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On the efficiency of some supplemented (α1,α2,…,αR)-resolvable block designs
Statistics & Probability Letters 57 (2002) 291–299
On the e ciency of some supplemented (1; 2; : : : ; R)-resolvable block designs Danuta Kachlicka, Iwona Mejza ∗ Department of Mathematical and Statistical Methods, Agricultural University of Poznan, Wojska Polskiego 28, PL-60-637 Poznan, Poland Received August 2001; received in revised form December 2001
Abstract Statistical properties of designs with two kinds of treatments: basic and supplementary, are examined. The basic treatments are arranged randomly in an (1 ; 2 ; : : : ; R )-resolvable block design. This basic design is orthogonally supplemented by some orthogonal addition of the supplementary treatments. Mixed linear models of observations following two- or three-step randomizations are considered. The 5nal design under these models is generally balanced and that allows obtaining its stratum e ciency factors for c 2002 Elsevier Science B.V. All rights reserved. both cases. Keywords: (1 ; 2 ; : : : ; R )-Resolvability; E ciency balance; General balance; Multistratum analysis; Nested block design; Orthogonal block structure; Partial e ciency balance; Supplemented block design
1. Introduction Supplemented block designs are often used in plant-breeding experiments and in many other 5elds of research with control or standard treatments. There are situations in which an experimental material for certain treatments is limited. So, such treatments (supplementary treatments, say) are treated di;erently than the rest of treatments (basic treatments, say) in the experiments. Supplemented block designs are widely described in many papers, for instance in Cali=nski and Ceranka (1974), Puri et al. (1977), Nigam et al. (1981), Nigam and Puri (1982), Kageyama (1993), etc. We assume the design for the basic treatments (called basic design) to be an (1 ; 2 ; : : : ; R )resolvable block design (cf. Cali=nski and Kageyama, 2000). There are nested block (NB) designs (cf. Morgan, 1996; Cali=nski and Kageyama, 2000) in which superblocks are divided into blocks. The blocks have constant size but the number of units may ∗
Corresponding author. Fax: +48-61-8487-146. E-mail addresses: [email protected] (D. Kachlicka), [email protected] (I. Mejza).
c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 0 6 4 - 0
292
D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299
vary from superblock to superblock. In the superblocks there are 1 ; 2 ; : : : ; R replicates of the basic treatments, respectively. The basic design is assumed to be orthogonal supplemented at the superblock level by groups of the supplementary treatments in such a way as to be proper. This property and adequate randomization of experimental units give an orthogonal block structure of the 5nal supplemented block design. Thus it may be analyzed using the techniques for multistratum experiments with orthogonal block structure (cf. Nelder, 1965). In the paper we will be considering connected e ciency balanced (EB) (1 ; 2 ; : : : ; R )-resolvable block designs in general as well as BIB (1 ; 2 ; : : : ; R )-resolvable designs as the basic designs. In both cases, the 5nal design will be a connected and partially e ciency balanced (PEB) (1 ; 2 ; : : : ; R )resolvable block design. The de5nition of EB designs can be found for instance in Cali=nski and Kageyama (2000). Especially for PEB designs, anyone can dip into Nigam and Puri (1982) and for BIB designs into Cochran and Cox (1957). The de5nition of the orthogonal block structure of a design may be found in Nelder (1965).
2. Construction of the design and its properties 2.1. The basic design Let us consider as the basic design, say D1 , any (1 ; 2 ; : : : ; R )-resolvable block design with parameters v1 ; b1 ; r1 ; k1 ; n1 which denote in turn: the number of basic treatments, the number of blocks, the number of treatment replications, the vector of block sizes and the number of observations (n1 = v1 r1 ). The blocks of D1 can be separated into R superblocks of b(1) ; b(2) ; : : : ; b(R) (¿ 1) blocks of sizes k (1) ; k (2) ; : : : ; k (R) so that k1 = [k (1) 1b(1) ; k (2) 1b(2) ; : : : ; k (R) 1b(R) ] ;
(1)
where 1x is the vector of ones of order x. Additionally, each superblock contains every basic treatment h (¿ 1) times, h = 1; 2; : : : ; R, and so 1 + 2 + · · · + R = r1 . Let D(h) and N(h) ; h = 1; 2; : : : ; R, denote the subdesign of the hth superblock and its incidence matrix, respectively. Then . . . N1 = [N(1) ..N(2) .. : : : ..N(R) ] is the incidence matrix of D1 design. Now 1 will be the multiplicity of nonzero eigenvalue 1 of matrix M0 of D1 , where M0 =
1 1 N1 k1− N1 − 1v1 1v1 r1 v1
and k1− = (k1 )−1 , k1 = diag(k (1) Ib(1) ; k (2) Ib(2) ; : : : ; k (R) Ib(R) ) and Ix is an identity matrix of order x. In the paper we assume that D1 is a (0; 1 ; 0)-EB design (cf. Cali=nski and Kageyama, 2000). It means that a complete set of 1 (=v1 − 1) orthogonal contrasts connected with the basic treatments
D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299
293
corresponds to a common e ciency factor 1 (=1 − 1 ). It can be shown that if the D(h) is also EB design for each h, then 1 =
R 1 h (h) ; r1
(2)
h=1
where
(h)
(h = 1; 2; : : : ; R) is nonzero eigenvalue with multiplicity 1 of matrix 1 1 (h) (h) M0h = N N − 1v1 1v1 h k (h) v1
of subdesign D(h) . Then, if matrix N1 is binary, D1 is any (1 ; 2 ; : : : ; R )-resolvable BIB design. 2.2. The supplemented block design Consider then design D1 that is supplemented by v2 additional treatments adding R q1 treatments to each block of D(1) , q2 treatments to each block of D(2) , and so on, with h=1 qh = v2 . The supplementary treatments are added to the blocks in an orthogonal way. Thus for the blocks of D(h) , if n(h) and n(h) addition, we will ∗ are the numbers of observations before and after the observation (h) (h) (h) (h) (h) have k = (n∗ =n )k , h = 1; 2; : : : ; R, so that k∗ = k∗ for each h. Let us put Rh=1 n(h) = n1 and R ∗(h) h=1 n∗ =n∗ , where n1 is given above and n∗ denotes the number of observations in a supplemented block design (say D∗ ). The parameters of D∗ will be the following v∗ = v1 + v2 ;
b∗ = b1 ;
The structure of design D∗ we N(1) N(2) 0 Jq1 ×b(1) Jq2 ×b(2) ND ∗ = 0 . .. . . . 0
r∗ = [r1 1v1 ; rv 2 ] ;
k∗ = k∗ 1b∗ ;
n∗ = b∗ k∗ = 1v∗ r∗ :
can describe by the incidence matrix · · · N(R) ··· 0 ··· 0 ; . .. .. .
