6.1 INTRODUCTION

In Chapter 1 and Section 5.1, it was indicated that when a sequence of treatments are applied on an experimental unit over several periods and if no washout period can be left between successive periods, the analysis of the design has to be carried by taking into account the residual effects in the assumed linear model. It was further indicated that the direct effect of the treatment is the effect of the treatment manifested in the period of its application and ith-order residual effect is the residual effect of the treatment in the ith period after the treatment’s application is discontinued. While it is possible that the residual effects of all orders may be the same and this will be discussed in Section 8.2, in most designs, the residual effects of second and higher orders will be negligible and cross-over designs with first-order residual effects (CODWR) will be discussed in this chapter.

Since the treatments are applied to the experimental units without a break in CODWR, it becomes desirable to make allowance for a unit to be untreated in some periods. In the most general case of CODWR, one can consider b experimental units used over a k period experiment where in each period a unit may or may not receive one of the v treatments 1, 2, …, v. The setting can be written as a k × b array using v treatments. Let N = (n_ij) be a k × b matrix where n_ij = 1 or 0 according as the jth unit receives a treatment or not in the ith period. If n_ij = 1, let d(i, j) denote the treatment applied to the jth unit in the ith period. Mercado (1976) noted that observations may be taken in the ith period on the jth unit whether or not that unit receives a treatment in the period under consideration, and he classified the designs as generalized residual effects designs-I (GRED-I), if the observations are taken on the units only in the periods with n_ij = 1, and as generalized residual effects designs-II (GRED-II), if the observations are taken in all periods on each unit.

The analysis of GRED-I will be discussed in Section 6.2 and will be specialized to the case when N = J_k,b. We will also briefly sketch the analysis of GRED-II designs and indicate the estimability of direct and residual effects in that section. The designs will then be characterized as BRED and partially balanced cross-over designs with m associate classes (PBCOD(m)) depending on the incidence structure of the treatments in the rows and columns of the designs in Sections 6.3 and 6.4. Numerical examples will be given in Section 6.5. The analysis when units are random will be given in Section 6.6. Finally, some concluding remarks will be given in Section 6.7.

6.2 Analysis of CODWR

Consider a GRED in a k × b array using v treatments and let N = (n_ij) be the k × b matrix indicating the pattern of treating or not treating a unit in a particular period. When n_ij = 1, let Y_ij be the observation recorded in the ith period on the jth unit for GRED-I. Let

(6.2.1)

where μ is the general mean; ρ_i is the ith period effect; γ_j is the jth unit effect; τ_{d(i, j)} is the direct effect of treatment d(i, j), the treatment applied on the jth unit in the ith period; δ_{d(i–1, j)} is the first-order residual effect of the treatment d(i – 1,  j); and e_ij are random errors distributed according to IIN (0, σ²). When there is no treatment in the cell (i – 1,  j), we take δ_{d(i–1, j)} = 0. Clearly, δ_{d(0, j)} = 0.

Let be the direct effects-periods, direct effects-units, residual effects-periods, residual effects-units, and direct effects-residual effects incidence matrices, respectively.

For the CODWR,

where “–” denotes an untreated cell, the matrices N, L, M, L^*, M^*, and S are, respectively, given by

Let u_i be the number of units observed during the ith period, w_j be the number of periods in which the jth unit is observed, r_g be the number of observations on cells receiving the gth treatment, and s_h be the number of cells observed preceded by cells receiving the hth treatment. Let u′ = (u₁, u₂, …, u_k), w′ = (w₁, w₂, …, w_b), r′ = (r₁, r₂, …, r_v), and s′ = (s₁, s₂, …, s_v). The following relations are then obvious:

(6.2.2)

Let G be the grand total of all observations, P_i be the total of observations in the ith period, C_j be the total of observations on the jth unit, T_g be the total of observations in which the gth treatment occurs, and R_h be the total of observations in which the residual effect of the hth treatment occurs. Let P′ = (P₁, P₂, …, P_k), C′ = (C₁, C₂, …, C_b), T′ = (T₁, T₂, …, T_v), R′ = (R₁, R₂, …, R_v), ρ′ = (ρ₁, ρ₂, …, ρ_k), γ′ = (γ₁, γ₂, …, γ_b), τ′ = (τ₁, τ₂, …, τ_v), and δ′ = (δ₁, δ₂, …, δ_v). The normal equations can then be easily verified to be

