 2 1
1 1σ ( )R RR−= −V I J , 2 12 2σ ( )R b bb−= ⊗ −V I I J ,
1 1
2 13 3 1σ ( )Rb k kk −= ⊗ −V I I J , , (3.3)
2
24 4 2σ (Rb k kk −= ⊗ −V I I J
2
1 )
Rb k k kk −= ⊗ ⊗ −V I I I J1 1 2 2
2 1 16 6 1 2σ ( ) ( )Rb k k k kk k− −= ⊗ − ⊗ −V I I J I J
1 )
2 3 3
2 15 5 3σ ( ), ,
2 1 1 3 3
2 17 7 1 3σ ( ) (Rbk k k k kk k− −= ⊗ − ⊗ −V I I J I J
and Cov( , )f f ′ =η η 0 for all f ≠ f’. Thus, the considered dispersion structure of the linear model has the form
7'
1Cov( ) f f f e n
fσ
=
= +∑ 2y D V D I . (3.4)
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Resolvable Incomplete Split-Plot x Split-Block Designs 409
It can be shown that the dispersion matrix (3.4) can be written as 7
0Cov( ) γ f f
f ==∑y P
where 2
0γ σe= , 2 21 1 2 3 1γ σ σebk k k= + , 2 2
2 1 2 3 2γ σ σek k k= + , 2 23 2 3 3γ σ σek k= + ,
24 1 3 4γ σ ek k= + 2σ 2, 2
5 1 5γ σ σek= + , 26 3 6
2γ σ σek= + , 27 7
2γ σ σe= + , (3.5)
and {Pf }, f = 0, 1,…, 7, are a set of pairwise orthogonal matrices summing up to the identity matrix. In this case, they have forms:
10 nn−=P J , 1 ' 1
1 1 2 3 1 1( ) nbk k k n− −= −P D D , 1 2 3n Rbk k kJ = ,
1 ' 1 '2 1 2 3 2 2 1 2 3 1( ) ( )k k k bk k k− −= −P D D 1 D D
,D
' ,D
,
,
4 .
,
1 ' 1 '3 2 3 3 3 1 2 3 2 2( ) ( )k k k k k− −= −P D D D (3.6)
( ) 1 ' 14 1 3 4 4 1 2 3 2 2( )k k k k k− −= −P D D D
1 ' 1 '5 1 5 5 1 3 4 4( )k k k− −= −P D D D D
1 ' 1 ' 1 ' 1 '6 3 6 6 1 3 4 4 2 3 3 3 1 2 3 2 2( ) ( ) ( )k k k k k k k k− − − −= − − +P D D D D D D D D
1 ' 1 ' 1 '7 1 5 5 3 6 6 1 3 4( )n k k k k− − −= − − +P I D D D D D D
The range space { }fℜ P of is termed the f-th stratum with being an orthogonal projection into this stratum. It follows that the considered design has an orthogonal block structure (cf. Nelder, 1965a, 1965b). So, the model can be analyzed using the methods developed for multistratum experiments (cf. Houtman and Speed, 1983, Kachlicka and Mejza, 1996). In this approach, the overall analysis is split into so called stratum analyses. In this case, we have 7 main strata, mentioned above in the Introduction, in which stratum analyses may be performed. The statistical analyses of submodels related to the different strata are based on algebraic properties of stratum information matrices for the treatment combinations, which are defined as
fP fP
f f ′=A ΔP Δ , f = 0, 1,…, 7. (3.7)
We assume that presented RSPSB designs are generally balanced (cf. Houtman and Speed, 1983). This property can be checked by the criterion (cf. Mejza, 1992)
fA f'A = f'A fA
Then the efficiency of an estimation of treatment combination comparisons (called also contrasts) in strata is identified by stratum efficiency factors (noted by εfi). The stratum efficiency factors are eigenvalues of the information matrices , f = 1, 2,…,7 with
respect to , where fA
vRδ =r I vR=r 1 is the vector of replications of the treatment combinations in the RSPSB designs. Moreover, the contrast i′c τ is estimable in the f–th stratum, if the following relation is fulfilled
f i fi iRε=A p p , for f = 1, 2, ..., 7; i = 1, 2, ..., v – 1 (3.8)
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410 I. Mejza & K. Ambroży where pi are common orthogonal eigenvectors of the matrices , corresponding to nonzero eigenvalues,
fA
fiε ( 0 1fiε< ≤ ), 1 2 7... 1i i iε ε ε+ + + = , ' 'i i iiR δ′ =p p , i (i R=c p 'iiδ denotes Kronecker delta). The variance of the f-th stratum BLUE of the contrast is
equal to Var(
i′c τ
i
∧′c τ ) = γ /f fiε , f = 1, 2, ..., 7; i = 1, 2, ..., v – 1. Using a generally balanced
design, we can control an amount of information concerning contrasts that is included into particular strata (cf. Houtman and Speed, 1983). It means that by proper choice of theoretical design, we can express suggestions concerning the precision of estimating particular contrasts. The contrasts , i = 1, 2,..., v – 1, are connected with the comparisons among main effects of the considered factors and interaction effects among them.