0
(3)
· · · JqR ×b(R)
where Jx1 ×x2 = 1x1 1x2 . For design D∗ we have the matrix 1 1 M0∗ = r∗− ND∗ ND ∗ − 1v r ; k∗ n∗ ∗ ∗ −1 ∗ where r∗− = (r ∗ ) ∗ and r∗ = diag(r1 ; r2 ; : : : ; rv∗ ). This matrix has eigenvalues i with∗multiplicities ∗
i such that i i = v∗ − 1, exception taken of the null eigenvalue arising from M0 1v∗ = 0. The distinct eigenvalues are ordered increasingly, i.e., as 0 = 0∗ ¡ 1∗ ¡ · · · ¡ m∗ ∗−1 ¡ 1.
Theorem 1. The design D∗ with incidence matrix given in (3) belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; : : : ;
∗m∗ −1 ; 0)-EB designs with parameters: 0∗ = 0;
∗0 = 1 + v2 − R;
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D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299
1∗ =
R 1 (h) k h (h) ; r1 k∗
∗1 = v1 − 1;
h=1
∗
∗
2 ; 3 ; : : : ; m∗ ∗ −1 ;
∗2 ; ∗3 ; : : : ; ∗m∗ −1 :
(4)
where ∗ m −1
i=2
R 1
i i = qh (r1 − h ) r1 k∗
∗ ∗
and
∗ m −1
∗i = R − 1:
i=2
h=1
It follows from the way of the orthogonal supplementation of the superblocks in design D1 and from the algebraic properties of both designs, before and after the observation addition. Remark 1. Let L ∗ = [1=(R − 1)]
m∗ −1 i=2
∗i i∗ . When 1 = 2 = · · · = R (=); then r1 = R and
R 1 v2 qh = : L = Rk∗ Rk∗
∗
(5)
h=1
Remark 2. Additionally; from q1 = q2 = · · · = qR = q we get v2 = Rq. Then formula (5) reduces to L ∗ = 2∗ = q=k∗ corresponding to ∗2 = R − 1 (cf. Kachlicka and Mejza; 2000). In this case D∗ belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; 0)-EB designs; so the number of e ciency classes reduces. In other words, D∗ is a PEB design with m∗ e ciency classes (PEB(m∗ )). It means that all ∗i orthogonal contrasts (and their functions) from the ith class, i = 0; 1; : : : ; m∗ − 1, are estimable in the so-called intra-block analysis with e ciency factor equal to i∗ = 1 − i∗ . Furthermore, L∗ = 1 − L ∗ will be called “the average e ciency factor” for the contrasts from all the ith classes with i ¿ 1.
3. Examples of constructions Example 1. Consider a supplemented 1 1 1 0 0 1 (1) (2) N N 0 0 ND∗ = J2×6 02×4 = 01×6 J1×4 1 1 1 1 0 0
block design D∗ with incidence matrix: 1 0 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 : 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1
(6)
D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299
295
Thus the N(h) ; h = 1; 2; are incidence matrices of BIB designs with parameters: for D(1) :
v1 = 4;
b(1) = 6;
for D(2) :
v1 = 4; b(2) = 4;
k (1) = 2;
1 = 3; 2 = 3;
k (2) = 3;
n(1) = 12; n(2) = 12;
(1) = 1=3; (2) = 1=9;
1 = 3;
1 = 3:
. Hence; D1 with incidence matrix N1 = [N(1) ..N(2) ] is 3-resolvable BIB design and its parameters are as follows: v1 = 4;
b = 10;
R = 2;
2 1 1 = h (h) = 2=9; r1
r1 = 6;
k = [216 ; 314 ] ;
n1 = 24;
1 = v1 − 1 = 3:
h=1
Design D1 was supplemented by three (v2 = 3) additional treatments in such a way that two treatments (q1 = 2) were added to each block of D(1) and the third supplementary treatment (q2 = 1) to each block of D(2) . It may be shown that D∗ belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; 0)-EB designs with incidence matrix (6) and parameters: v∗ = 7;
b∗ = 10;
0∗ = 0;
∗0 = 2;
. r∗ = [614 ..[6; 6; 4]] ; 1 1∗ = ; 8
∗1 = 3;
k∗ = 4; 3 2∗ = ; 8
n∗ = 40
∗2 = 1:
It means that in the intra-block analysis from all orthogonal contrasts, two are estimated with full e ciency (0∗ = 1 − 0∗ = 1), three of them with e ciency factor 1∗ (=1 − 1∗ = 7=8), and one contrast with e ciency factor 2∗ (=1 − 2∗ = 5=8). One may expect the D∗ design to be orthogonal with respect to the two orthogonal contrasts de5ned by the vectors with coe cients [0; 0; 0; 0; 1; −1; 0] and [1; 1; 1; 1; −1; −1; −2] which correspond to the comparison of the supplementary treatment e;ects from the 5rst superblock and the comparison between the group of the basic treatment e;ects and the group of the supplementary treatment e;ects, respectively. Moreover, D∗ is PEB with respect to the remaining orthogonal contrasts. Example 2. We now consider a D∗ design with the following incidence matrix: (1) N N(2) N(3) J4×7 04×7 04×7 ∗ ND = 01×7 J1×7 01×7 04×7 04×7 J4×7
(7)
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with
1 1 0 (1) (3) N =N = 1 0 0 0
0 1 1 0 1 0 0
0 0 1 1 0 1 0
0 0 0 1 1 0 1
1 0 0 0 1 1 0
1 0 1 0 ; 0 0 1
0 1 0 0 0 1 1
1 1 1 (2) N = 1 1 1 0
1 1 1 1 1 0 1
1 1 1 1 0 1 1
1 1 1 0 1 1 1
1 1 0 1 1 1 1
1 0 1 1 1 1 1
0 1 1 1 : 1 1 1
N(1) ; N(2) and N(3) are incidence matrices of BIB designs with parameters, respectively: for D(1) :
