Information Matrices and Optimality Values for various Block Designs

The International Journal Of Engineering And Science (IJES)
|| Volume || 4 || Issue || 11 || Pages || PP -28-32|| 2015 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 28
Information Matrices and Optimality Values for
various Block Designs
T. Shekar Goud and N.Ch. Bhatra Charyulu
Department of Statistics, University College of Science, Osmania University, Hyderabad-7
--------------------------------------------------------ABSTRACT-----------------------------------------------------------
The conditions of an experiment allow the possibility of simultaneous existence of a number of experimental
designs. To choose an appropriate design, is easy to analyze and satisfies optimal properties, developed by
Kiefer (1959) based on the Information matrix of the experimental design. In this paper an attempt is made to
obtain the information matrices for CRD, RBD, LSD and BIBD’s and are illustrated in case of CRD, RBD and
LSD and for different parameters of BIBD’s.
Key words: CRD, RBD, LSD, BIBD.
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Date of Submission: 31 October 2015 Date of Accepted: 25 November 2015
----------------------------------------------------------------------------------------------------------------
I. INTRODUCTION
An optimality criterion is a criterion which summarizes how good a design is and it is maximized or
minimized by an optimal design. To estimate the treatment effects with a certain number of restrictions on
allotment of treatments to the experimental units, and also to estimate how much information would have been
lost by using a smaller number of restrictions, the optimality criteria can be used. Optimality criteria can be
used to select best among the available designs belongs to a particular class of experimental design satisfying
the conditions, and easy to analyze and are generally optimal only for a specific statistical model.
Kiefer (1959) developed a useful criterion for finding optimum designs based on the Information
matrix. The information matrix is proportional to the inverse of the variance–covariance matrix of the least
squares estimates of the linear parameters of the model. Some of the alphabetical optimality criteria are: A-, C-,
D-, E-, G- optimality criteria.
DEFINITION 1.1: Let D be the class of designs, X be any design belongs to D , is said to be optimal if
 [ Var(X) ]   [ Var(X*) ] for any X, X* D (1.1)
where  is a criterion function of information matrix used for estimating the parameters in the model.
DEFINITION 1.2: A design X  D is said to be A-optimum in the class of designs D if
Trace [ (X′X)-1
]X  Min { Trace [ (X′X)-1
]X*} for any X, X* D (1.2)
DEFINITION 1.3: A design X  D is said to be C-optimum in the class of designs D, if
[ λ1 / λn ] X  [ λ1 / λn ] X* 0 for any X, X* D (1.3)
where λ1 and λn are the largest and the smallest eigen values of X’X.
DEFINITION 1.4: A design X  D is said to be D-optimum in the class of designs D, if
| (X′X)-1
| X  Inf | (X′X)-1
| X*. for any X, X* D (1.4)
Where | (X′X)-1
| indicates the Determinant of (X′X)-1
matrix.
DEFINITION 1.5: A design X  D is said to be E-optimum in the class of designs D, if
max [(X′X)-1 ] X   max [(X′X)-1 ] X*
.
for any X, X* D (1.5)
DEFINITION 1.6: A design X D is said to be G-optimum in the class of designs D if
Min {Var( )(ˆ xY )X }  Min Max { Var( )(ˆ xY )X*} for any X, X* D (1.6)
2. INFORMATION MATRICES FOR VARIOUS DESIGNS
In this section an attempt is made to obtain information matrices and optimality values for the
experimental designs, Completely Randomized Design, Randomized Block Design, Latin Square Design and
Balanced Incomplete Block Design and are presented below with suitable examples.
2.1 COMPLETELY RANDOMIZED DESIGN: The statistical model of Randomized Block Design yij =  + I
+ ij can be expressed in a general linear model as Y = X +  , Where Y = [Y11,..Y1j, ..Y1n1 |…| Yi1,..Yij,…Yin2 |
…| Yk1,..Ykj,…Yknk ]  vector of responses, β = [µ | α1…αi...αk]  where µ is the mean, αi is effect due to treatment