(6.2.3)

where , , , , and are the least squares estimators of μ, ρ, γ, τ, and δ, respectively. Note that we are using C as a column vector and it is not a C-matrix. Continuing the notation of C_α|β,γ,… as introduced in the last chapter, the reduced normal equations after eliminating , , and are

(6.2.4)

where

(6.2.5)

(6.2.6)

(6.2.7)

(6.2.8)

(6.2.9)

and

(6.2.10)

The reduced normal equations for estimating and from Equation (6.2.4) after eliminating and are, respectively,

(6.2.11)

and

(6.2.12)

where

(6.2.13)

(6.2.14)

(6.2.15)

and

(6.2.16)

The ANOVA table is then given in Table 6.2.1. Tests of significance and confidence intervals on the contrasts of direct effects as well as residual effects of the treatments can then be made by standard procedures. The detailed results will be discussed for BRED in Section 6.3.

Source	d.f.	S.S.	M.S.	F
Periods (ig units, direct and residual effects)	k – 1
Units
(el periods, ig direct and residual effects)	R(Cγ|ρ)	Q C Qγ|ρ
Direct effects
(el periods and units, ig residual effects)	R(Cτ|ρ,γ)	Q C Qτ|ρ,γ
Residual effects
(el periods, units, direct effects)	R(Cδ|ρ,γ,τ)	Q C Qδ|ρ,γ,τ	MSr	MSr/MSe
Direct effects
(el periods, units, and residual effects)	R(Cτ|ρ,γ,δ)	Qτ|ρ,γ,δ	MSd	MSd/MSe
Error SSe	υ*	SSe	MSe
Total	n – 1

Q_γ|ρ = C – N′{D(u)}^-1P, SS_e = total SS – period SS – unit SS – dir. eff. (ig res. eff.) SS – res. eff. SS, n = , υ* = (n – 1)–{(k – 1) + R(C_γ|ρ) + R(C_τ|ρ,γ) + R(C_δ|ρ,γ,τ)}, ∑⁺ denotes summation over all treated cells, and R(⋅) denotes the rank of the matrix in parenthesis.

The equality of direct effects as well as residual effects will be tested by using the F statistics of Table 6.2.1 in the usual way. Most of the CODWR discussed in the literature have N = J_k,b, and every treatment occurs t times in each row of the design so that L = tJ_v,k and b = tv. Such designs will hence worth be called regular CODWR. Such designs are also called uniform designs across periods. Designs with k, a multiple of v, and every treatment occurs equally often in the columns are called uniform designs across units. A design is uniform if it is uniform in periods and units.

Clearly, for a regular CODWR, we get

(6.2.17)

(6.2.18)

and

(6.2.19)

From Equations (6.2.13), (6.2.14), (6.2.17), (6.2.18), and (6.2.19), it is clear that if MM^*′ is expressed in terms of MM′ and M^* M^*′, the dispersion matrices of direct and residual effects will be functions of MM′, M^*M^*′, and S, and it is easy to obtain their structures from the structures of MM′, M^* M^*′, and S. To this end, the following lemma is useful:

Regular CODWR satisfying the requirements of Lemma 6.2.1 can be characterized as BRED and PBCOD(m) based on MM′, M^* M^*′, and S. These results will be discussed in Sections 6.3 and 6.4.

For GRED-II, observations will be taken in treated and untreated cells. The model for the response from an untreated cell will contain the period and unit effects parameters and will not contain direct effects parameters. If the unit is treated preceding the untreated cell, then the model also contains the residual effects parameters. Here, N = J_k,b, L, M, and S will be the same as earlier, but L^*, M^* matrices will be different. The C-matrices and ANOVA table will be the same as GRED-I.

Noting that a linear parameter function is estimable for a linear model,

(6.2.24)

iff

(6.2.25)

It can be verified that for all CODWR designs, the linear parametric contrast ρ₁ - ρ_j is not estimable, whereas all other contrasts ρ_i - ρ_j are estimable for j = 1, 2, 3, …, k;  2 ≤ i < j.

Consider a CODWR

(6.2.26)

If all 16 observations are nonmissing, then Rank(C_τ|ρ,γ,δ) = Rank(C_δ|ρ,γ,τ) = 3 and all independent contrasts of treatment effects as well as residual effects are estimable.