i′c τ
4. Construction of resolvable split-plot × split-block designs
Statistical properties of the RSPSB designs follow mainly from crossed and nested treatment structures in an experiment and also from the constructing method of the design considered. Treatment structures of the factors impose a number and a structure of units (plots) in the experiment (see section 2). It can be noticed that, inside each block in the superblocks of RSPSB designs, there are five plot sizes (the row, the column I, the column II, the whole plot and the subplot), so there are five levels of a precision with which the effects of the various factors are estimated. The precision is strictly connected with efficiency of the estimation of the contrasts of the treatment combinations. Also, statistical properties of subdesigns used in the construction for the factors generate statistical properties of the resulting design.
Let NA ( A As R b× ), NB ( B Bt R b× ) and NC ( C Cw R b× ) be incidence matrices of resolvable designs for the row treatments, the column I treatments and the column II treatments with respect to blocks, respectively. They are as follows
1 2[ ]AA A A A R=N N N N… ,
1 2[BB B B B R=N N N N… ] , (4.1)
1 2[ ]CC C C C R=N N N N…
where RA (> 1), RB (> 1) and RC (> 1) are numbers of superblocks each of bA , bB and bC blocks with block sizes , 1 1 Abk=k 1 2 2 Bbk=k 1 and . Hence, ,
and 3 3 Cbk=k 1 1A A An R b k=
2B B Bn R b k= 3C C Cn R b k= are numbers of units in the subdesigns, respectively. The RSPSB design for v = stw treatment combinations in Rb blocks grouped into R
superblocks is described by the following incidence matrices: v×R incidence matrix (with respect to the superblocks), v×Rb incidence matrix (with
respect to the blocks), v×Rbk1 incidence matrix 1 ′=N ΔD1 2
3
2 ′=N ΔD
3 ′=N ΔD (with respect to the rows), v×Rbk2 incidence matrix (with respect to the columns I), v×Rbk2k3 incidence matrix
4 ′=N ΔD4
55 ′=N ΔD (with respect to the columns II), v×Rbk1k2 incidence matrix (with respect to the whole plots). The matrix N1 can be written as
6 6′=N ΔD
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Resolvable Incomplete Split-Plot x Split-Block Designs 411
1 1 1 2[ ]R′= =N ΔD r r r , where rd (= 1v) denotes the replication vector of the treatment combinations in the d-th superblock, therefore, 1 v R′=N 1 1 . In the designing experiment the incidence matrix
2 2 21 22 2[ ]R′= =N ΔD N N N (4.2)
plays the most important role, where N2d is the incidence matrix with respect to blocks inside the d-th (d = 1, 2, …, R) superblock. It can be noted that 1 2R Rb vR= = =Ν 1 N 1 r 1 , and , 1 1 2 3v Rbk k k′ = =N 1 m 1 2 1 2 3v Rbk k k′ = =N 1 k 1 , where m is the vector of superblock sizes, and k is the vector of block sizes in the RSPSB design. Other incidence matrices, besides the matrix N1, result from the matrix N2, but their general forms are not unique. However, corresponding to them concurrence matrices i i′N N , i = 3, 4, 5, 6, are unique (see (4.4)).
