Combined Intra- and Inter- Block Analysis of Balanced Ternary Designs

Sobita Sapam; B. K. Sinha; N.K. Mandal

doi:10.2991/jsta.2018.17.1.7

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Volume 17, Issue 1, March 2018, Pages 91 - 100

Combined Intra- and Inter- Block Analysis of Balanced Ternary Designs

Authors

Sobita Sapamsobita1@yahoo.com

Guest Lecturer, Department of Statistics, Manipur University, Canchipur, Imphal-795003, India

B. K. Sinhabikassinha1946@gmail.com

ISI, Kolkata [Retired Professor of Statistics], 203, B. T. Road, Kolkata- 700 108, India

N.K. Mandalmandalnk2001@yahoo.co.in

Professor of Statistics (Retired), Calcutta University, 35, B.C. Road, Kolkata-700 019, India

Received 31 December 2016, Accepted 29 March 2017, Available Online 31 March 2018.

DOI: 10.2991/jsta.2018.17.1.7 How to use a DOI?
Keywords: Balanced ternary design; Intra-block analysis; Inter-block analysis; Combined intra- and inter-block analysis
Abstract: In the present paper an attempt has been made to study the inter–and intra-block estimation of treatment effects contrasts with random block effects in the context of balanced ternary designs. The analysis is illustrated through examples.
Copyright: Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Balanced n-ary designs were introduced by Tocher (1952). In such a design each entry in the incidence matrix can take any value of the ‘n’ possible values, usually, 0,1,2,.., (n−1). If n=3, we get a ternary design. Billington(1984,1989), Meena and Mangla(2006), Sharma, Singh & Roshni(2013) give results on the construction of balanced ternary designs.

In the present paper an attempt has been made to study the inter–and intra-block estimation of treatment effects contrasts with random block effects in the context of balanced ternary designs. The term recovery of inter-block information was first coined by Yates (1939). Generally, the experimental units are partitioned to eliminate the heterogeneity of the experimental units (eus) in such a way that the eus within blocks are as homogeneous as possible. Because of the variability of the block totals, the initial estimates of the treatment contrasts were based on contrasts within blocks neglecting the information contained in block totals. If, however, the block totals are not that heterogeneous, the loss of information on treatment contrasts can be recovered by treating block totals as observations. According to Rao(1947), “If the ratio of the variances for inter-and intra-block comparisons is sufficiently greater than 1, these estimates from incomplete block designs will, of course, be considerably more accurate.”

Bose(1975) worked on combined intra-and inter-block estimation of treatment effects in incomplete block designs.Searle(1986) dealt with the case of block effects being treated as random;estimators of treatment effects contrasts were derived, along with their variances, and the results were applied to balanced incomplete blocks to yield an expression for the inter-and intra- block estimator of a treatment effects contrast.

The organization of the paper is as follows: In Section 2 , we introduce some useful definitions and preliminaries. Next in Section 3 , we deal with two useful models viz., Fixed Effects Additive Model [FEAM] and Mixed Effects Additive Model [MEAM] in the context of Ternary Designs. The data analysis under these two models are carried out separately in this section. In section 4 , combined data analysis is presented.

2. Definitions and Preliminaries

A Balanced Ternary Design (BTD) with parameters [v,b,r, k,t, s, q,ρ₁, ρ_q, f₁₁, f_1q and f_qq] is an arrangement of ‘v’ treatments in ‘b’ blocks – each of size ‘k’- whose incidence matrix is given by N = (n_ij), where n_ij is the number of times the i-th treatment occurs in the j-th block and n_ij = 0,1,q(≥ 2) and the elements of the matrix N_v×b = (n_ij) satisfy the following conditions:

(i)
n_ij = 0 if the i-th treatment does not occur in the j-th block,
- =1, if the i-th treatment occurs in the j-th block exactly once,
- =q, if the i-th treatment occurs in the j-th block exactly q(≥2) times;
(ii)
every treatment is replicated r times in the entire design;
(iii)
each block contains a set of ‘t’ treatments exactly once each and another distinct set of ‘s’ treatments exactly qtimes each so that k = t + qs, t + s < v;
(iv)
ρ₁ is the number of blocks that any arbitrarily specified treatment occurs once in each such block;
(v)
ρ_q is the number of blocks that any arbitrarily specified treatment occurs q times in each such block;
(vi)
f₁₁ is the number of blocks that any pair of arbitrarily specified treatments occur together once each exactly in each of these blocks;
(vii)
f_1q is the number of blocks that any arbitrarily specified treatment occurs once and any other arbitrarily specified treatment occurs q(≥2) times in each of these blocks;
(viii)
f_qq is the number of blocks that any two arbitrarily specified treatments occur each q times in each of these blocks.