v1 = 7;
b(1) = 7;
1 = 3;
k (1) = 3;
n(1) = 21;
for D(2) :
v1 = 7;
b(2) = 7;
2 = 6;
k (2) = 6; n(2) = 42;
for D(3) :
v1 = 7;
b(3) = 7;
3 = 3;
(1) = 2=9;
1 = 6;
(2) = 1=36;
1 = 6:
n(3) = 21; (3) = 2=9; 1 = 6: . . It can be seen that D1 with incidence matrix N1 = [N(1) ..N(2) ..N(3) ] is (3; 6; 3)-resolvable design belonging to the class of (0; 1 ; 0)-EB designs with parameters: v1 = 7;
b = 21;
R = 3;
3 1 h (h) = 1=8; 1 = r1
r1 = 12;
k (3) = 3;
k = [317 ; 617 ; 317 ] ;
n1 = 84;
1 = v1 − 1 = 6:
h=1
Adding to each block of D(1) four (q1 = 4) supplementary treatments, to each block of D(2) one (q2 = 1) other treatment and to each block of D(3) four (q3 = 4) also di;erent treatments, D∗ design is obtained with incidence matrix (6). It belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; ∗3 ; 0)-EB designs with parameters: . v∗ = 16; b∗ = 21; r∗ = [1217 ..719 ] ; k∗ = 7; n∗ = 147; 0∗ = 0;
∗0 = 7;
It can be shown that 3 i=2
∗i i∗ =
1∗ =
5 ; 84
∗1 = 6;
2∗ =
5 ; 14
∗2 = 1;
4 3∗ = ; 7
∗3 = 1:
3 3 1 13 and qh (r1 − h ) =
∗i = R − 1 = 2: r1 k∗ 14 i=2 h=1
One may expect that all orthogonal contrasts will be estimated in the intra-block analysis. Seven of them will be estimated with full e ciency (0∗ = 1), from these six correspond to comparisons among additional treatment e;ects inside the 5rst superblock and the third superblock and one contrast concerns comparison between the two groups of basic and supplementary treatments. Besides this, six contrasts containing basic treatments will be estimated with e ciency factor 1∗ (=79=84). However, the two remaining contrasts among supplementary treatment e;ects from di;erent superblocks will be estimated with two di;erent e ciency factors equal to 2∗ (=9=14) and 3∗ (=3=7). The average e ciency of D∗ design with respect to this group of last contrasts is equal to L∗ (=15=28).
D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299
297
4. Remarks on the analysis Since units have to be randomized before they enter the experiment, randomization models with three or four strata are suitable. The model with three strata, say model I, is obtained when the two-step randomization (blocks → plots) is performed in the experiment, while the three-step randomization (superblocks → blocks → plots) leads to the model with four strata, say model II. In both models we have a zero stratum (0) generated by the vector of ones, intra-block stratum (1) and inter-block stratum (2). Additionally, in model II, the inter-superblock stratum (say, 3) appears. The choice of the model depends actually on the number of sources of variation that should be controlled in an experiment. However, it should be underlined, that model II can be adapted only when each superblock has the same number of blocks, i.e. b(1) = b(2) = · · · = b(R) = b∗ =R. Then the superblocks can have number of plots equal to K = (b∗ =R)k∗ . This condition need not be ful5lled in the 5rst model. From the T -step (T = 2 or 3) randomization of units performed in the experiment, and the assumption of additivity between treatments and plots, follows a mixed linear model for the (n∗ × 1) vector of observations y (cf. Mejza and Mejza, 1989; Cali=nski and Kageyama, 2000). This model is of the form: E(y) = P ; Cov(y) = V( ), where P (n∗ × v∗ ) is the design matrix for v∗ treatments, and (v∗ × 1) is the vector of 5xed treatment e;ects. According to the orthogonal block structure of the design, we have dispersion matrix V( ) = Tf=0 f Pf where f ¿ 0 and the {Pf } are a family of known pairwise orthogonal projectors adding up to the identity matrix (cf. Houtman and Speed, 1983). The range space R{Pf } of Pf ; 0 6 f 6 T , is termed the fth stratum of the model and the { f } are unknown strata variances. One may check that statistical properties of D∗ design are strictly connected with algebraic properties of some information matrices for the treatment contrasts (except matrix A0 which is associated with the treatment mean) associated with the strata. These matrices are as follows: for model I: A1 = r∗ − k∗−1 ND∗ ND ∗ ;
r(A1 ) = v∗ − 1;
1 A2 = k∗−1 ND∗ ND ∗ − n− ∗ r∗ r ∗ ;
r(A2 ) = v1 + R − 2;
(8)
for model II: A1 = r∗ − k∗−1 ND∗ ND ∗ ;
r(A1 ) = v∗ − 1;
A2 = k∗−1 ND∗ ND ∗ − K −1 NR NR ; 1 A3 = K −1 NR NR − n− ∗ r∗ r ∗ ;
r(A2 ) = v1 − 1;
r(A3 ) = R − 1;
(9)