In the paper, the designing three-factor experiment is based on the ordinary Kronecker product of the submatrices (4.1), (cf. Gupta, 1985, Gupta and Mukerjee, 1989). Then, the incidence matrix (4.2) is as follows
2 =N AN ⊗ BN ⊗ . (4.3) CN
From (4.3) and according to the treatment structures, concurrence matrices have the forms:
1 1 v vR′ =N N 1 1′ , 2 2 ,A A B B C C′ ′ ′ ′= ⊗ ⊗N N N N N N N N
3 3 ,A s B B C CR′ ′= ⊗ ⊗N N I N N N N′ 4 4 ,A A B t C CR′ ′ ′= ⊗ ⊗N N N N I N N (4.4)
5 5 ,A A B t C wR R′ ′= ⊗ ⊗N N N N I I 6 6 A s B t C CR R′ ′= ⊗ ⊗N N I I N N .
Then, the RSPSB design has R (= ) superblocks, each composed of b (= ) blocks of equal sizes k (= ). The number of replications of the v (= stw)
treatment combinations is equal to R.
A B CR R R
A B Cb b b 1 2 3k k k
Let CA, CB and CC be C-matrices for the treatments in the subdesigns. They are, respectively:
11A A s AR k −
A′= −C I N N with nonzero eigenvalues 1 2 1, ,..., sμ μ μ − with respect to , A sR I
12B B t B BR k − ′= −C I N N with nonzero eigenvalues 1 2 1, ,..., tξ ξ ξ − with respect to B tR I ,
13C C w CR k −
C′= −C I N N with nonzero eigenvalues 1 2 1, ,..., wψ ψ ψ − with respect to . C wR I (4.5)
In the paper, we assume that all C-matrices correspond to the so-called connected subdesigns. The generalization to the disconnected designs is possible.
Using (3.6) and (4.4)-(4.5) the information matrices (3.7) can be expressed by
01 2 3
v vR
bk k k′=A 1 1 , , 1 =A 0
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412 I. Mejza & K. Ambroży
21 2 3
1 [ ]A A B B C C v vR
k k k b′ ′ ′= ⊗ ⊗ −A N N N N Ν N 1 1′ ,
32 3
1A B B Ck k C′ ′= ⊗ ⊗A C N N N N , (4.6)
41 3
1A A B C Ck k
′ ′= ⊗ ⊗A N N C N N , 51
1A A B t CR
k′= ⊗ ⊗A N N I C ,
63
1A B Ck
′= ⊗ ⊗A C C N NC C, 7 A B tR= ⊗ ⊗A C I C .
Following algebraic properties of the stratum information matrices (4.5)-(4.6), and from the relation (3.8), one can notice that
• in the inter-superblock stratum (f = 1), no contrasts are estimable (A1 = 0). It follows from the facts the RSPSB design is connected and the incidence matrix has the form
′=m 1 1 (it is of an orthogonal design); 11 v Rn− ′=N r
• in the inter-block stratum (f = 2), from the fact 1 12 n n Rk− −′ ′≠ =N rk 11 (it is not of an
orthogonal design) it appears that all (v –1) contrasts are estimable in this stratum. Generally, efficiency factors corresponding to these contrasts are equal to
ε2i = 1 – μh (for the row treatments),
ε2i = 1 – ξm (for the column I treatments),
ε2i = 1 – ψg (for the column II treatments),
ε2i = (1 – μh)(1 –ξm) (for interaction contrasts of type A × B),
ε2i = (1 – μh)(1 – ψg) (for interaction contrasts of type A × C),
ε2i = (1 – ξm)(1 – ψg) (for interaction contrasts of type B × C),
ε2i = (1 – μh)(1 – ξm)(1 – ψg) (for interaction contrasts of type A × B × C)
• in the row-stratum (f = 3), from the fact that 11A A An R k− ′≠N 11 (it is not of an
orthogonal subdesign) it follows that all (s – 1) row treatment contrasts are estimated with stratum efficiency factor equal to ε3i = μh. If AA s b′=N 1 1 , then, ε3i = 1, and thus, all row treatment contrasts are estimated with full efficiency. Additionally, in this stratum we may expect estimability of the interaction contrasts of types A × B, A × C and A × B × C. Efficiency factors corresponding to them can be expressed by ε3i = μh(1 – ξm), ε3i = μh(1 – ψg) and ε3i = μh(1 – ξm)(1 – ψg), respectively;
• in the column I-stratum (f = 4), from the fact 12B B Bn R k− ′≠N 11 (it is not of an
orthogonal subdesign) it follows that all (t – 1) column I treatment contrasts are estimated