Then it is seen that the design parameters [v, b, r, k,t, s,q, ρ₁,ρ₂, f₁₁,f_1q and f_qq]satisfy the following conditions:
(ix)
bk=vr, (x) r =ρ₁+qρ_q, (xi) k = t + qs, (xii) ρ₁(k−1)=(v−1)(f₁₁+f_1q),
(xiii)
ρ_q(k−q)=(v−1)(f_1q+qf_qq), q(≥2).

Further, it follows that
(xiv)
λ=f11+2qf1q+q2fqq=∑j=1b for i ≠ i′ = 1, 2, …, v,
(xv)
Δ=ρ1+q2ρq=∑j=1bnij2 for each i=1, 2, …, v,
(xvi)
NN′=[Δ⋯λ⋮⋱⋮λ⋯Δ]⋅.

3. The FEAM & MEAM Models and ANOVA Tables

3.1. Mixed Effects Additive Model [MEAM] and Related ANOVA Table

First we consider Linear MEAM. Let y_iju be the observation corresponding to u-th experimental unit in the j-th block to which the i-th treatment is applied. We then stipulate the following model

(1)yiju=μ+τi+aj+eiju;i=1,2,...,v;j=1,2,...,b;u=1,2,...,k

where μ is the general mean effect, τ_i is the fixed treatment effect, a_j is the random block effect with mean zero and variance σb2, e_iju’s are random errors assumed to be iid with mean zero and variance σe2. The random errors e_iju’s and the random block effects a_j’s are supposed to be uncorrelated and normally distributed.

Under MEAM, we want to explore the contribution of block average contrasts towards estimation of treatment effects contrasts. These are referred to as ‘inter-block’ estimates of treatment contrasts. Therefore the source of information on treatment contrasts can be obtained from the (b-1) block mean contrasts. Besides these, the within block observational contrasts are known to yield ‘intra-block’ estimates of treatment effects contrasts.

Notations:

Denote by B¯1,...,B¯b the block means for b blocks of the design. Next, we introduce elementary block mean contrasts below. Since the blocks are proper and binary, under a mixed effects model, the block mean contrasts will involve only the treatment effects contrasts in their model expectations. This will provide information [and related inference] on treatment effects in terms of their contrasts. This is referred to as Inter-Block Analysis. We start as follows:

D1=B¯1−B¯2,.....,Db−1=B¯1−B¯b , where D=(D₁, … , D_b−1)^′ which follows N_b−1 (ζ., Σ(D) with ζj′−1=E(Dj′−1)=E(B¯1−B¯j′), j′=2,3,...,b.

Explicitly,

E(D1)=E(B¯1−B¯2)=1k[∑i=1vni1τi−∑i=1vni2τi]=1k[∑i=1v(ni1−ni2)τi]

and so on;

E=(Db−1)=E(B¯1−B¯b)=1k[∑i=1vni1τi−∑i=1vnibτi]=1k[∑i=1v(ni1−nib)τi].

Since block sizes are same, it is clear from the above expression that E(D_i) is a treatment contrast for each i.

Var(D1)=Var(B¯1−B¯2)=Var(B¯1)+Var(B¯2) [since all the blocks are independent]=2k2(k2σb2+kσe2) [sincevar(Bi)=k2σb2+kσe2]=2(σb2+σe2/k).

Hence Var(Dj′−1)=2(σb2+σe2/k)for j′=2,3,...,b. .

Cov(D1,D2)=Cov(B¯1−B¯2,B¯1−B¯3)=Var(B¯1)−Cov(B¯1,B¯2)−Cov(B¯1,B¯3)+Cov(B¯2,B¯3)=σb2+σe2/k−0−0+0 [since var(Bi)=k2σb2+kσe2]=σb2+σe2/k.

Hence

Disp(D)=[2(σb2+σe2/k⋯σb2+σe2k⋮⋱⋮(σb2+σe2k⋯2(σb2+σe2/k]having order (b−1)×(b−1).=(σb2+σe2k)[I+J]=σ*2W=∑(D),say.

Here σ*2=(σb2+σe2k), I is the identity matrix of order (b−1) and J is the matrix of all elements 1’s; W=[I + J].