where ND∗ is the (treatments vs. blocks) incidence matrix given by (3) and NR is the (treatments vs. superblocks) incidence matrix given by NR = [r∗(1) ; r∗(2) ; : : : ; r∗(R) ], where r∗(h) ; h = 1; 2; : : : ; R, is the replication vector for the basic treatments and the supplementary treatments in the hth superblock. We point out that matrices A1 and A2 , as well in (8) as in (9), are connected with analyses in the intra-block and inter-block strata, respectively. However, matrix A3 appears in the inter-superblock stratum only. One may check that D∗ design is generally balanced, i.e. the information matrices mutually commute with respect to r− in each model. It means that the same orthogonal contrasts are estimable
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D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299 Table 1 Stratum e ciency factors in model I Type
Number of contrasts
Intra-block stratum
Inter-block stratum
A B C
v1 − 1 v2 − R
∗2 ; ∗3 ; : : : ; ∗m∗ −1 or R − 1 1
1 − 1∗ 1 1 − 2∗ ; : : : ; 1 − m∗ ∗ −1 or 1 − L∗ 1
1∗ — 2∗ ; : : : ; m∗ ∗ −1 or L∗ —
D
Table 2 Stratum e ciency factors in model II Type
Number of contrasts
Intra-block stratum
Inter-block stratum
Inter-superblock stratum
A B C
v1 − 1 v2 − R
∗2 ; ∗3 ; : : : ; ∗m∗ −1 or R − 1 1
1 − 1∗ 1 1 − 2∗ ; : : : ; 1 − m∗ ∗ −1 or 1 − L∗ 1
1∗ — —
— — 2∗ ; : : : ; m∗ ∗ −1 or L∗ —
D
—
in one or more strata of the model (cf. Houtman and Speed, 1983; Mejza, 1992). Those contrasts can be divided into four types, say, A, B, C and D. A type relates to comparisons among e;ects of basic treatments, B type—among e;ects of the supplementary treatments within each superblock and C type—among e;ects of the supplementary treatments from di;erent superblocks. Lastly, D type corresponds to comparison between group of the basic treatments and group of the supplementary treatments. One can expect that in D∗ contrasts of type A will belong to one e ciency class, as well as contrasts of types B and D. But contrasts of type C can belong to di;erent e ciency classes. Information about orthogonal contrasts in the strata and their e ciency factors for both models are shown in Tables 1 and 2. Statistical analysis of the considered design under both models can be mainly performed in the intra-block stratum, since all orthogonal contrasts are estimable in that stratum. Thus all hypotheses, both general and particular, for treatments are testable in that stratum. For these contrasts that are estimable in two strata, methods for combining information may be used (cf. Nelder, 1968; Mejza, 1978; Cali=nski and Kageyama, 2000). Acknowledgements The work was partially supported by grant KBN 3 P06A 017 22. References Cali=nski, T., Ceranka, B., 1974. Supplemented block designs. Biometrical J. 16, 299–305. Cali=nski, T., Kageyama, S., 2000. Block Designs: A Randomization Approach, Vol. I: Analysis. Lecture Notes in Statistics, Vol. 150. Springer, New York.
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Cochran, W.G., Cox, G.M., 1957. Experimental Designs. Wiley, New York. Houtman, A.M., Speed, T.P., 1983. Balance in designed experiments with orthogonal block structure. Ann. Statist 11, 1069–1085. Kachlicka, D., Mejza, I., 2000. Supplemented -resolvable block designs. Bibl. Fragmenta Agron., T. 7, 59 –76 (in Polish). Kageyama, S., 1993. The family of block designs with some combinatorial properties. Discrete Math. 116, 17–54. Mejza, I., Mejza, S., 1989. A note on model building in block designs. Biul. Oceny Odmian 21–22, 187–201. Mejza, S., 1978. Use of inter-block information to obtain uniformly better estimators of treatment contrasts. Math. Operationsforsch. Statist. Ser. Statistics 9, 335–341. Mejza, S., 1992. On some aspects of general balance in designed experiments. Statistica LII 2, 264–278. Morgan, J.P., 1996. Nested designs. In: Ghosh, S., Rao, C.R. (Eds.), Handbook of Statistics, Vol. 13: Design and Analysis of Experiments. Elsevier, Amsterdam, pp. 939–976. Nelder, J.A., 1965. The analysis of randomized experiments with orthogonal block structure. Proc. Roy. Soc. London Ser. A 283, 147–178. Nelder, J.A., 1968. The combination of information in generally balanced designs. J. Royal Statist. Soc. Ser. B 30, 303–311. Nigam, A.K., Puri, P.D., 1982. On partially e ciency balanced designs—II. Commun. Statist. Theory Methods 11 (24), 2817–2830. Nigam, A.K., Gupta, V.K., Narain, P., 1981. Monograph on supplemented block designs. Indian Agricultural Statistics Research Institute (ICAR), Libr. Avenue, New Delhi, India. Puri, P.D., Nigam, A.K., Narain, P., 1977. Supplemented block designs. Sankhya B 39, 189–195.