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Resolvable Incomplete Split-Plot x Split-Block Designs 413 with the stratum efficiency factor equal to ε4i = ξm. If
BB t b′=N 1 1 , then, ε4i =1, and thus, all column I treatment contrasts are estimated with full efficiency. Additionally, in this stratum we may expect estimability of interaction contrasts of types A × B, B × C and A × B × C. Efficiency factors corresponding to them we may express by ε4i = (1 – μh)ξm, ε4i = ξm(1 – ψg) and ε4i = (1 – μh)ξm(1 – ψg), respectively;
• in the column II-stratum (f = 5), from the fact that 13CC Cn R k− ′≠N 11 (it is not of an
orthogonal subdesign) it appears that all (w – 1) column II treatment contrasts are estimated with the stratum efficiency factor ε5i = ψg. If CC w b′=N 1 1 , then, ε5i =1, and thus, all column II treatment contrasts are estimated with full efficiency. Additionally, in this stratum we may expect of estimability of interaction contrasts of types A × C, B × C and A × B × C. Efficiency factors corresponding to them can be expressed by ε5i = (1 – μh)ψg, ε5i = ψg and ε5i = (1 – μh)ψg, respectively;
• in the whole plot-stratum (f = 6), from the facts 11 1 and
12
−
A A An R k− ′≠N 1
B B Bn R k ′≠N 11 it follows that all (s – 1)(t – 1) interaction contrasts of type A × B are estimated with stratum efficiency factors which can be calculated from ε6i = μhξm. If both
AN and BN are incidence matrices of orthogonal subdesigns, then ε6i = 1, and thus, all interaction contrasts of type A × B are estimated with full efficiency. Additionally, in this stratum we may expect of estimability of interaction contrasts of type A × B × C. Efficiency factors for them are calculated from the formula ε6i = μhξm(1 – ψg);
• an analysis within the subplot-stratum (f = 7) is connected with an estimation of (s – 1)(w – 1) interaction contrasts of type A × C and (s – 1)(t – 1)(w – 1) contrasts of type A × B × C only. It is evident from the considered crossed and nested treatment structures of the RSPSB design. Hence, its efficiency with respect to these contrasts is strictly connected with the subdesigns for the row and column II treatments only. Generally, stratum efficiency factors for these contrasts are equal to ε7i = μhψg. It follows from the both facts
11
−A A An R k ′≠N 1 ≠N1 and 1
3− ′11 . When they are incidence matrices of orthogonal
subdesigns then all interaction contrasts of type A × C and contrasts of type A × B × C are estimated with full efficiency in this stratum, i.e. ε7i = 1.
C C Cn R k
5. Example
To illustrate the theory presented in the paper, consider some 4 × 4 × 4 factorial experiment. Let us assume that the row treatments correspond to levels of nitrogen fertilization (s = 4), the column I treatments correspond to varieties of wheat (t = 4) and the column II treatments correspond to doses of an application of a chemical preparation - growth regulator (w = 4). According to an accessible material structure of the experiment a resolvable split-plot × split-block design can be used for v = stw = 64 treatment combinations.
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414 I. Mejza & K. Ambroży
As a generating (theoretical) design D for the row treatments, the column I treatments, and the column II treatments, we used the same resolvable balanced incomplete block (BIB) design with the incidence matrix (= =N AN BN = ) = , where ,
and are incidence matrices within superblocks. They are: CN 1 2 3[ ]N N N 1N
2N 3N
Superblock 1 Superblock 2 Superblock 3
1
1 01 00 10 1
⎡ ⎤⎢ ⎥⎢=⎢⎢ ⎥⎣ ⎦
N ⎥⎥
⎥⎥
’ , 2
1 00 11 00 1
⎡ ⎤⎢ ⎥⎢=⎢⎢ ⎥⎣ ⎦
N 3
1 00 10 11 0
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
N .