Information matrices and optimality values For various block …
(with ‘k’ treatments), ε = [ε 11,.. ε 1j, .. ε 1n1 |…| ε i1, ..ε ij,… ε in2 | …| ε k1,.. ε kj,… ε knk ]  is the vector of random
errors, follows NI(0, 2
) and X is the design matrix where
Note: Augment the conditions  I = 0 in X under H0 to estimate the parameters in the model.
2.2 RANDOMIZED BLOCK DESIGN: The general linear model for a Randomized Block Design is Y=X+ 
Where Y = [Y11,..Y1j, ..Y1n |…| Yi1,..Yij,…Yin | …| Yk1,..Ykj,…Ykn ]  vector of responses, β = [µ | α1…αi...αk | β1
… βj…βn]  where µ is the mean, αi is effect due to ith
treatment, βj is effect due to jth
block (with ‘k’ treatments
and ‘n’ blocks ), ε = [ε 11,.. ε 1j, .. ε 1n |…| ε i1, ..ε ij,… ε in | …| ε k1,.. ε kj,… ε kn ] is the vector of random errors
and are follows NI(0, 2
Note: Augment the conditions i=0, j=0 in X under H0 to estimate the parameters in the model.
2.3 LATIN SQUARE DESIGN: The general linear model for a Latin Square Design is Y = X  +  , where Y
= [y111…y1jk…y1vv | … |yi11…yijk…yivv| … |yv11…yvjk…yvvv] ′ is the vector of observations where yijk is the
observation belongs to ith
row, jth
column and kth
treatment ( design has ‘v’ rows, ‘v’ columns and ‘v’
treatments ),  = [ µ | α1 ... αi ... αv | β1… βj … βv | γ1… k … γv ]′ where µ is the mean αi ,βj , k are the ith
row,
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jth
column and k is kth
treatment effects respectively.  = [ε11 1…ε1jk…ε1vv|…| εi11… εijk ….εivv |…|εv11…εvjk…
εvvv ]′ is the vector of random errors follows NI(0, 2
Note: Augment the conditions i=0, j=0, k=0 in X under H0 to estimate the parameters in the model.
EXAMPLE 2.1: Consider an experimental design conducted in two way blocked design, the responses are
presented below.
3.10 (C) 5.95 (F) 1.75 (A) 6.40 (E) 3.85 (B) 5.30 (D)
4.80 (B) 2.70 (A) 3.30 (C) 5.95 (F) 3.70 (D) 5.40 (E)
3.00 (A) 2.95 (B) 6.70 (E) 5.95 (D) 7.75 (F) 7.10 (C)
6.40 (E) 5.80 (D) 3.80 (B) 6.55 (C) 4.80 (A) 9.40 (F)
5.20 (F) 4.85 (C) 6.60 (D) 4.60 (B) 7.00 (E) 5.00 (A)
4.25 (D) 6.65 (E) 9.30 (F) 4.95 (A) 9.30 (C) 8.40 (F)
The optimality values are evaluated in case of CRD, RBD and LSD and are presented in Table 2.2 , by
assuming the design as CRD by ignoring the blocking, RBD by ignoring the columns as blocks and LSD by
consider the two way blocking.
A- Opt E- Opt G- Opt I- Opt
CRD 0.02040 7.8729 0.0029 0.0204
RBD 1.6875 - 0.0156 1.6875
LSD 2.5185 - 0.0123 2.518
Table 2.2
2.4 BALANCED INCOMPLETE BLOCK DESIGN: The General Linear Model for a Balanced Incomplete
Block Design is Y = Xβ + ε. It can be expressed as :
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0.100.010.101
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.............
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0.010.010.011
X
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XX '

Then its X’X is
The various optimality values for the different parameters of BIBD are evaluated and presented in the
following Table 2.3.
No V B r k  E=v/rk A-opt D-opt E-opt G-opt I-opt
1 4 6 3 2 1 0.666 1.6660 0.020800 6 0.416 1.66
2 4 4 3 3 2 0.888 3.1100 0.111100 9 0.77 3.11
3 9 12 4 3 1 0.750 2.7500 0.000012 12 0.3055 2.75
4 9 36 8 2 1 0.560 1.2050 0.000001 16 0.1338 1.205
5 9 12 3 4 1 0.750 4.0909 0.000355 11 0.4545 4.0909
6 9 12 8 6 5 0.930 2.6875 0.00000031 48 0.2986 2.6875
7 6 15 5 2 1 0.600 1.3500 0.000097 10 0.225 1.35
8 5 10 4 2 1 0.625 1.4580 0.001540 8 0.2916 1.458
9 6 15 10 4 6 0.900 1.2750 0.000024 40 0.2125 1.275
10 13 13 3 4 1 1.083 6.0660 0.000016 15 0.4666 6.066
11 13 13 4 4 1 0.812 4.06250 8503056 16 0.3125 4.0625
12 11 11 5 5 2 0.880 3.3730 0.000000 25 0.3066 3.373
13 5 10 4 2 1 0.625 1.4583 0.00154 8 0.2916 1.4583
14 5 5 4 4 3 0.937 4.0625 0.0625 16 0.8125 4.0625
15 5 10 6 3 3 0.833 1.3888 0.00068 18 0.2777 1.3888
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16 6 15 6 2 1 0.500 1.350 0.00009 10 0.225 1.350
17 6 10 5 3 2 0.800 1.733 0.00027 15 0.2888 1.733
18 6 6 5 5 4 0.960 5.040 0.0400 25 0.840 5.040
19 6 6 5 5 6 1.440 4.971 0.0285 35 0.828 4.970
20 7 7 3 3 1 0.777 3.111 0.0017 9 0.4444 3.111
21 7 7 4 4 2 0.875 3.111 0.00097 16 0.4444 3.111
22 7 21 6 2 1 0.583 1.283 0.00001 12 0.1832 1.283
23 8 28 7 2 1 0.571 1.238 0.00007 14 0.154 1.238
24 8 14 7 4 3 0.857 1.785 0.0000021 28 0.2231 1.785
25 8 8 7 7 6 0.979 7.020 0.0200 49 0.8775 7.020
26 10 18 9 5 4 0.888 7.020 0.0000001 45 0.702 7.020
27 10 15 9 6 5 0.925 2.268 0.00000007 54 0.2268 2.2608
28 11 11 6 6 3 0.916 3.361 0.00000047 36 0.3055 3.361
29 11 55 1
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2 1 0.550 1.161 0.000000000
001
20 0.1055 1.161
30 12 44 1
1
3 2 0.727 1.252 0.000000000
0000965
39 0.1043 1.252
Table 2.3
Acknowledgements: The authors grateful to the UGC for providing fellowship under BSR-RFMS and the
authors are also thankful to the editor for the improvisation of the manuscript.
REFERENCES
[1] Chakravarthi M.C. (1962): Mathematics of design and analysis of experiments, Asia Publishing
House, Bombay.
[2] Kiefer J (1959): Optimum experimental designs, J.R. stat. Soc., B. 21, 272-319.
[3] Raghava Rao D (1970): Construction and combinatorial problems in design of experiments, John
Wiley, New York.

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