On the e ciency of some supplemented (1; 2; : : : ; R)-resolvable block designs Danuta Kachlicka, Iwona Mejza ∗ Department of Mathematical and Statistical Methods, Agricultural University of Poznan, Wojska Polskiego 28, PL-60-637 Poznan, Poland Received August 2001; received in revised form December 2001
Abstract Statistical properties of designs with two kinds of treatments: basic and supplementary, are examined. The basic treatments are arranged randomly in an (1 ; 2 ; : : : ; R )-resolvable block design. This basic design is orthogonally supplemented by some orthogonal addition of the supplementary treatments. Mixed linear models of observations following two- or three-step randomizations are considered. The 5nal design under these models is generally balanced and that allows obtaining its stratum e ciency factors for c 2002 Elsevier Science B.V. All rights reserved. both cases. Keywords: (1 ; 2 ; : : : ; R )-Resolvability; E ciency balance; General balance; Multistratum analysis; Nested block design; Orthogonal block structure; Partial e ciency balance; Supplemented block design
1. Introduction Supplemented block designs are often used in plant-breeding experiments and in many other 5elds of research with control or standard treatments. There are situations in which an experimental material for certain treatments is limited. So, such treatments (supplementary treatments, say) are treated di;erently than the rest of treatments (basic treatments, say) in the experiments. Supplemented block designs are widely described in many papers, for instance in Cali=nski and Ceranka (1974), Puri et al. (1977), Nigam et al. (1981), Nigam and Puri (1982), Kageyama (1993), etc. We assume the design for the basic treatments (called basic design) to be an (1 ; 2 ; : : : ; R )resolvable block design (cf. Cali=nski and Kageyama, 2000). There are nested block (NB) designs (cf. Morgan, 1996; Cali=nski and Kageyama, 2000) in which superblocks are divided into blocks. The blocks have constant size but the number of units may ∗
Corresponding author. Fax: +48-61-8487-146. E-mail addresses: [email protected] (D. Kachlicka), [email protected] (I. Mejza).
c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 0 6 4 - 0
292
D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299
vary from superblock to superblock. In the superblocks there are 1 ; 2 ; : : : ; R replicates of the basic treatments, respectively. The basic design is assumed to be orthogonal supplemented at the superblock level by groups of the supplementary treatments in such a way as to be proper. This property and adequate randomization of experimental units give an orthogonal block structure of the 5nal supplemented block design. Thus it may be analyzed using the techniques for multistratum experiments with orthogonal block structure (cf. Nelder, 1965). In the paper we will be considering connected e ciency balanced (EB) (1 ; 2 ; : : : ; R )-resolvable block designs in general as well as BIB (1 ; 2 ; : : : ; R )-resolvable designs as the basic designs. In both cases, the 5nal design will be a connected and partially e ciency balanced (PEB) (1 ; 2 ; : : : ; R )resolvable block design. The de5nition of EB designs can be found for instance in Cali=nski and Kageyama (2000). Especially for PEB designs, anyone can dip into Nigam and Puri (1982) and for BIB designs into Cochran and Cox (1957). The de5nition of the orthogonal block structure of a design may be found in Nelder (1965).
2. Construction of the design and its properties 2.1. The basic design Let us consider as the basic design, say D1 , any (1 ; 2 ; : : : ; R )-resolvable block design with parameters v1 ; b1 ; r1 ; k1 ; n1 which denote in turn: the number of basic treatments, the number of blocks, the number of treatment replications, the vector of block sizes and the number of observations (n1 = v1 r1 ). The blocks of D1 can be separated into R superblocks of b(1) ; b(2) ; : : : ; b(R) (¿ 1) blocks of sizes k (1) ; k (2) ; : : : ; k (R) so that k1 = [k (1) 1b(1) ; k (2) 1b(2) ; : : : ; k (R) 1b(R) ] ;
(1)
where 1x is the vector of ones of order x. Additionally, each superblock contains every basic treatment h (¿ 1) times, h = 1; 2; : : : ; R, and so 1 + 2 + · · · + R = r1 . Let D(h) and N(h) ; h = 1; 2; : : : ; R, denote the subdesign of the hth superblock and its incidence matrix, respectively. Then . . . N1 = [N(1) ..N(2) .. : : : ..N(R) ] is the incidence matrix of D1 design. Now 1 will be the multiplicity of nonzero eigenvalue 1 of matrix M0 of D1 , where M0 =
1 1 N1 k1− N1 − 1v1 1v1 r1 v1
and k1− = (k1 )−1 , k1 = diag(k (1) Ib(1) ; k (2) Ib(2) ; : : : ; k (R) Ib(R) ) and Ix is an identity matrix of order x. In the paper we assume that D1 is a (0; 1 ; 0)-EB design (cf. Cali=nski and Kageyama, 2000). It means that a complete set of 1 (=v1 − 1) orthogonal contrasts connected with the basic treatments
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293
corresponds to a common e ciency factor 1 (=1 − 1 ). It can be shown that if the D(h) is also EB design for each h, then 1 =
R 1 h (h) ; r1
(2)
h=1
where
(h)
(h = 1; 2; : : : ; R) is nonzero eigenvalue with multiplicity 1 of matrix 1 1 (h) (h) M0h = N N − 1v1 1v1 h k (h) v1
of subdesign D(h) . Then, if matrix N1 is binary, D1 is any (1 ; 2 ; : : : ; R )-resolvable BIB design. 2.2. The supplemented block design Consider then design D1 that is supplemented by v2 additional treatments adding R q1 treatments to each block of D(1) , q2 treatments to each block of D(2) , and so on, with h=1 qh = v2 . The supplementary treatments are added to the blocks in an orthogonal way. Thus for the blocks of D(h) , if n(h) and n(h) addition, we will ∗ are the numbers of observations before and after the observation (h) (h) (h) (h) (h) have k = (n∗ =n )k , h = 1; 2; : : : ; R, so that k∗ = k∗ for each h. Let us put Rh=1 n(h) = n1 and R ∗(h) h=1 n∗ =n∗ , where n1 is given above and n∗ denotes the number of observations in a supplemented block design (say D∗ ). The parameters of D∗ will be the following v∗ = v1 + v2 ;
b∗ = b1 ;
The structure of design D∗ we N(1) N(2) 0 Jq1 ×b(1) Jq2 ×b(2) ND ∗ = 0 . .. . . . 0
r∗ = [r1 1v1 ; rv 2 ] ;
k∗ = k∗ 1b∗ ;
n∗ = b∗ k∗ = 1v∗ r∗ :
can describe by the incidence matrix · · · N(R) ··· 0 ··· 0 ; . .. .. .
