On the efficiency of some non-orthogonal split-plot×split-block designs with control treatments

Journal of Statistical Planning and Inference 142 (2012) 752–762

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On the efficiency of some non-orthogonal split-plot  split-block designs with control treatments Katarzyna Ambroz˙y, Iwona Mejza n ´ , Poland Department of Mathematical and Statistical Methods, Poznan University of Life Sciences, Wojska Polskiego 28, 60-637 Poznan

a r t i c l e in f o

abstract

Article history: Received 28 September 2010 Received in revised form 22 September 2011 Accepted 26 September 2011 Available online 29 September 2011

Many split-plot  split-block (SPSB) type experiments used in agriculture, biochemistry or plant protection are designed to study new crop plant cultivars or chemical agents. In these experiments it is usually very important to compare test treatments with the socalled control treatments. It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design. In the paper we propose a non-orthogonal SPSB design for consideration. Two cases of the design are presented here, i.e. when its incompleteness is connected with a crossed treatment structure only or with a nested treatment structure only. It is assumed the factors’ levels connected with the incompleteness of the design are split into two groups: a set of test treatments and a set of control treatments. The method of constructions involves applying augmented block designs for some factors’ levels. In a modelling data obtained from such experiments the structure of experimental material and appropriate randomization scheme of the different kinds of units before they enter the experiment are taken into account. With respect to the analysis of the obtained randomization model the approach typical to the multistratum experiments with orthogonal block structure is adapted. The proposed statistical analysis of linear model obtained includes estimation of parameters, testing general and particular hypotheses defined by the (basic) treatment contrasts with special reference to the notion of general balance. & 2011 Elsevier B.V. All rights reserved.

Keywords: Split-plot  split-block (SPSB) design Non-orthogonal SPSB design Augmented block design Control treatment General balance

1. Introduction 1.1. Terminology and notation We have to design an agricultural three factor experiment with control treatments and a modelling data obtained from it. With respect to technical reasons an incomplete (non-orthogonal) split-plot  split-block (shortly, SPSB) design have been taken into account. The complete (orthogonal) SPSB design is the most widely used in an agriculture research (e.g. LeClerg et al., 1962; Wadas et al., 2005; Federer and King, 2007). In particular, it is very useful in field experiments when certain treatments such as types of cultivation, application of irrigation water, etc., may be necessary to arrange them in strips (rows or columns) across each block. In the SPSB designs there are three kinds of strips to accommodate independently levels (treatments) of three factors (say, A, B and C). Usually it is assumed that two first factors (A and B) are crossed as in a split-block design (strip-plot design) while the factors B and C are nested as in a split-plot design

n

Corresponding author. Fax: þ 48 61848 7140. E-mail addresses: [email protected] (K. Ambroz˙y), [email protected] (I. Mejza).

0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.09.013

K. Ambroz˙y, I. Mejza / Journal of Statistical Planning and Inference 142 (2012) 752–762

753

(e.g. Gomez and Gomez, 1984). In practice however there are situations in which an experimental material for certain treatments is limited. Then a numbers of levels of one or more factors are larger than a numbers of appropriate for them strips within each block. It follows usually such treatments have less number of replications than the rest of treatments. In such situations it is convenient to plan non-orthogonal experiments in incomplete SPSB designs in advance. In particular a special attention is paid to: (1) optimal utilizing the structure of available experimental material, (2) possibilities of applying the control treatments to the factors, and (3) possibilities of the incomplete design in such a way that the most interesting contrasts can be estimated with efficiencies as high as possible. 1.2. General purpose In the paper we are interested in incomplete SPSB designs which have orthogonal block structure (OBS) and general balance (GB) property only. The OBS property means that in a model of observations the variance–covariance matrix can be expressed in the form given in (2). Next, in designs with the GB property all information matrices connected with treatment combinations fulfill commutative condition (5). It allows joining information about the contrasts from different strata in which they are estimable, if it at all possible (e.g. Calin´ski and Kageyama, 2000). The traditional (complete) SPSB design possesses both properties. With incomplete data sets, however, it may be difficult to fulfill all the conditions of general balance (Houtman and Speed, 1983; Mejza, 1992). General methodology of complete and incomplete SPSB designs, i.e. designing; modelling and statistical analysis under mixed linear model is presented in Ambroz˙y and Mejza (2003, 2004, 2008). Some constructing methods of incomplete SPSB experiment designs using, for example, group divisible partially balanced incomplete block designs or resolvable block designs are given in Ambroz˙y and Mejza (2004, 2006), Mejza and Ambroz˙y (2007). In the present paper two cases of the SPSB design are considered, i.e. when its incompleteness is connected with crossed treatment structure only or with nested structure only. Additionally we assumed that control treatments are included in the levels of some factors. In a designing the considered three factor experiment some augmented block designs (called also supplemented block designs) are helpful. They play role of generating subdesigns, which statistical properties can retain orthogonal block structure of the SPSB design, but allow unbalanced treatment structure. The augmented block designs belong to a wide family of known in literature block designs that have been often used in a planning one and more factor experiments, especially in a research with additional treatments (called control or standard treatments), see e.g. Calin´ski and Ceranka (1974), Puri et al. (1977), Singh and Dey (1979), Mejza (1996, 1998), Federer (2005), Federer and Arguillas (2006), and Federer and King (2007). In the modelling data obtained from SPSB type experiments we took into account a cross-nested structure of an experimental material and a four-step randomization scheme of the different kinds of units. With respect to the analysis of the obtained randomization model with seven strata we adapted the approach, typical to multistratum experiments with orthogonal block structure (cf. Nelder, 1965a, 1965b). Additional information about modelling observations by this way can be found in e.g. Mejza and Mejza (1984), Calin´ski (1994), Mejza (1994), and Calin´ski and Kageyama (2000). 2. Material and methods 2.1. Experimental material structure Consider an (s  t  w)—experiment in which the first factor, say A, has s levels A1, A2, y, As, the second factor, say B, has t levels B1, B2, y, Bt and the third factor, say C, has w levels C1, C2, y, Cw. Let v( ¼stw) be the number of all treatment combinations. Generally, an experimental SPSB design structure is the following: we draw b blocks in such a way that each block has a row–column structure with k1 rows and k2 columns of the first order, shortly, columns I. Then each column I has to be split into k3 columns of the second order, shortly, columns II. Here the rows correspond to the levels of the factor A, termed also as row treatments or A treatments, the columns I correspond to the levels of the factor B, called also column I treatments or B treatments and the columns II are to accommodate the levels of the factor C termed as column II treatments or C treatments. It should be noticed that the row treatments and the column I treatments are in a split-block design (crossed treatment structure) while the column II treatments are in a split-plot design in a relation to the column I (or row) treatments (nested treatment structure). Case 1. First we consider a situation that SPSB design is incomplete with respect to the A and B treatments only, i.e. k1 o s, k2 ot, k3 ¼w. Additionally we assume the A treatments consist of two groups (s¼s1 þs2) called test A and control A treatments, respectively, and also the B treatments consist of two groups (t¼t1 þt2) called test B and control B treatments, respectively. Case 2. We consider here a situation that SPSB design is incomplete with respect to the B and C treatments only, i.e. k1 ¼s, k2 ot, k3 ow. Additionally we assume the B treatments consist of two groups (t ¼t1 þt2) called test B and control B treatments, respectively, and also the C treatments consist of two groups (w ¼w1 þw2) called test C and control C treatments, respectively. In both cases we assume the incomplete SPSB designs are proper, i.e. their block sizes (k1k2k3) are all equal (cf. Calin´ski and Kageyama, 2000).