Suppose the observation on unit 1 is missing in the first period. As GRED-I, Rank(C_δ|ρ,γ,τ) = 4 and Rank(C_τ|ρ,γ,δ) = 3, indicating that all contrasts of direct effects are estimable and all individual residual effects parameters are estimable. If the design is considered as GRED-II, then Rank(C_τ|ρ,γ,δ) = Rank(C_δ|ρ,γ,τ) = 4 and all individual direct effects and residual effects parameters are estimable. This result is true in general for the residual effects when the missing observation is not in the last period and is always true for direct effects parameters for GRED-II. An intuitive justification for the phenomenon is that we assume the empty cell is providing a carryover effect δ₀, which is identically zero, and as a GRED-II, it produces τ₀, which is identically zero. Since all contrasts are estimable, δ_i - δ₀ ≡ δ_i, τ_i - τ₀ ≡ τ_i are estimable.

Another interesting observation of Mercado (1976) is that for the design (6.2.26), if two cells (1, 1) and (2, 2) are missing, then individual δ_i are nonestimable and only contrasts δ_i - δ_j are estimable. If cells (1, 1) and (2, 3) are missing, all individual δ_i′s are estimable. The estimation of linear parametric functions of τ_i and δ_i for any GRED-I and/or GRED-II is an interesting and challenging problem.

6.3 Bred

If MM′, M^* M^*′, S, and MM^*′ are completely symmetric matrices, then C_τ|ρ,γ, C_δ|ρ,γ, and F₁ are completely symmetric and it follows that C_τ|ρ,γ,δ, C_δ|ρ,γ,τ are completely symmetric. In that case, the design is variance balanced in the estimation of the contrasts of direct effects as well as residual effects of the treatments.

In view of Lemma 6.2.1 under the condition stated there, it is sufficient that MM′, M^* M^*′, and S are completely symmetric in order that C_τ|ρ,γ,δ and C_δ|ρ,γ,τ be completely symmetric and the design is variance balanced for the estimation of the contrasts of direct effects as well as residual effects of the treatments. MM′ is completely symmetric if M is the incidence matrix of a BIBD discussed in Section 5.6, and thus, the b columns of the residual effects design form a BIBD of block size k. M^* M^*′ is completely symmetric if M^* is the incidence matrix of a BIBD and thus the b columns of the residual effects design omitting the last row form a BIBD of block size k − 1. Finally, S will be completely symmetric if every ordered pair of treatments (i, j), for i ≠ j, occurs together in successive rows equally often, say, υ times. These considerations lead us to the following definition:

1. Every column contains distinct treatments.
2. Every treatment occurs t times in each row.
3. For every pair of distinct treatments i and j, the number of columns in which i occurs with j in the last row is the same as the number of columns in which j occurs with i in the last row.
4. The b columns form a BIBD with parameters v₁ = v, b₁ = b, r₁ = tk, k₁ = k, λ₁ = λ.
5. The b columns excluding the last row form a BIBD with parameters v₂ = v, b₂ = b, r₂ = t (k − 1), k₂ = k − 1, λ₂ = μ.
6. For every pair of distinct treatments i and j, the number of columns in which the ordered pair of treatments (i, j) occurs in successive rows is υ.

The parameters of a BRED are v, k, b, t, λ, μ, and υ, and they satisfy the following relations:

(6.3.1)

The design

can be easily verified to be a BRED with parameters v = 4 = b = k,  t = 1, λ = 4, μ = 2, and υ = 1. The design

can be seen to be a BRED with parameters v = 3 = k, b = 6, t = 2, λ = 6, μ = 2, and υ = 2.

These two examples have a complete replication of all treatments in the columns. Williams (1949) showed that a BRED can always be constructed with one or two Latin squares depending on whether v is even or odd, and the construction methods will be given in Section 9.4.

Patterson (1952) gave BREDs where k < v, and some of his construction methods will be discussed in Section 9.5. An example of a BRED with v = 7, b = 14, k = 4, t = 2, r = 8, λ = 4, μ = 2, and υ = 1 is

From Definition 6.3.1 and Lemma 6.2.1, we get

(6.3.2)

(6.3.3)

(6.3.4)

and

(6.3.5)

Thus, for the BRED, Equations (6.2.13) and (6.2.14) simplify to

(6.3.6)

and

(6.3.7)

To get the vectors of adjusted yield, let be the sum of the column totals where treatment i is in the last row and put . After some simplification, one gets

(6.3.8)

(6.3.9)

Thus,

(6.3.10)

and

(6.3.11)

Clearly,

(6.3.12)

and

(6.3.13)

These variances indicate that _i – _j is estimated more accurately than _i – _j. The ANOVA table given in 6.2.1 can be easily completed to test the significance of direct effects as well as residual effects of the treatments.