0
(3)
· · · JqR ×b(R)
where Jx1 ×x2 = 1x1 1x2 . For design D∗ we have the matrix 1 1 M0∗ = r∗− ND∗ ND ∗ − 1v r ; k∗ n∗ ∗ ∗ −1 ∗ where r∗− = (r ∗ ) ∗ and r∗ = diag(r1 ; r2 ; : : : ; rv∗ ). This matrix has eigenvalues i with∗multiplicities ∗
i such that i i = v∗ − 1, exception taken of the null eigenvalue arising from M0 1v∗ = 0. The distinct eigenvalues are ordered increasingly, i.e., as 0 = 0∗ ¡ 1∗ ¡ · · · ¡ m∗ ∗−1 ¡ 1.
Theorem 1. The design D∗ with incidence matrix given in (3) belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; : : : ;
∗m∗ −1 ; 0)-EB designs with parameters: 0∗ = 0;
∗0 = 1 + v2 − R;
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1∗ =
R 1 (h) k h (h) ; r1 k∗
∗1 = v1 − 1;
h=1
∗
∗
2 ; 3 ; : : : ; m∗ ∗ −1 ;
∗2 ; ∗3 ; : : : ; ∗m∗ −1 :
(4)
where ∗ m −1
i=2
R 1
i i = qh (r1 − h ) r1 k∗
∗ ∗
and
∗ m −1
∗i = R − 1:
i=2
h=1
It follows from the way of the orthogonal supplementation of the superblocks in design D1 and from the algebraic properties of both designs, before and after the observation addition. Remark 1. Let L ∗ = [1=(R − 1)]
m∗ −1 i=2
∗i i∗ . When 1 = 2 = · · · = R (=); then r1 = R and
R 1 v2 qh = : L = Rk∗ Rk∗
∗
(5)
h=1
Remark 2. Additionally; from q1 = q2 = · · · = qR = q we get v2 = Rq. Then formula (5) reduces to L ∗ = 2∗ = q=k∗ corresponding to ∗2 = R − 1 (cf. Kachlicka and Mejza; 2000). In this case D∗ belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; 0)-EB designs; so the number of e ciency classes reduces. In other words, D∗ is a PEB design with m∗ e ciency classes (PEB(m∗ )). It means that all ∗i orthogonal contrasts (and their functions) from the ith class, i = 0; 1; : : : ; m∗ − 1, are estimable in the so-called intra-block analysis with e ciency factor equal to i∗ = 1 − i∗ . Furthermore, L∗ = 1 − L ∗ will be called “the average e ciency factor” for the contrasts from all the ith classes with i ¿ 1.
3. Examples of constructions Example 1. Consider a supplemented 1 1 1 0 0 1 (1) (2) N N 0 0 ND∗ = J2×6 02×4 = 01×6 J1×4 1 1 1 1 0 0
block design D∗ with incidence matrix: 1 0 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 : 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1
(6)
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Thus the N(h) ; h = 1; 2; are incidence matrices of BIB designs with parameters: for D(1) :
v1 = 4;
b(1) = 6;
for D(2) :
v1 = 4; b(2) = 4;
k (1) = 2;
1 = 3; 2 = 3;
k (2) = 3;
n(1) = 12; n(2) = 12;
(1) = 1=3; (2) = 1=9;
1 = 3;
1 = 3:
. Hence; D1 with incidence matrix N1 = [N(1) ..N(2) ] is 3-resolvable BIB design and its parameters are as follows: v1 = 4;
b = 10;
R = 2;
2 1 1 = h (h) = 2=9; r1
r1 = 6;
k = [216 ; 314 ] ;
n1 = 24;
1 = v1 − 1 = 3:
h=1
Design D1 was supplemented by three (v2 = 3) additional treatments in such a way that two treatments (q1 = 2) were added to each block of D(1) and the third supplementary treatment (q2 = 1) to each block of D(2) . It may be shown that D∗ belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; 0)-EB designs with incidence matrix (6) and parameters: v∗ = 7;
b∗ = 10;
0∗ = 0;
∗0 = 2;
. r∗ = [614 ..[6; 6; 4]] ; 1 1∗ = ; 8
∗1 = 3;
k∗ = 4; 3 2∗ = ; 8
n∗ = 40
∗2 = 1:
It means that in the intra-block analysis from all orthogonal contrasts, two are estimated with full e ciency (0∗ = 1 − 0∗ = 1), three of them with e ciency factor 1∗ (=1 − 1∗ = 7=8), and one contrast with e ciency factor 2∗ (=1 − 2∗ = 5=8). One may expect the D∗ design to be orthogonal with respect to the two orthogonal contrasts de5ned by the vectors with coe cients [0; 0; 0; 0; 1; −1; 0] and [1; 1; 1; 1; −1; −1; −2] which correspond to the comparison of the supplementary treatment e;ects from the 5rst superblock and the comparison between the group of the basic treatment e;ects and the group of the supplementary treatment e;ects, respectively. Moreover, D∗ is PEB with respect to the remaining orthogonal contrasts. Example 2. We now consider a D∗ design with the following incidence matrix: (1) N N(2) N(3) J4×7 04×7 04×7 ∗ ND = 01×7 J1×7 01×7 04×7 04×7 J4×7
(7)
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with
1 1 0 (1) (3) N =N = 1 0 0 0
0 1 1 0 1 0 0
0 0 1 1 0 1 0
0 0 0 1 1 0 1
1 0 0 0 1 1 0
1 0 1 0 ; 0 0 1
0 1 0 0 0 1 1
1 1 1 (2) N = 1 1 1 0
1 1 1 1 1 0 1
1 1 1 1 0 1 1
1 1 1 0 1 1 1
1 1 0 1 1 1 1
1 0 1 1 1 1 1
0 1 1 1 : 1 1 1
N(1) ; N(2) and N(3) are incidence matrices of BIB designs with parameters, respectively: for D(1) :
v1 = 7;
b(1) = 7;
1 = 3;
k (1) = 3;
n(1) = 21;