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2.2. Randomization model In the paper we consider a randomization model of observations, which the form and properties are strictly connected with the performed randomization processes in the experiment. The randomization scheme for both cases of the SPSB design consists of four randomization steps performed independently, that is by permuting blocks, rows, columns I and columns II. This leads to the mixed linear model of observations of the form (cf. Ambroz˙y and Mejza, 2003, 2004, 2008): y ¼ D0 s þ

6 X

EðyÞ ¼ D0 s,

D0f nf þe,

ð1Þ

f ¼1

where y is the n-dimensional vector of lexicographically ordered observations, n¼ bk1k2k3, D0 (n  v) is a known design matrix for v treatment combinations, D01 (n  b), D02 (n  bk1), D03 (n  bk2), D04 (n  bk2k3), D05 (n  bk1k2), D06 (n  n), are the design matrices for blocks, rows (within blocks), columns I (within blocks), column II (within columns I), whole plots (within blocks) and subplots (within whole plots) respectively, s(v  1) is the vector of fixed treatment combination effects, n1(b  1), n2(bk1  1), n3(bk2  1), n4(bk2k3  1), n5(bk1k2  1), n6(n  1), e(n  1) are random effect matrices of blocks, rows, columns I, columns II, whole plots, subplots and technical errors, respectively. The dispersion structure of the linear model (1) can be written as VðgÞ ¼

6 X

gf Pf ,

ð2Þ

f ¼0

where gf are the nonnegative variance components and {Pf} are a family of known pairwise orthogonal, symmetric and P 0 idempotent projectors adding up to the identity matrix (cf. Houtman and Speed, 1983), i.e. 6f ¼ 0 Pf ¼ In , Pf ¼Pf, PfPf ¼Pf, 0 0 PfPf0 ¼0, f ,f ¼0,1,y,6; f af. The forms of the matrices Pf for the SPSB design are given in Ambroz˙y and Mejza (2003). It can be shown that the ranks of them are as follows: rankðP0 Þ ¼ 1,

rankðP1 Þ ¼ b1,

rankðP4 Þ ¼ bk2 ðk3 1Þ,

rankðP2 Þ ¼ bðk1 1Þ,

rankðP5 Þ ¼ bðk1 1Þðk2 1Þ,

rankðP3 Þ ¼ bðk2 1Þ,

rankðP6 Þ ¼ bðk1 1Þk2 ðk3 1Þ:

ð3Þ

The range space R{Pf} of Pf, f¼0,1, y, 6, is termed the fth stratum of the model and {gf} are unknown stratum variances. From (2) and the properties of the projectors Pf it follows that considered SPSB design has an orthogonal block structure (OBS) as defined by Nelder (1965a, 1965b). So, the model (1) can be analysed using the methods developed for multistratum experiments. Here we have zero stratum (0) generated by the vector of ones, inter-block stratum (1), interrow (within the block) stratum (2), inter-column I (within the block) stratum (3), inter-column II stratum (4) (within the column I), inter-whole plot (within the block) stratum (5), and inter-subplot (within the whole plot) stratum (6). 2.3. Stratum analyses In both cases of the non-orthogonal SPSB design we have 6 mentioned above main strata in which stratum analyses may be performed. The statistical analyses of submodels related to the different strata are based on algebraic properties of 0 stratum information matrices for treatment combinations, which are defined as Af ¼ DPfD , f¼ 0,1, y, 6. In the considered SPSB designs they are following (cf. Ambroz˙y and Mejza, 2003): A0 ¼ n1 rr0 , A2 ¼ ðk2 k3 Þ