For Williams’ BREDs consisting of one or two Latin squares, MC = GJ_v,1 and

(6.3.14)

and

(6.3.15)

The ANOVA table for these designs is given in Table 6.3.1.

Here,

(6.3.16)

and

(6.3.17)

In the literature, strongly balanced residual effects are considered where every treatment precedes itself on units the same number of times it precedes other treatments. With v = 4 treatments, k = 8 periods, and b = 16 units, the following design is uniform on both periods and units and is strongly balanced as every pair of treatments occurs four times in successive periods:

There is a vast literature on the optimality of BRED for estimating direct effects and residual effects contrasts considering different classes of competing designs, and the interested reader is referred to Bose and Dey (2009), Cheng and Wu (1980), Hedayat and Afsarinejad (1975, 1978), Hedayat and Yang (2003, 2004), Kunert (1983, 1984), Kushner (1997, 1998, 1999), Laska, Meisner, and Kushner (1983), Shah, Bose, and Raghavarao (2005), and Stufken (1996).

Source	d.f.	S.S.	M.S.	F
Periods	v – 1
Units	tv – 1
Direct effects (ig residual effects)	v – 1
Residual effects (el direct effects)	v – 1		MSr	MSr/MSe
Direct effects (el residual effects)	v – 1		MSd	MSd/MSe
Error	(v – 1) (tv – 3)	a+	MSe
Total	tv 2 – 1

a⁺ = total SS – period SS – units SS – direct effects (ig res. eff.) SS – residual effects (el dir. eff.) SS.

We will conclude this section by giving a simple result on optimality. In a class of designs, a design is optimal for all optimality criterion, that is, universal optimal, for the estimation of the effects of a factor, if the C-matrix for estimating these effects eliminating all other factors is completely symmetric with maximum trace. In the class of uniform CODWR with k = v, b = tv, N = J_k,b, without loss of generality, assume that the last period has treatments 1, 2, …, v occurring in that order t times. Then, we verify

(6.3.18)

(6.3.19)

(6.3.20)

and hence, the trace of C_τ|ρ,γ,δ and C_δ|ρ,γ,τ is maximum iff is minimum. Since ∑ _i ≠ js_ij = t(v − 1)v, is minimum iff s_ij is constant for i ≠ j, and those designs are Williams’ BRED.

6.4 PBCOD (m)

PBCOD (m) were introduced by Blaisdell and Raghavarao (1980) generalizing the concept of BRED in which there are m types of variances for the estimated elementary contrasts of direct effects as well as residual effects of treatments. For defining these designs, the concepts of association scheme and partially balanced incomplete block design (PBIBD) are required and are introduced in the following definitions:

Given an association scheme, we define m, v × v matrices for i = 1, 2, …, m, where

(6.4.1)

In addition, put B₀ = I_v ⋅ B₀, B₁, …, B_m are known as the association matrices of the m-class association scheme, and they satisfy the relations

(6.4.2)

where being the Kronecker delta equal to 1 if i = j and 0 otherwise. From Equation (6.4.2), clearly, matrix multiplication is closed in the set of linear functions of B₀, B₁, …, B_m, and the regular or (g-inverse) of a linear function of B₀, B₁, …, B_m is again a linear function of the association matrices.

Using an association scheme, we define a PBIBD in Definition 6.4.2:

Although the analysis of these designs is quite straightforward, the algebra is rather tedious and lengthy and will not be presented here. It depends in expressing all involved matrices in terms of linear combinations of the association matrices and defining the g-inverse of , where ∑ a_iB_iJ_v,1 = 0_v,1, as , satisfying

(6.4.5)

We solve the following simultaneous equations to determine b₀, b₁, …, b_m:

(6.4.6)

Defining the efficiency, E_d (or E_r) of PBCOD (m) as a/kt where 2σ²/a is the average variance of the elementary contrasts of direct (or residual) treatment effects, Raghavarao and Blaisdell (1985) proved the following:

This section will be concluded with the note that there is a need of developing efficient PBCOD (m) to increase the bag of CODWR to meet all requirements of experimenters in this area.