for D(2) :
v1 = 7;
b(2) = 7;
2 = 6;
k (2) = 6; n(2) = 42;
for D(3) :
v1 = 7;
b(3) = 7;
3 = 3;
(1) = 2=9;
1 = 6;
(2) = 1=36;
1 = 6:
n(3) = 21; (3) = 2=9; 1 = 6: . . It can be seen that D1 with incidence matrix N1 = [N(1) ..N(2) ..N(3) ] is (3; 6; 3)-resolvable design belonging to the class of (0; 1 ; 0)-EB designs with parameters: v1 = 7;
b = 21;
R = 3;
3 1 h (h) = 1=8; 1 = r1
r1 = 12;
k (3) = 3;
k = [317 ; 617 ; 317 ] ;
n1 = 84;
1 = v1 − 1 = 6:
h=1
Adding to each block of D(1) four (q1 = 4) supplementary treatments, to each block of D(2) one (q2 = 1) other treatment and to each block of D(3) four (q3 = 4) also di;erent treatments, D∗ design is obtained with incidence matrix (6). It belongs to the class of ( ∗0 ; ∗1 ; ∗2 ; ∗3 ; 0)-EB designs with parameters: . v∗ = 16; b∗ = 21; r∗ = [1217 ..719 ] ; k∗ = 7; n∗ = 147; 0∗ = 0;
∗0 = 7;
It can be shown that 3 i=2
∗i i∗ =
1∗ =
5 ; 84
∗1 = 6;
2∗ =
5 ; 14
∗2 = 1;
4 3∗ = ; 7
∗3 = 1:
3 3 1 13 and qh (r1 − h ) =
∗i = R − 1 = 2: r1 k∗ 14 i=2 h=1
One may expect that all orthogonal contrasts will be estimated in the intra-block analysis. Seven of them will be estimated with full e ciency (0∗ = 1), from these six correspond to comparisons among additional treatment e;ects inside the 5rst superblock and the third superblock and one contrast concerns comparison between the two groups of basic and supplementary treatments. Besides this, six contrasts containing basic treatments will be estimated with e ciency factor 1∗ (=79=84). However, the two remaining contrasts among supplementary treatment e;ects from di;erent superblocks will be estimated with two di;erent e ciency factors equal to 2∗ (=9=14) and 3∗ (=3=7). The average e ciency of D∗ design with respect to this group of last contrasts is equal to L∗ (=15=28).
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4. Remarks on the analysis Since units have to be randomized before they enter the experiment, randomization models with three or four strata are suitable. The model with three strata, say model I, is obtained when the two-step randomization (blocks → plots) is performed in the experiment, while the three-step randomization (superblocks → blocks → plots) leads to the model with four strata, say model II. In both models we have a zero stratum (0) generated by the vector of ones, intra-block stratum (1) and inter-block stratum (2). Additionally, in model II, the inter-superblock stratum (say, 3) appears. The choice of the model depends actually on the number of sources of variation that should be controlled in an experiment. However, it should be underlined, that model II can be adapted only when each superblock has the same number of blocks, i.e. b(1) = b(2) = · · · = b(R) = b∗ =R. Then the superblocks can have number of plots equal to K = (b∗ =R)k∗ . This condition need not be ful5lled in the 5rst model. From the T -step (T = 2 or 3) randomization of units performed in the experiment, and the assumption of additivity between treatments and plots, follows a mixed linear model for the (n∗ × 1) vector of observations y (cf. Mejza and Mejza, 1989; Cali=nski and Kageyama, 2000). This model is of the form: E(y) = P ; Cov(y) = V( ), where P (n∗ × v∗ ) is the design matrix for v∗ treatments, and (v∗ × 1) is the vector of 5xed treatment e;ects. According to the orthogonal block structure of the design, we have dispersion matrix V( ) = Tf=0 f Pf where f ¿ 0 and the {Pf } are a family of known pairwise orthogonal projectors adding up to the identity matrix (cf. Houtman and Speed, 1983). The range space R{Pf } of Pf ; 0 6 f 6 T , is termed the fth stratum of the model and the { f } are unknown strata variances. One may check that statistical properties of D∗ design are strictly connected with algebraic properties of some information matrices for the treatment contrasts (except matrix A0 which is associated with the treatment mean) associated with the strata. These matrices are as follows: for model I: A1 = r∗ − k∗−1 ND∗ ND ∗ ;
r(A1 ) = v∗ − 1;
1 A2 = k∗−1 ND∗ ND ∗ − n− ∗ r∗ r ∗ ;
r(A2 ) = v1 + R − 2;
(8)
for model II: A1 = r∗ − k∗−1 ND∗ ND ∗ ;
r(A1 ) = v∗ − 1;
A2 = k∗−1 ND∗ ND ∗ − K −1 NR NR ; 1 A3 = K −1 NR NR − n− ∗ r∗ r ∗ ;
r(A2 ) = v1 − 1;
r(A3 ) = R − 1;
(9)