1

A1 ¼ ðk1 k2 k3 Þ1 N1 N01 n1 rr0 , N2 N02 ðk1 k2 k3 Þ1 N1 N01 ,

A3 ¼ ðk1 k3 Þ1 N3 N03 ðk1 k2 k3 Þ1 N1 N01 , 1

A4 ¼ k1 N4 N04 ðk1 k3 Þ1 N3 N03 , 1

A5 ¼ ðk1 k2 k3 Þ1 N1 N01 þ k3 N5 N05 ðk2 k3 Þ1 N2 N02 ðk1 k3 Þ1 N3 N03 , 1

1

A6 ¼ ðk1 k3 Þ1 N3 N03 k1 N4 N04 k3 N5 N05 þ rd , 0

0

ð4Þ

0

0

0

where N1( ¼ DD1), N2(¼ DD2), N3( ¼ DD3), N4( ¼ DD4) and N5( ¼ DD5) are treatments vs. blocks, treatments vs. rows, treatments vs. columns I, treatments vs. columns II, and treatments vs. whole plots incidence matrices, respectively, r ¼ N11b ¼ N21bk1 ¼N31bk2 ¼N41bk1k2 is a vector of replicates of v treatment combinations, rd ¼ diag [r1, r2, y, rv] and 1x is the x-dimensional vector of ones. We assume the considered non-orthogonal SPSB designs are generally balanced (cf. Houtman and Speed, 1983). The GB property can be checked by the general criterion (cf. Mejza, 1992) Af rd Af 0 ¼ Af 0 rd Af

0

f or f ,f ¼ 1,2,. . .,6, f af

0

and

rd ¼ diag½1=r 1 ,1=r 2 ,. . .,1=r v :

ð5Þ

It means that if all information matrices (4) fulfill commutative condition (5) then the design is generally balanced. It allows finding common set of rd-orthonormal eigenvectors for all the information matrices (4).

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Table 1 ANOVA in the fth stratum, f ¼1,2, y, 6. Source of variation

DF

SS

E(MS)

‘‘Treatments‘‘(f)

nTf ¼rank(Af) nEf ¼ nf  nTf nf ¼ rank(Pf)

SSTf

gf þ n1 Tf s Afs gf gf þ n1 f s Afs

Error (f) Total (f)

0

SSEf SSYf

0

Let efh denote an eigenvalue of the matrix Af with respect to rd, corresponding to an eigenvector sh, f¼0,1,y, 6, h¼ 1, 2,y,v. The eigenvalues efh satisfy the following relations: 0 r efh r1

6 X

8h o v ðe0v ¼ 1, e0h ¼ 0Þ,

ef h ¼ 1 for ho v;

f ¼ 0,1,. . .,6;

h ¼ 1,2,. . .,v:

f ¼1

The eigenvectors of the form sh ¼ ai  bj  cm

for h ¼ 1, 2,. . .,v;

i ¼ 1,2,. . .,s;

j ¼ 1,2,. . .,t;

m ¼ 1,2,. . .,w

ð6Þ

are rd-orthonormal, i.e., satisfy the following conditions: s0h rd sh ¼ 1

and

s0h rd sh0 ¼ 0

0

0

for h,h ¼ 1,2,. . .,v,hah

ð7Þ

Since Af1v ¼0, f 40, the last eigenvector may be chosen as sv ¼n  1/21v. We can note that any vector ph ¼ rdsh such that the eigenvector sh satisfies the condition Af sh ¼ ef h rd sh

for f ¼ 1,2. . .,6

and

h ¼ 1,2,. . .,v1

ð8Þ

p0h

defines an orthogonal (basic) contrast s (cf. Pearce et al., 1974). These contrasts are strictly connected with comparisons among the main effects of the considered factors and interaction effects. Eigenvalues efh of matrices (4), f¼1,2y,6; h¼1,2,y,v  1 with respect to rd are interpreted as stratum efficiency factors for a set of orthogonal (basic) contrasts (noted by p0h s). 0 b 0h s)¼ gf/efh, for f¼ 1,2y,6 and h ¼1,2,y,v  1. The variance of the fth stratum BLUE of the contrast phs is equal to Var(p As well, both the orthogonal contrasts and stratum efficiency factors corresponding to these contrasts for each case of the SPSB design considered independently are given in Sections 3.3 - 3.4. In the paper we do not consider estimating any contrast (a function of some basic contrasts). Information about that you can find in e.g. Graybill (1961). 2.3.1. Testing hypotheses If the normality of the random variables of the model (1) is assumed it is easy to construct an exact test of the 0 hypothesis H0f: s Afs ¼0, f¼ 1,2y,6, relating to all the contrasts estimable in the fth stratum (ANOVA). In particular we are interested in testing hypothesis for any contrasts, Hn0f : c0 s ¼ 0, estimable in the fth stratum. Using the basic contrasts, it is 0 easy to express (cf. Graybill, 1961) also general hypothesis, H0: W s ¼0, where rank(W)¼ o denotes the number of the all basic contrasts estimable in the fth stratum, connecting with main or interaction effects of the factors. The necessary sum P bÞf 2 ,f¼1,2y,6 while the sum of of squares for treatments in Table 1 can be obtained from the formula SSTf ¼ h ef h ½ðp0h s squares for errors are as follows 0 SSEf ¼SSYf  SSTf, where SSYf ¼y Pfy. They are sufficient to build the appropriate F-tests. In Table 1 notation ‘‘Treatments‘‘(f) means a set of all basic contrasts which are estimable in the fth stratum. Ranks of matrices Pf are given in (3). Rank of each matrix Af depends on a design matrix for the treatment combinations and is equal to the number of basic contrasts estimable in the fth stratum. A general hypothesis connected with one factor or an interaction can be verified in the fth stratum when all basic contrasts of one type are estimable in that stratum (cf. Graybill, 1961). 3. Some construction methods of non-orthogonal SPSB designs In the paper we consider two cases of a construction of the augmented SPSB designs both using traditional method based on Kronecker’s product of matrices. Then an incidence matrix with regard to blocks for the SPSB designs is of the form N1 ¼ NA  NB  NC ,