6.5 Numerical Example

The SAS program for analyzing a CODWR will be illustrated with the following BRED example. In the program, we denote Q_γ|ρ, Q_τ|ρ,γ, Q_δ|ρ,γ, Q_τ|ρ,γ,δ, and Q_δ|ρ,γ,τ by Q₁, Q₂, Q₃, Q₄, and Q₅, respectively. Further, C_γ|ρ, C_τ|ρ,γ, C_δ|ρ,γ, C_τ|ρ,γ,δ, and C_δ|ρ,γ,τ will be denoted by C₁, C₂, C₃, C₄, and C₅, respectively. The symbols T, R, r, s, L^*, and M^* will be denoted as trt, rtrt, smallr, smalls, lstar, and mstar, respectively.

The following SAS program provides the necessary output.

data a; input row column dirtrt$ restrt$ response @@; cards;
1 1 A . 720 1 2 B . 1020 1 3 C . 640 1 4 A . 1640 1 5 B . 2080 1 6 C . 960
2 1 B A 900 2 2 C B 820 2 3 A C 720 2 4 C A 1560 2 5 A B 1560 2 6 B C 1120
3 1 C B 720 3 2 A C 920 3 3 B A 800 3 4 B C 2090 3 5 C A 1784 3 6 A B 900
;
data row;set a;proc sort;by row;
proc univariate noprint; var response; output
out=rowsum sum=p;by row;
data rowsum;set rowsum;

data column;set a;proc sort;by column;
proc univariate noprint; var response; output
out=colsum sum=c;by column;
data colsum;set colsum;

data dirtrt;set a;proc sort;by dirtrt;
proc univariate noprint; var response; output
out=dirtrtsum sum=trt;by dirtrt;
data dirtrtsum;set dirtrtsum;

data restrt;set a;proc sort;by restrt;
proc univariate noprint; var response; output
out=restrtsum sum=rtrt;by restrt;
data restrtsum;set restrtsum;if restrt= ' ' then delete;

data grandtotal;set a;
proc univariate noprint; var response; output
out=grandtotal sum=grandtotal n=grandn;
data grandtotal;set grandtotal;

data sqobs;set a;responsesq=response*response;
proc univariate noprint; var responsesq; output
out=responsesq sum=respsq;
data responsesq;set responsesq;
proc iml;

/*****input data***/
n=J(3,6,1);l=J(3,3,2);m=j(3,6,1);lstar=shape({0 2 2},3,3);
mstar={1 0 1 1 1 0,1 1 0 0 1 1, 0 1 1 1 0 1};
s={0 2 2, 2 0 2, 2 2 0};u=J(3,1,6); w=J(6,1,3);
smallr=J(3,1,6);smalls=J(3,1,4);k=3;*number of rows in the design;
treatment={1,2,3};
/***end of data input***/

use rowsum;read all var {p} into p ; use colsum;read all var {c} into c ;
use dirtrtsum;read all var {trt} into trt ; use restrtsum;read all var {rtrt} into rtrt ;
use grandtotal;read all var {grandtotal} into grandtotal ;
use grandtotal;read all var {grandn} into grandn ;
use responsesq;read all var {respsq} into respsq ;

q1=c-n`*inv(diag(u))*p;
c1=diag(w)-n`*inv(diag(u))*n;
c2=diag(smallr)-l*inv(diag(u))*l`-(m-l*inv(diag(u))*n)*ginv(c1)*(m-l*inv(diag(u))*n)`;
q2=trt-l*inv(diag(u))*p-(m-l*inv(diag(u))*n)*ginv(c1)*(c-n`*inv(diag(u))*p);
c3=diag(smalls)-lstar*inv(diag(u))*lstar`-(mstar-lstar*inv(diag(u))*n)*ginv(c1)*(mstar-lstar*inv(diag(u))*n)`;
q3=rtrt-lstar*inv(diag(u))*p-(mstar-lstar*inv(diag(u))*n)*ginv(c1)*
(c-n`*inv(diag(u))*p);
f1=s-l*inv(diag(u))*lstar`-(m-l*inv(diag(u))*n)*ginv(c1)*(mstar-lstar*inv(diag(u))*n)`;
c4=c2-f1*ginv(c3)*f1`; q4=q2-f1*ginv(c3)*q3;
c5=c3-f1`*ginv(c2)*f1; q5=q3-f1`*ginv(c2)*q2;