Now let us define D^* =kD, where D= (D₁, D, … , D_b−1)^′ and k is the constant size of the blocks. Then D^* relates to the elementary contrasts among block totals. Our model based on block total contrasts is given by:

(2)(D*, Aθ, k2σ*2W)

where A is the coefficient matrix of θ of order (b−1)×(v−1) and θ = (θ₁, θ₂, … ,θ_v−1)^′ with θ_i′−1 = τ₁−τ_i′, i′ =2,3,…,v;

E(D*)=[∑i=1v(ni1−ni2)τi∑i=1v(ni1−ni3)τi⋮∑i=1v(ni1−nib)τi]=Aθ,

where

A=[n22−n21n32−n31...nv2−nv1n23−n21n33−n31...nv3−nv1......................................n2b−n21n3b−n31...nvb−nv1]

(3)Disp(D*)=Disp(kD)=k2∑(D)=k2σ*2W

The Normal Equations for the model (2) are given by

(4)A′W−1Aθ=A′W−1D*

We may refer to Kshirsagar (1983) for the derivation of (4).

Therefore, BLUE θ of is

(5)θ^=(A′W−1A)−1A′W−1D* where W−1=(I−/bJ)

Further, E(θ^)=E(A′W−1A)−1(A′W−1D*)= θ

and

(6)Disp(θ^)=D{(A′W−1A)−1A′W−1D*} = k2σ*2(A′W−1A)−1

The following ANOVA Table is in order.

Remark 1.

It must be noted that in the above ANOVA Table, Mean Squares (due to) Error [MSE] provides unbiased estimation of k2σ*2=k2(σb2+σe2k) with respect to Model (2) . This is because the error variance based on block totals/means under MEAM is a linear combination of pure error variance [σe2] and random block effects variance [σb2].

Remark 2.

When in the model (1) the block effects are treated as fixed effects, the model will change to one involving the fixed block effects and hence no “extra” information will be available for estimation of treatment effects contrasts from the block totals or block total contrasts. This is discussed in the next subsection in the framework of (Fixed effects additive model (FEAM).

3.2 Fixed Effects Additive Model [FEAM]& related ANOVA Table

We now turn back to the same model as in (1) taking a_j as the fixed block effect [for each block] so that we are in the framework of FEAM. We first develop a general theory for estimation of treatment effects contrasts based on ‘within block contrasts’. For this, we closely follow the use of C-matrix in the set-up of block designs. We refer to any standard book on ANOVA-based analysis of block designs.

Recall that in its most general set-up of a Balanced Ternary Design, block size ‘k’ has the representation: k = t + qs so that in effect, there are t distinct treatments each with a single appearance and s distinct treatments each with replication number q in each block of the design. Since q≥2, there are (q−1) ‘pure’ errors arising out of these replications for each of the s treatments and hence there are s(q−1) degrees of freedom attributed towards estimation of error variance [σe2]. The computation of corresponding sums squaresis pretty straightforward. Besides, we also have (t + s – 1) treatment contrasts from within each block. Together, they contribute towards a collection of b(t+s−1) treatment contrasts and only (v−1) treatment contrasts are linearly independent. We are thus led to the identity

n−1 = b(t+sq) – 1=bsq + bt – 1 = b(t+s−1) + bs(q−1) + (b−1) where

(i)
b(t+s−1) = number of within block observational contrasts leading to treatment contrasts and sources of error [for estimation of error variance σe2];
(ii)
bs(q−1) = number of within block observational contrasts leading to pure errors [again for estimation of error variance σe2];
(iii)
(b−1) = number of block total [or block mean] contrasts leading to estimation of treatment effects contrasts [under MEAM] and sources of random block effects error variance together with[observational] error variance [in other words, estimation of (σb2+σe23)].

We will now work with the set in (i) to provide intra-block estimates of treatment effects contrasts under FEAM which suggests that only such observational contrasts are relevant. We will end up with ANOVA Table based on Intra-block Analysis.

In standard notations, we refer to the ‘Reduced Normal Equations’, Cτ=Q where C = r I − NN′ / k and NN′ is as shown in (xvi) above. Further, Q=T − NB/k, T being the vector of treatment totals and B being the vector of block totals. The ternary design is balanced and therefore, the C-matrix is completely symmetric [having all diagonal elements equal and also all off-diagonal elements equal]. We have

C = [r − (Δ−λ)/k]I − λJ/k where r = constant treatment replication number; λ = f₁₁+2qf_1q+q²f_qq and Δ=ρ₁+q²ρ_q. Note that NN′1 has two representations: NN′1=k[ρ₁+qρ_q]1=rk1=[Δ + (v−1) λ]1 and, hence kC1=[rk − Δ −λ(v−1)]1=0. Further, Q′1 = 0.