ð9Þ

where NA, NB and NC are incidence matrices of a randomized complete block (RCB) design or/and some augmented block designs (called generating subdesigns). Other incidence matrices for the SPSB designs, N2, N3, N4 and N5 can be obtained by taking the rows (within blocks) as ‘‘blocks’’, the columns I (within blocks) as ‘‘blocks’’, the columns II (within columns I) as ‘‘blocks’’ and the whole plots (within blocks) as ‘‘blocks’’, respectively. Since general forms of these matrices are not unique, we present concurrence matrices which are the following: N1 N01 ¼ NA N0A  NB N0B  NC N0C ,

N2 N02 ¼ r dA  NB N0B  NC N0C ,

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N3 N03 ¼ NA N0A  rdB  NC N0C ,

N4 N04 ¼ NA N0A  rdB  rdC ,

N5 N05 ¼ rdA  rdB  NC N0C ,

ð10Þ

where h i rdA ¼ diag rA1 ,rA2 ,. . .,rAs ,

  rdB ¼ diag rB1 ,rB2 ,. . .,rBt ,

  rdC ¼ diag rC1 ,rC2 ,. . .,rCw :

Specified forms of matrices (10) for each case of the SPSB design we present in Sections 3.3 and 3.4, respectively. With n n respect to the considered in the paper problems two augmented block designs d1 and d2 as the generating subdesigns are taken into account. Then we give statistical consequences in further analysis of using them in the construction methods of both SPSB designs. n

3.1. The augmented block design d1

" # d~ 1 n Consider an augmented block design d1 ¼ ~ with the incidence matrix d2 2 3 0 1v~ 1 1 b~ 1 5 Ndn1 ¼ 4 Ib~ 2

ð11Þ

and the following parameters: b1 ¼ b~ 1 ¼ b~ 2 ,

n

k1 ¼ k~ 1 þ k~ 2 ¼ v~ 1 þ 1,

n

vn1 ¼ v~ 1 þ v~ 2 ,

n

rn1 ¼ ½r~ 1 ^r~ 1 0 ¼ ½b~ 1 10v~ 1 ^10v~ 2 0 0

0

ð12Þ

n

where vn1 , b1 , k1 and rn1 are the numbers of treatments, the blocks, the units in blocks and the vector of replications of the n treatments, respectively, in the d design, while v~ is a number of treatments in d~ design (here called test treatments), 1

1

1

v~ 2 is a number of treatments in d~ 2 design (here called control treatments), r~ 1 , r~ 2 denote the vectors of replications of both groups of treatments, b~ 1 , b~ 2 are the numbers of blocks in d~ 1 and d~ 2 designs and k~ 1 , k~ 2 denote their block sizes. Moreover, Ix n n n denotes x-dimensional unity matrix. Shortly, we can write down d1 ðvn1 ,b1 ,k1 ,rn1 Þ. The test treatments occur in a RCB design ~d ðv~ , b~ , k~ , r~ Þ. Each of the b~ blocks of the d~ design is supplemented by one different than others additional (control) 1

1

1

1

1

1

1

treatment , i.e. v~ 2 ¼ b~ 1 . It can be shown (cf. Singh and Dey, 1979; Nigam et al., 1981; Ceranka and Chudzik, 1984) that n n n d1 ðvn1 ,b1 ,k1 ,rn1 Þ is a partially efficiency balanced design with two efficiency classes. The information matrix Cdn1 ¼ ðrn1 Þd ðk1 Þ1 Ndn1 N0dn has two different eigenvalues calculated with respect to ðrn1 Þd . They are following: mn0 ¼ 1 and mn1 ¼ n

1

k~ 1 =k1 ¼ v~ 1 =ðv~ 1 þ 1Þ with multiplicities equal to rn0 ¼ v~ 1 and rn1 ¼ b~ 1 1, respectively. It can be shown that the first class of efficiency equal to mn0 is connected with comparisons: (1) between the test group and the control group of the treatments n

n

in d1 , (2) among the test treatments only. The second class of efficiency equal to mn1 refers to the comparisons among the control treatments only. n

3.2. The augmented block design d2 n

n

n

Let us consider now an augmented block design d2 ðvn2 ,b2 ,k2 ,rn2 Þ with the incidence matrix 2 3 ^1 N 5, Ndn2 ¼ 4 1v^ 2 10b^

ð13Þ

1

^ 1 and parameters: where the test treatments occur in a balanced incomplete block (BIB) design with an incidence matrix N ^v , b^ , r^ , k^ , l, where l ¼ r^ ðk^ 1Þ=ðv^ 1Þ. Each of the b^ blocks of the d^ design is supplemented by v^ additional (control) 1

1

1

1

1

1

1

1

treatments. Then an information matrix Cd^

1

1

2

of the d^ 1 design has got only one eigenvalue equal to e ¼ v^ 1 ðk^ 1 1Þ=

n k^ 1 ðv^ 1 1Þ ¼ lv^ 1 =r^ k^ 1 with multiplicity v^ 1 1. Therefore, it can be shown (cf. Calin´ski and Ceranka, 1974) the d2 is a partially efficiency balanced design with two efficiency classes and the following parameters:

vn2 ¼ v^ 1 þ v^ 2 ,

n b2 ¼ b^ 1 ,

n k2 ¼ k^ 1 þ v^ 2 ,

rn2 ¼ ½r^ 1 10v^ 1 ^b^ 1 10v^ 2 0 :

ð14Þ

n n n n The information matrix Cdn2 has two different eigenvalues x0 ¼ 1 and x1 ¼ ðk^ 2 þ k^ 1 eÞ=k2 with multiplicities equal to c0 ¼ v^ 2 n

n

and c1 ¼ v^ 1 1 respectively. It can be shown that the first class of efficiency equal to x0 is connected with comparisons: (1) n

between the test group and the control group of the treatments in d2 , (2) among the control treatments only. The second n class of efficiency equal to x1 refers to the comparisons among the test treatments only.