where ND∗ is the (treatments vs. blocks) incidence matrix given by (3) and NR is the (treatments vs. superblocks) incidence matrix given by NR = [r∗(1) ; r∗(2) ; : : : ; r∗(R) ], where r∗(h) ; h = 1; 2; : : : ; R, is the replication vector for the basic treatments and the supplementary treatments in the hth superblock. We point out that matrices A1 and A2 , as well in (8) as in (9), are connected with analyses in the intra-block and inter-block strata, respectively. However, matrix A3 appears in the inter-superblock stratum only. One may check that D∗ design is generally balanced, i.e. the information matrices mutually commute with respect to r− in each model. It means that the same orthogonal contrasts are estimable
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D. Kachlicka, I. Mejza / Statistics & Probability Letters 57 (2002) 291–299 Table 1 Stratum e ciency factors in model I Type
Number of contrasts
Intra-block stratum
Inter-block stratum
A B C
v1 − 1 v2 − R
∗2 ; ∗3 ; : : : ; ∗m∗ −1 or R − 1 1
1 − 1∗ 1 1 − 2∗ ; : : : ; 1 − m∗ ∗ −1 or 1 − L∗ 1
1∗ — 2∗ ; : : : ; m∗ ∗ −1 or L∗ —
D
Table 2 Stratum e ciency factors in model II Type
Number of contrasts
Intra-block stratum
Inter-block stratum
Inter-superblock stratum
A B C
v1 − 1 v2 − R
∗2 ; ∗3 ; : : : ; ∗m∗ −1 or R − 1 1
1 − 1∗ 1 1 − 2∗ ; : : : ; 1 − m∗ ∗ −1 or 1 − L∗ 1
1∗ — —
— — 2∗ ; : : : ; m∗ ∗ −1 or L∗ —
D
—
in one or more strata of the model (cf. Houtman and Speed, 1983; Mejza, 1992). Those contrasts can be divided into four types, say, A, B, C and D. A type relates to comparisons among e;ects of basic treatments, B type—among e;ects of the supplementary treatments within each superblock and C type—among e;ects of the supplementary treatments from di;erent superblocks. Lastly, D type corresponds to comparison between group of the basic treatments and group of the supplementary treatments. One can expect that in D∗ contrasts of type A will belong to one e ciency class, as well as contrasts of types B and D. But contrasts of type C can belong to di;erent e ciency classes. Information about orthogonal contrasts in the strata and their e ciency factors for both models are shown in Tables 1 and 2. Statistical analysis of the considered design under both models can be mainly performed in the intra-block stratum, since all orthogonal contrasts are estimable in that stratum. Thus all hypotheses, both general and particular, for treatments are testable in that stratum. For these contrasts that are estimable in two strata, methods for combining information may be used (cf. Nelder, 1968; Mejza, 1978; Cali=nski and Kageyama, 2000). Acknowledgements The work was partially supported by grant KBN 3 P06A 017 22. References Cali=nski, T., Ceranka, B., 1974. Supplemented block designs. Biometrical J. 16, 299–305. Cali=nski, T., Kageyama, S., 2000. Block Designs: A Randomization Approach, Vol. I: Analysis. Lecture Notes in Statistics, Vol. 150. Springer, New York.
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Cochran, W.G., Cox, G.M., 1957. Experimental Designs. Wiley, New York. Houtman, A.M., Speed, T.P., 1983. Balance in designed experiments with orthogonal block structure. Ann. Statist 11, 1069–1085. Kachlicka, D., Mejza, I., 2000. Supplemented -resolvable block designs. Bibl. Fragmenta Agron., T. 7, 59 –76 (in Polish). Kageyama, S., 1993. The family of block designs with some combinatorial properties. Discrete Math. 116, 17–54. Mejza, I., Mejza, S., 1989. A note on model building in block designs. Biul. Oceny Odmian 21–22, 187–201. Mejza, S., 1978. Use of inter-block information to obtain uniformly better estimators of treatment contrasts. Math. Operationsforsch. Statist. Ser. Statistics 9, 335–341. Mejza, S., 1992. On some aspects of general balance in designed experiments. Statistica LII 2, 264–278. Morgan, J.P., 1996. Nested designs. In: Ghosh, S., Rao, C.R. (Eds.), Handbook of Statistics, Vol. 13: Design and Analysis of Experiments. Elsevier, Amsterdam, pp. 939–976. Nelder, J.A., 1965. The analysis of randomized experiments with orthogonal block structure. Proc. Roy. Soc. London Ser. A 283, 147–178. Nelder, J.A., 1968. The combination of information in generally balanced designs. J. Royal Statist. Soc. Ser. B 30, 303–311. Nigam, A.K., Puri, P.D., 1982. On partially e ciency balanced designs—II. Commun. Statist. Theory Methods 11 (24), 2817–2830. Nigam, A.K., Gupta, V.K., Narain, P., 1981. Monograph on supplemented block designs. Indian Agricultural Statistics Research Institute (ICAR), Libr. Avenue, New Delhi, India. Puri, P.D., Nigam, A.K., Narain, P., 1977. Supplemented block designs. Sankhya B 39, 189–195.