Therefore, rank (C) = v−1, as expected and the Moore-Penrose g-inverse of C is easy to work out. Clearly, SS due to Treatment Contrasts =Q′C⁺Q and SS due to pure error [σe2] derives a contribution from SS within blocks [with b(k−1) df ] minus SS due to Treatment Contrasts [with v−1 df]. This SS has 2 components: SS due to pure error derived from within treatment q replications for s treatments in each block [with bs(q−1) df] and SS due to excess of total Within Block SS over SS due to Treatment Contrasts.

It must be noted that with respect to ANOVA Table 1 for Inter-block Analysis, Total Sum of Squares, shown as D^*′W⁻¹D^*, simplifies to ∑i(yi..−y¯...)2 which carries (b−1) df. Thus the splitting of Total SS as also of the total degrees of freedom (bk−1) is well understood.

Source of variation	d.f.	Sums of squares
Treatment contrasts	v−1	θ^′(A′W−1D*)
Error	b−v	By subtraction
Total	b−1	D^′W⁻¹D^

Table 1:

ANOVA table for Inter-block Analysis

Source of Variation	d.f.	SS
Pure Errors	bs(q−1)	∑i∑j∑u(yiju−y¯ij)2
Treatment Contrasts	v−1	Q′C⁺Q
Errors from within blocks	b(k−1) −bs(q−1)− (v−1)	by subtraction
Total Within Blocks	b(k−1)	∑i∑j∑u(yiju−y¯i..)2

Table 2:

ANOVA table for Intra-Block Analysis

At the last leg, we can work out combined inter- and intra-block estimates of the treatment contrasts. This we do with respect to an illustrative example involving growth of paddy in an agricultural experiment involving 3 treatments.

4. Combined Intra- and Inter-block estimates

Here we consider the BTD with parameters: v = k = 3, b = r = 6, q = 2, t = s = 1, ρ₁=2,ρ₂ = 2, f₁₁ = 0, f_1q = 1 and f_qq = 0. The design has the following block compositions:

B₁=(1,2,2), B₂=(1,3,3), B₃=(2,1,1), B₄=(2,3,3),B₅=(3,1,1), B₆=(3,2,2).

We assume the following figures for the yield of paddy [in suitable unit].

Block I: y₁ = 9.7, y_2,1 = 8.3, y_2,2 = 9.1;	Block II: y₁ = 9.3, y_3,1 = 8.4, y_3,2 = 9.0
Block III: y₂ = 9.3, y_1,1 = 9.4, y_1,2 = 10.2;	Block IV: y₂ = 9.6, y_3,1 = 8.5, y_3,2 =9.1,
Block V: y₃ = 9.3, y_1,1 = 9.4, y_1,2 = 10.6;	Block VI: y₃ = 8.2, y_2,1 = 8.4, y_2,2 = 9.6

Under the MEAM, we want to explore the contribution of block average contrasts towards treatment effect contrasts. Here we see that there are 5 linearly independent block average contrasts viz., Dj−1=B¯1−B¯j ; j=2,…, 6 and there are only 2 treatment contrasts viz., θ=(θ₁, θ₂)^′ with θ₁=τ₁−τ₂, θ ₂=τ₁−τ₃. Further, D ^* = 3D.

Here A=(−22−10−12−2101) and W = I + J of order 5. It turns out that

θ^1=[−D1*−D2*+D3*−2D4*+2D5*]/6=−0.55;θ^2=[−D1*−2D2*+2D3*−D4*+D5*]/6=−0.4166.Disp(θ^)=32(2112)σ*2=σb2+σe23.

ANOVA Table is displayed below. We find that estimated σ^*² is given by ERROR MSE/9 = 6.8234/27= 0.2527.

Hence, estimated Var(θ^1)=3 times estimated σ^*²= 0.7582 estimated Var(θ^2) .

Turning back to the Intra-block analysis based on FEAM, we have obtained:

Q₁ = 1.87, Q₂ = 0.07, Q₃ = −1.94; C = (1/3) [12 I – 4 J];
C+ = (1/12)[3I − J];
τ^=(0.4675,0.0175,−0.4850)’;
1. (a)
  Tr. SS=τ^′Q=1.8163 with 2 df.
2. (b)
  Error SS from within blocks [by subtraction] = 2.8772 with 4df.
3. (c)
  Further to this, Pure Error SS = 0.32 + 0.18 + 0.32 + 0.18 + 0.72 + 0.72 = 2.34 with 6 df It follows that (a) – (c) add up to:
4. (d)
  Total Within Blocks SS = SS=∑i[∑j∑uyiju2−ky¯i..2]=4.6935 with 12 df.

Below is the full ANOVA Table.