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3.3. SPSB designs with incomplete crossed treatment structure only (case 1) We accept the incidence matrix (9) of the SPSB design of the form N1 ¼ Ndn1  Ndn2  1w ,

ð15Þ

where vector 1w presents one block incidence matrix of the RCB design for the C treatments. In this case of the construction the matrices (11) and (13) define ways of an arrangement of the test and control A treatments (s ¼s1 þs2) and test and control B treatments (t ¼t1 þt2) in the SPSB design, respectively. The parameters of the final design are as follows: v ¼ stw,

n

n

b ¼ b1 b2 ,

n

n

k ¼ k1 k2 w,

r ¼ rn1  rn2  1w

ð16Þ

where v, b, k, r denote the number of treatment combinations, the number of blocks, a size of the blocks, a vector of replicates of the treatment combinations, respectively. In the case 1 of the SPSB design the concurrence matrices (10) assume the following forms: 2 3 2 3 ðr^ 1 lÞIt1 þ l1t1 10t1 r^ 1 1t1 10t2 b~ 1 1s1 10s1 1s1 10s2 0 4 5 4 5  1w 10 , N1 N1 ¼  w 1s2 10s1 Ib~ 1 k^ 1 1t2 10t1 b^ 1 1t2 10t2 2 3 ðr^ 1 lÞIt1 þ l1t1 10t1 r^ 1 1t1 10t2 0 n d 5  1w 10 , N2 N2 ¼ ðr1 Þ  4 w k^ 1 1t2 10t1 b^ 1 1t2 10t2 2 3 2 3 b~ 1 1s1 10s1 1s1 10s2 b~ 1 1s1 10s1 1s1 10s2 0 0 0 d n 5  ðr Þ  1w 1 , N4 N ¼ 4 5  ðrn Þd  Iw , N3 N3 ¼ 4 w 2 2 4 1s2 10s1 Ib~ 1 1s2 10s1 Ib~ 1 N5 N05 ¼ ðrn1 Þd  ðrn2 Þd  1w 10w ,

ð17Þ

where " ðrn1 Þd ¼

b~ 1 Is1 0

0 Is2

#

" ðrn2 Þd ¼

and

r^ 1 It1

0

#

: b^ 1 It2

0

Algebraic properties, eigenvalues efh with respect to rd and their multiplicities corresponding to proper contrasts p0h s, f¼1,2y,6; h¼ 1,2,y,v  1), of the information matrices (4) with (17) are presented in a special form (18). Earlier it is convenient to introduce an abbreviation to describe the property of balance of the SPSB design. In the case 1 of the SPSB design the efficiency factors efh correspond to the following orthogonal contrasts p0h s among effects of: the test A treatments (AT), the control A treatments (AC), both groups of the test and control A treatments (AT vs: AC ), the test B treatments (BT), the control B treatments (BC), both groups of the test and control B treatments (BT vs: BC ), the C treatments (C), interaction of type A  B, including: AT  BT , AC  BT , ðAT vs: AC Þ  BT , AT  BC , AC  BC , ðAT vs: AC Þ  BC , AT  ðBT vs: BC Þ, AC  ðBT vs: BC Þ, (AT vs: AC )  (BT vs: BC ), interaction A  C, including: AT  C, AC  C, ðAT vs: AC Þ  C, interaction B  C, including: BT  C, BC  C, ðBT vs: BC Þ  C, interaction A  B  C, including: AT  BT  C, AC  BT  C, ðAT vs: AC Þ  BT  C, AT  BC  C, AC  BC  C, ðAT vs: AC Þ  BC  C, AT  ðBT vs: BC Þ  C, AC  ðBT vs: BC Þ  C, (AT vs: AC )  ðBT vs: BC Þ  C. Using the above abbreviations we can express the statistical properties for the case 1 of the SPSB design in Table 2. 3.4. SPSB designs with incomplete nested treatment structure only (case 2) We accept the incidence matrix (9) of the SPSB design of the form N1 ¼ 1s  Ndn2  Ndn1 ,

ð18Þ

where the vector 1s presents one block incidence matrix of the RCB design for the A treatments. In this case of the construction the matrices Ndn2 and Ndn1 define ways of a distribution of the test and control B treatments (t ¼t1 þt2) and the test and control C treatments (w¼w1 þw2) in the SPSB design, respectively. The parameters of the final design are as follows: v ¼ stw,

n

n

b ¼ b2 b1 ,

n n

k ¼ sk2 k1 ,

r ¼ 1s  rn2  rn1 ,

ð19Þ

where v, b, k, r denote the number of treatment combinations, the number of blocks, a size of the blocks, a vector of replicates of the treatment combinations, respectively. In the case 2 of the SPSB design the concurrence matrices (10) have the forms 2 3 2 3 ðr^ 1 lÞIt1 þ l1t1 10t1 r^ 1 1t1 10t2 b~ 1 1w1 10w1 1w1 10w2 0 0 54 5, N1 N1 ¼ 1s 1s  4 1w 10 I~ k^ 1t 10 b^ 1t 10 1

2

t1

1

2

t2

2

w1

b1

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Table 2 Stratum efficiency factors of the SPSB design (case 1). Types of the contrasts

Numbers of the contrasts

T

A AC

r0 1 rn1

AT vs: AC BT

1

(1)

(2)

1mn1

m0 ¼ 1 mn1

n

BC BT vs: BC C

w 1

A B T

A  ðB vs: B Þ AC  BT A  ðB vs: B Þ ðAT vs: AC Þ  BT

cn1

T

A B

C

C

xn1

1x1

1 n

mn1 ð1xn1 Þ

n

mn1 xn1 mn1

ð1mn1 Þx1 1mn1

n

xn1

1x1

1

n

ðr0 1Þðc0 1Þ 1mn1

rn1 ðcn0 1Þ C

ðA vs: A Þ  B

C

mn1

c0 1

1

1

1

n

ðAT vs: AC Þ  ðBT vs: BC Þ AT  C AC  C T

n

ð1mn1 Þð1x1 Þ

n

AC  BC T

1 n

rn1 cn1 rn1

T

ðrn0 1Þðw1Þ rn1 ðw1Þ w 1

C

ðA vs: A Þ  C BT  C

(6)

xn1 xn0 ¼ 1

ðr0 1Þc1 rn0 1

C

(5)