It may be noted that the subdesign D is the resolvable BIB design with the parameters:
( )v s t w= = = = 4 3
) 2 2
, , ( )A B CR R R R= = = =
( A B Cb b b b= = = = , 1 2 3( )k k k k= = = = .
Its matrix, 1k
R ′= −C I NN , is of the form:
12
3 1 1 1 3 1 1 11 3 1 1 1 3 1 11( ) 31 1 3 1 1 1 3 121 1 1 3 1 1 1 3
A B C
− − −⎡ ⎤ ⎡⎢ ⎥ ⎢
⎤⎥− − −⎢ ⎥ ⎢= = = = − =
⎢ ⎥ ⎢⎥⎥− − −
⎢ ⎥ ⎢− − −⎣ ⎦ ⎣
C C C C I
⎥⎦
. (5.1)
The matrix (5.1) has one nonzero eigenvalue 1 2 31
v(k ) /k(v )
ε −= =
− with multiplicity 3
(calculated with respect to 3I4). Hence, 2 / 3h m gε μ ξ ψ= = = = , for h = 1, 2, 3; m = 1, 2, 3; g = 1, 2, 3.
We get the RSPSB design for the considered three-factor experiment according to the incidence matrix (4.3). Then, the resulting design has 27 superblocks (R = 27), each composed of 8 blocks (b = 8) of equal sizes, k ( 1 2 3k k k= ) = 8. The number of all units is equal to n = 1728. The number of the treatment combinations is equal to v = stw = 64 (> k). They are equally replicated and the vector of replications is 6427=r 1 . The incidence matrix for superblocks has the form 1 v R′=N 1 1 , so it is of the orthogonal design.
The layout (before randomization) of the 4 × 4 × 4 factorial experiment, arranged in the resolvable SPSB design, consists of 216 blocks of the form { , | , | , }i i j j l lA A B B C C′ ′ ′ , where , in which the row treatments ; ;i i ; j j l l′ ′ ′ 1 2 3 4i, i ; j, j ; l, l , , ,′ ′ ′ =< < < ,i iA A ′ are
allocated to the rows, the column I treatments ,j jB B ′ are allocated to the columns I, and the column II treatments ,l lC C ′ are allocated to the columns II. For example, after randomization one of the blocks, inside the first superblock, has the treatment combinations as in Figure 2.
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Resolvable Incomplete Split-Plot x Split-Block Designs 415
Figure 2. An example of a single block inside the first superblock in the RSPSB design (after randomization)
B1 B2
C2 C1 C1 C2
A2 A1
From the algebraic properties of the information matrices (4.6), we have obtained stratum efficiency factors of the RSPSB design (Table1).
Table 1. Stratum efficiency factors of the RSPSB design
Strata Type of contrast Df
f = 1 f = 2 f = 3 f = 4 f = 5 f = 6 f = 7
A s – 1 = 3 1/3 2/3
B t – 1 = 3 1/3 2/3
C w – 1 = 3 1/3 2/3
A × B (s – 1)(t – 1) = 9 1/9 2/9 2/9 4/9
A × C (s – 1)(w – 1) = 9 1/9 2/9 2/9 4/9
B × C (t – 1)(w – 1) = 9 1/9 2/9 6/9
A × B × C (s – 1)( t – 1)( w – 1) = 27 1/27 2/27 2/27 6/27 4/27 12/27
Summing up, it can be noticed that in the generated RSPSB design, information about the orthogonal contrasts is split into two strata (for the contrasts of main effects of the factors A, B and C) or into more strata (for the interaction contrasts). Statistical inferences (estimates and tests) about them can be obtained using the information separately from one stratum only or performing for them the combined estimation and testing based on information from all these strata in which they are estimable (e.g. Hedges and Olkin, 1985, Caliński and Kageyama, 2000).
Acknowledgement
The Authors would like to thank two referees for their valuable suggestions and comments.
References
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