periodss= (p`*inv(diag(u))*p)-(grandtotal**2)/grandn;
unitss=(q1`*ginv(c1)*q1); unitdf=round(trace(ginv(c1)*c1));
dirigresss=q2`*ginv(c2)*q2; dirigresdf=round(trace(ginv(c2)*c2));
reseldirss=q5`*ginv(c5)*q5; reseldirdf=round(trace(ginv(c5)*c5));
direlresss=q4`*ginv(c4)*q4; direlresdf=round(trace(ginv(c2)*c4));
totalss=respsq-grandtotal**2/grandn;
errorss=totalss-periodss-unitss-dirigresss-reseldirss;
error_df=grandn-1-(k-1)-unitdf-dirigresdf-reseldirdf;

error_ms=errorss/error_df;
direlresms=direlresss/direlresdf;
reseldirms=reseldirss/reseldirdf;
reseldirf=reseldirms/(errorss/error_df);
direlresf=direlresms/(errorss/error_df);direct_pvalue=1-probf(direlresf,direlresdf,error_df);residual_pvalue=1-probf(reseldirf,reseldirdf,error_df);
direct=ginv(c4)*q4; residual=ginv(c5)*q5;
ginvc4=ginv(c4); ginvc5=ginv(c5);

print direct_pvalue residual_pvalue error_ms error_df ;
print treatment direct residual; print ginvc4 ginvc5; quit;

The p-value for testing the equality of all direct effects of treatments is given at (a1) and is 0.0048. This p-value is significant, indicating that the direct effects of all treatments are not the same. The p-value for testing that the residual effects of the treatments are the same is given at (a2) and is 0.064. This indicates that there is no evidence against the null hypothesis that all residual effects are the same. The least squares estimators of the direct treatment effects are given in column (a5) and for residual effects in column (a6). is given at (a7) and is given at (a8). These g-inverses and the error MS given at (a3) will provide the standard error for contrasts of the direct effects as well as residual effects. For example,

The t-value for testing the hypothesis τ₁ = τ₂ is

The two-sided p-value for this t-value with error df, 6, given at (a4) is 0.001957168 and is significant. With this method, one can test the significance for the contrasts of interest.

6.6 Analysis With Unit (or Subject) Effects Random

In this section, we consider the analysis of CODWR, assuming that the unit (or subject) effects are random, N = J_k,b, u_i = b, and w_j = k. We assume γ_j are distributed IIN and are independently distributed with e_ij. Without loss of generality, we assume and . We now have

(6.6.1)

From the model (6.6.1), using weighted least squares, we estimate τ and δ by τ′ and δ′, respectively, given by

(6.6.2)

and

(6.6.3)

Let w = 1/σ² and . We combine the estimator given by Equations (6.6.2) and (6.2.3) to get the estimator of τ, , given by

(6.6.4)

Similarly, combining the estimator of δ given by Equations (6.6.3) and (6.2.3), we get the estimator of δ, , given by

(6.6.5)

The dispersion matrices of these estimators are

(6.6.6)

and

(6.6.7)

To get the estimators (6.6.4)–(6.6.7), we need to estimate σ² and . To this end, let us introduce the following notation:

,

The subjects’ (or units or columns) sum of squares SS₀, adjusted for periods, direct and residual treatment effects, is given by

Then,

(6.6.8)

(6.6.9)

From the estimators and , we estimate w and w′ and complete the analysis.

6.7 Concluding Remarks

In Section 6.2, the general analysis was given for CODWR when observations are taken only on treated cells (i.e., GRED-I). The analysis of GRED-II when observations are taken on treated and untreated cells can be easily developed based on the standard procedures and is left as an exercise to the interested reader. Some results on the estimability of parameter functions were also discussed in that section.

To emphasize the interest on the estimability of parametric function of τ′s and δ′s, we will give another example.

Mercado (1976) considered a 7 × 7 design for testing seven treatments by treating only 28 cells as given in the following:

When the design was considered as GRED-I, he observed that the matrix C_δ|ρ,γ,τ has rank 7 and the matrix C_τ|ρ,γ,δ has rank 6. When the design was considered as GRED-II, both the matrices C_δ|ρ,γ,τ and C_τ|ρ,γ,δ have rank 7. The average variance of all estimated elementary contrasts of the direct and residual effects is the following:

If the experimenter decides to leave more than one cell in a BRED untreated, the pattern of untreated cells determines the estimable functions as noted in Section 6.2. An analogous situation in the case of Latin square designs was discussed by Dodge and Shah (1977).

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