1

n

T

C

n

1x1

1

T

(4)

1

cn1 cn0 1

T

(3)

n

1 1mn1

mn1 1

cn1 ðw1Þ n ðc0 1Þðw1Þ

1

BC  C ðBT vs: BC Þ  C

w 1

1

AT  BT  C

ðrn0 1Þc1 ðw1Þ

AC  BT  C A B C

rn1 cn1 ðw1Þ n ðrn0 1Þðc0 1Þðw1Þ

ðAT vs: AC Þ  BC  C

ðc0 1Þðw1Þ

T

n

C

C

1

n

xn1

1x1 1mn1

mn1 1 1

n

C

1mn1

r1 ðcn0 1Þðw1Þ n

A B C

mn1

(1)–(6) the numbers of the strata (see Section 2.2).

2 N2 N02

¼ Is  4

N3 N03

¼ 1s 10s

N4 N04

¼ 1s 10s

ðr^ 1 lÞIt1 þ l1t1 10t1

r^ 1 1t1 10t2

k^ 1 1t2 10t1

b^ 1 1t2 10t2

2  ðr2 Þ  4 n

d

b~ 1 1w1 10w1

1w1 10w2

1w2 10w1

Ib~ 1

3

2

54

d

n

 ðr2 Þ  ðr1 Þ

d

,N5 N05

1w2 10w1

Ib~ 1

b~ 1 1w1 10w1

1w1 10w2

1w2 10w1

Ib~ 1

3 5,

5,

¼ Is  ðr2 Þ  4 n

1w1 10w2

3

2 n

b~ 1 1w1 10w1

d

3 5,

ð20Þ

where " n

d

ðr1 Þ ¼

b~ 1 Is1

0

0

Is2

#

" and

n

d

ðr2 Þ ¼

r^ 1 Iw1 0

0

#

: b^ 1 Iw2

Algebraic properties, eigenvalues efh with respect to rd and their multiplicities corresponding to proper contrasts p0h s, f¼1,2y,6; h¼ 1,2,y,v1, of the information matrices (4) with (21) are given in (22). It can be shown that in the case 2 of the SPSB design the efficiency factors efh correspond to the following orthogonal contrasts p0h s among effects of: the A treatments (A), the test B treatments (BT), the control B treatments (BC), both groups of the test and control B treatments ðBT vs: BC Þ, the test C treatments (CT), the control C treatments (CC), both groups of the test and control C treatments (C T vs: C C ), interaction of type A  B, including: A  BT, A  BC, A  ðBT vs: BC Þ, interaction A  C, including: A  CT, A  CC, A  ðC T vs: C C Þ, interaction B  C, BT  C T ,

including:

BC  C T ,

(BT vs: BC )  ðC T vs: C C Þ, C

C

T

ðBT vs: BC Þ  C T ,

interaction C

C

A  B  C, T

T

BT  C C , including: C

C

BC  C C ,

ðBT vs: BC Þ  C C ,

A  BT  C T , T

C

A  BC  C T , T

C

BT  ðC T vs: C C Þ,

BC  ðC T vs: C C Þ,

A  ðBT vs: BC Þ  C T , T

C

A  BT  C C ,

A  B  C , A  ðB vs: B Þ  C , A  B  ðC vs: C Þ, A  B  ðC vs: C Þ, A  ðB vs: B Þ  ðC vs: C Þ. Similarly to case 1 of the SPSB design, we can describe considered properties for the case 2 of the SPSB design in Table 3.

K. Ambroz˙y, I. Mejza / Journal of Statistical Planning and Inference 142 (2012) 752–762

759

Table 3 Stratum efficiency factors of the SPSB design (case 2). Types of the contrasts A BT

Numbers of the contrasts

(2)

s1

BT vs: BC CT CC

1

C T vs: C C

1

n

1 n

xn1

1x1

1

n

T

ðs1Þðc0 1Þ s1

1

ðs1Þðrn0 1Þ s1

C

A  ðC vs: C Þ A  CC

1 1

ðs1Þrn1

BT  C T

1mn1

mn1 1

cn1 ðrn0 1Þ cn0 ðrn0 1Þ

BC  C T

1

rn0 1 cn1 rn1

ðBT vs: BC Þ  C T BT  C C

1 n

xn1 ð1mn1 Þ

ð1x1 Þð1mn1 Þ

mn1 mn1 mn1

ðc0 1Þr1

1mn1

ðBT vs: BC Þ  C C

rn1

1mn1

BT  ðC T vs: C C Þ BC  ðC T vs: C C Þ

cn1 cn0 1

1

ðBT vs: BC Þ  ðC T vs: C C Þ

1

1

A  BT  C T

ðs1Þc1 ðrn0 1Þ

A  BC  C T A  ðBT vs: BC Þ  C T

ðs1Þðc0 1Þðrn0 1Þ ðs1Þðrn0 1Þ

A  BT  C C

ðs1Þc1 rn1

C

B C

C

C

AB C

n

T

1

1 1

n

1

n

n

ð1x1 Þð1mn1 Þ

C

ðs1Þðc0 1Þrn1 ðs1Þrn1

A  B  ðC vs: C Þ A  BC  ðC T vs: C C Þ

ðs1Þðc0 1Þ s1

A  ðBT vs: BC Þ  ðC T vs: C C Þ

1mn1

mn1 mn1 mn1 1

n

ðs1Þc1

xn1 ð1mn1 Þ 1mn1

n

A  ðBT vs: BC Þ  C C T

n

n

C

(6)

mn0 ¼ 1 mn1

1mn1

ðs1Þc1

A  ðBT vs: BC Þ A  CT

(5)

1

n

A  BC

(4)

xn1 xn0 ¼ 1

1x1

rn0 1 rn1

T

(3)

1

cn1 cn0 1

BC

AB

(1)

1

n

1

(1)–(6) the numbers of the strata (see Section 2.2).