From Intra-block Analysis :

σe2 estimated = Error SS from the above 2 sources / (6+4)= [2.8772 +2.34]/ 10=0.5217.Finally, based on Intra-Block Analysis, θ^1=τ^1−τ^2=0.4500;; θ^2=τ^1−τ^3=0.9525 with respective estimated variances computed as 12times σe2 estimated = 0.2608. Again, from Inter-block Analysis, 9σ*2=9(σb2+σe23) is estimated by the Error MS = 6.8234/3 = 2.2745. Therefore, σb2 estimated = [2.2745 – 3 × 0.5217]/9= 0.0788.

Combined Intra- and Inter-block Estimates of Treatment Contrasts

Since the observational contrasts and the block average contrasts are independent, the combined intra-and Inter-block estimation of treatment effects is given by

θ^c=(/W1θ^1(WBC))+(/W2θ^1(BBC))(/W11)+(/W21).

Hence, θ^1,c=[(−0.55/0.7582)+(0.4500/0.2608)]/[(1/0.7582)+(1/0.2608)]=0.1940; θ^2,c=[(0.4166 / 0.7582) + (0.9525/0.2608)]/[(1/0.7582) + (1/0.2608)]= 0.8153. Further, estimated variance is the same for both and it is given by 0.1940.

Remark 3:

As a final point, it may be noted that it is a trivial exercise to display Combined ANOVA Table for Splitting of Total SS with (bk−1) = 17 df. This follows from Tables 3 and 4.

Source of variation	d.f.	Sums of squares
Treatment contrasts	v−1=2	θ^′(A′W−1D*)=1.8099
Error	b−v=3	6.8234 [By subtraction]
Total	b−1=5	D^′W⁻¹D^=8.6333

Table 3:

ANOVA Table for Inter-block Analysis

Source of Variation	Df	SS
Treatment Contrasts	2	1.8163
Pure Errors	6	2.34
Errors from Within blocks		2.8772
Total Within blocks	12	4.6935

Table 4:

ANOVA Table for Intra-Block Analysis

4. Acknowledgement

We express our sincere thanks for administrative co-operation from Dr. K. K. Singh Meitei, Head of the Department of Statistics, Manipur University, Imphal, during exchange visits of the authors. The authors are also grateful to an anonymous referee for his constructive suggestions which helped to improve the presentation of the article.

References

[1]RC Bose, Combined intra-and Inter-block estimation of treatment effects in incomplete block designs, A survey of Statistical designs and linear models, North Holland publishing company, 1975.

[2]EJ Billington, Balanced n-ary designs: A combinatorial survey and some new results, ARS Combinatoria, Vol. 17A, 1984, pp. 37-72.

[3]EJ Billington, Designs with repeated elements in blocks:A survey and some recent results, Congressus Numerantium, Vol. 68, 1989, pp. 123-143.

[4]AM Kshirsagar, A course in Linear Models, M. Dekker, 1983.

[5]MR Satam and MS Despande, Construction of ternary group divisible designs, J. Ind. Soc. Agril. Statist, Vol. 61, No. 3, 2007, pp. 361-363.

[6]CR Rao, General methods of analysis in incomplete block designs, Journal of the American Statistical Association, Vol. 42, 1947, pp. 541-561.

[7]HL Sharma, RN Singh, and R Tiwari, Balanced ternary, ternary group divisible and nested ternary group divisible designs, Ind. Soc. Agril.Statist, Vol. 67, No. 3, 2013, pp. 339-344.

[8]SR Searle, Inter- and intra- block estimation of treatment effects in randomized blocks. Technical report BU-897- M (1986) in the Biometrics Unit.

[9]KD Tocher, The design and analysis of block experiments (with discussion), J. Roy. Statist. Soc, Vol. 14, 1952, pp. 45-100. Ser. B

[10]F Yates, The recovery of inter block information in varietal trials arranged in three dimensional lattices, Ann. Eugenics, Vol. 9, 1939, pp. 136-156.

[11]F Yates, The recovery of inter block information in balanced incomplete block designs, Annals of Eugenics, Vol. 10, 1940, pp. 317-325.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 17 - 1
Pages: 91 - 100
Publication Date: 2018/03/31
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.2018.17.1.7 How to use a DOI?
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

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TY  - JOUR
AU  - Sobita Sapam
AU  - B. K. Sinha
AU  - N.K. Mandal
PY  - 2018
DA  - 2018/03/31
TI  - Combined Intra- and Inter- Block Analysis of Balanced Ternary Designs
JO  - Journal of Statistical Theory and Applications
SP  - 91
EP  - 100
VL  - 17
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.1.7
DO  - 10.2991/jsta.2018.17.1.7
ID  - Sapam2018
ER  -

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