4. Application To illustrate above considerations we have taken into account an example with experimental data generated from field trials carried out in 1987–1993 in Gorzyn Experimental Station belonging to Poznan University of Live Sciences in Poland (Koziara, 1996). Three-factorial field experiments were set up in orthogonal split-plot  split-block (SPSB) designs with four replicates but different number of units in some years depending on factors considered in experiments. We applied the case 2 of the incomplete (non-orthogonal) experimental SPSB design (see Section 3.4) to analyse data of grain yield of winter triticale. Here we present efficiency of that design for defined groups of contrasts of treatment combination parameters (comparisons among main effects and among interaction effects) for three factor experiment with the following factors: A—water variant: non irrigated (A1), irrigated (A2), s¼2, B—nitrogen fertilization at levels: 0 kg N/ha (B1), 50 kg N/ha (B2), 50þ50 kg N/ha (B3), 50þ 50þ 50 kg N/ha (B4), t¼4 and C—winter triticale varieties: Bolero (C1), Malno (C2), Ugo (C3), Almo (C4), MAH—1288 (C5) and Presto (C6), w¼6. With respect to research subject the level B1 is treated as the control B treatment (t2 ¼1) while B2, B3, B4 are the test B treatments (t1 ¼ 3). Also the varieties are considered depending on yield years and so, the levels C1, C2, C3 (1988–1989) are assumed to be the control C treatments (w1 ¼3) whereas C4, C5, C6 (1990–1992) are the test C treatments (w2 ¼3). Above-mentioned treatments occur in the experiment in accordance with the incidence matrix (18) where incidence matrices (13) and (11) for factors B and C, respectively, are of the form: 2 3 1 1 1 2 3 6 7 1 1 0 61 1 17 6 7 61 0 17 61 1 17 6 7 7 Ndn2 ¼ 6 ð21Þ 7 and Ndn1 ¼ 6 61 0 07 40 1 15 6 7 6 7 40 1 05 1 1 1 0

0

1

760

K. Ambroz˙y, I. Mejza / Journal of Statistical Planning and Inference 142 (2012) 752–762

It means the A treatments (s¼2) occur in the RCB design with one block incidence matrix. The test B treatments are in the BIB design (with parameters: v^ 1 ¼ t 1 ¼ 3, b^ 1 ¼ 3, r^ 1 ¼ 2, k^ 1 ¼ 2, l ¼1, e ¼ v^ 1 ðk^ 1 1Þ=k^ 1 ðv^ 1 1Þ ¼ 3=4) which is then augmented by one control B treatment (t ¼1). Next, the test C treatments occur in the RCB design with parameters: v~ ¼ w ¼ 3, b~ ¼ 3, 1

2

1

1

r~ 1 ¼ 3, k~ 1 ¼ 3. That design is augmented by three control C treatments, each one in the separate block (w2 ¼3). From (14) and n n (12) parameters of the subdesigns d2 and d1 are in turn: vn2 ¼ t ¼ 4, vn1 ¼ w ¼ 6,

n

b2 ¼ 3,

n

k2 ¼ 3,

n

rn2 ¼ ½2103 ^30 rn1 ¼ ½3103 ^103 0 :

n

b1 ¼ 3,

k1 ¼ 4,

ð22Þ

Finally the parameters (19) of the considered SPSB design have the form v ¼ 48,

b ¼ 9,

r ¼ 12  ½2103 ^30  ½3103 ^103 0 :

k ¼ 24,

Below we present a layout of the experiment using the following abbreviation. Let fA1 ,A2 9 Bj ,:::, Bj0 9 C m ,:::, C m0 g, where 0 0 0 0 j,j ¼1,2,3,4; jaj ; m,m ¼1,2,3,4,5,6; mam , denote a block in which the A treatments are allocated to the rows, the B treatments are allocated to the columns I and the C treatments are allocated to the columns II. The layout (before randomization) of the (2  4  6)-factorial experiment arranged in the incomplete SPSB design is as follows: fA1 ,A2 9B1 ,B2 ,B4 9C 1 ,C 2 ,C 3 ,C 5 g

fA1 ,A2 9B1 ,B2 ,B4 9C 1 ,C 2 ,C 3 ,C 6 g

fA1 ,A2 9B1 ,B2 ,B4 9C 1 ,C 2 ,C 3 ,C 4 g

fA1 ,A2 9B1 ,B3 ,B4 9C 1 ,C 2 ,C 3 ,C 4 g

fA1 ,A2 9B1 ,B3 ,B4 9C 1 ,C 2 ,C 3 ,C 5 g

fA1 ,A2 9B1 ,B3 ,B4 9C 1 ,C 2 ,C 3 ,C 6 g

fA1 ,A2 9B2 ,B3 ,B4 9C 1 ,C 2 ,C 3 ,C 4 g

fA1 ,A2 9B2 ,B3 ,B4 9C 1 ,C 2 ,C 3 ,C 5 g

fA1 ,A2 9B2 ,B3 ,B4 9C 1 ,C 2 ,C 3 ,C 6 g

The eigenvalues with their multiplicities of the information matrices Cdn2 and Cdn1 (calculated with respect to ðrn2 Þd ¼ diag½2, 2, 2, 3 and ðrn1 Þd ¼ diag½3, 3, 3, 1, 1, 1, respectively) are the following: Cdn2 : x0 ¼ 1

with c0 ¼ v^ 2 ¼ 1,

xn1 ¼ ðk^ 2 þ k^ 1 eÞ=kn2 ¼ 5=6 with cn1 ¼ v^ 1 1 ¼ 2,

Cdn1 : mn0 ¼ 1

with rn0 ¼ v~ 1 ¼ 3,

and

n

n

mn1 ¼ k~ 1 =kn1 ¼ 3=4 with rn1 ¼ b~ 1 1 ¼ 2:

ð23Þ

From (23) we can easily calculate stratum efficiency factors efh of the SPSB design and make a choice of basic contrasts corresponding to them (see also Ambroz˙y and Mejza, 2004). They are generated by the eigenvectors (6) with respect to # " # " 2I3 03 3I3 03 rd ¼ I2  :  03 3 03 I3 Table 4 Stratum efficiency factors for the SPSB design in the example. Types of the contrasts

Numbers of the contrasts

(1)

(2)

(3)

(4)

(5)

(6)

A BT

1 2 1

0 1/6 0

1 0 0

0 5/6 1

0 0 0

0 0 0

0 0 0

2 2 1

0 1/4 0

0 0 0

0 0 0

1 3/4 1

0 0 0

0 0 0

2 1

0 0

1/6 0

0 0

0 0

5/6 1

0 0

A  ðC T vs: C C Þ

2 2 1

0 0 0

0 1/4 0

0 0 0

0 0 0

0 0 0

1 3/4 1

BT  C T

4

0

0

0

1

0

0

ðBT vs: BC Þ  C T

2

0

0

0

1

0

0

4

1/24

0

5/24

3/4

0

0

BT vs: BC CT CC C T vs: C C A  BT A  ðBT vs: BC Þ A  CT A  CC

T

B C

C

ðBT vs: BC Þ  C C

2

0

0

1/4

3/4

0

0

BT  ðC T vs: CCÞ

2

0

0

0

1

0

0

ðBT vs: BC Þ  ðC T vs: C C Þ

1

0

0

0

1

0

0

4

0

0

0

0

0

1

A  ðBT vs: BC Þ  CT

2

0

0

0

0

0

1

A  BT  C C

4

0

1/24

0

0

5/24

3/4

A  ðBT vs: BC Þ  CC

2

0

0

0

0

1/4

3/4

A  BT  ðC T vs: C C Þ

2

0

0

0

0

0

1

A  ðBT vs: BC Þ  ðC T vs: C C Þ

1

0

0

0

0

0

1

T

AB C

T

(1)–(6) the numbers of the strata (see Section 2.2).

K. Ambroz˙y, I. Mejza / Journal of Statistical Planning and Inference 142 (2012) 752–762

761

In the example the rd-orthonormal eigenvectors, having the form (6) for h¼24(i 1)þ 6(j 1)þm; i¼1,2; j¼ 1,2,3,4; m¼1,2,3,4,5,6, where pffiffiffi pffiffiffi a1 ¼ ½1,10 = 2, a2 ¼ ½1,10 = 2, pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi b1 ¼ ½1,1,0,00 = 4, b2 ¼ ½1,1,2,00 = 12, b3 ¼ ½1,1,1,20 = 18, b4 ¼ ½1,1,1,10 = 9, pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi c1 ¼ ½1,1,0,0,0,00 = 6, c2 ¼ ½1,1,2,0,0,00 = 18, c3 ¼ ½1,1,1,3,3,30 = 36, p ffiffiffi p ffiffiffi p ffiffiffiffiffiffi ð24Þ c4 ¼ ½0,0,0,1,1,00 = 2, c5 ¼ ½0,0,0,1,1,20 = 6, c6 ¼ ½1,1,1,1,1,10 = 12 satisfy the condition (7). Note that the eigenvectors (6) with (22), where s0h rd sv ¼ 0, hov, v¼stw¼48, define the basic 0 contrasts phs, where ph ¼rdsh. In Table 4 the efficiency of the considered incomplete design for the estimation of the basic contrasts is given (cf. the general properties given in Table 3). 5. Discussion From Table 4 it can be seen there are many contrasts estimated with efficiency factors equal to 1, e.g. in the row stratum (2)—one contrast of type A among main effects of water variant, in the column II stratum (4)—four contrasts of type BT  C T between interaction effects of test B treatments (Nitrogen fertilization) and test C treatments (Almo, MAH—1288, Presto), and so on. In these strata only we can estimate and analyse mentioned above contrasts as in a traditional SPSB design. There are some basic contrasts also which are estimated in two or three strata, e.g. four contrasts of type BT  C C between interaction effects of test B treatments (Nitrogen fertilization) and control C treatments (Bolero, Malno, Ugo). 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 these strata in which they are estimable (Calin´ski and Kageyama, 2000). 6. Conclusions Two augmented SPSB designs for three factor experiments are described here. They differ in an arrangement of control factor treatments on the experimental material (i.e. on rows or/and columns I or/and columns II in blocks). Both methods of the construction give us generally balanced incomplete SPSB designs with orthogonal block structure. From the statistical point of view these properties are very useful. They allow us to estimate efficiency factors for some sets of the contrasts before starting the experiment and then perform stratum analyses for them. Only two generating designs from a large class of augmented block designs are applied in the paper. It follows from the fact that usually control treatments are to be replicated one or more times and test treatments depending on a research problem. It is easy to widen our interest on other designs from that class, using, for instance, partially efficiency balanced designs or resolvable block designs for test treatments. For that part we could not compare both final designs (case 1 and case 2) with respect to efficiency but from (18) and (22) it is possible to note one fact. The case 2 of the SPSB design (with incomplete nested treatment only) is more efficient than the case 1 of the SPSB design (with incomplete crossed treatment structure only) for estimating some comparisons of the treatment combinations, i.e. in the case 2 more sets of the contrasts of the same type are estimated with full efficiency ( ¼1) as in a complete (traditional) SPSB design (see also Table 2).

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