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Chapter 6
Chapter 6
Balanced Incomplete Block Design (BIBD)
The designs like CRD and RBD are the complete block designs. We now discuss the balanced incomplete block design (BIBD) and the partially balanced incomplete block design (PBIBD) which are the incomplete block designs.
A balanced incomplete block design (BIBD) is an incomplete block design in which
· b blocks have the same number k of plots each and
· every treatment is replicated r times in the design.
·
Each treatment occurs at most once in a block, i.e., or 1 where is the number of times the jth treatment occurs in ith block,
·
Every pair of treatments occurs together is of the b blocks.
Five parameters denote such design as .
The parameters are not chosen arbitrarily.
They satisfy the following relations:
and for all Obviously cannot be a constant for all j. So the design is not orthogonal.
Example of BIBD
Now we see how the conditions of BIBD are satisfied.
Even if the parameters satisfy the relations, it is not always possible to arrange the treatments in blocks to get the corresponding design.
The necessary and sufficient conditions to be satisfied by the parameters for the existence of a BIBD are not known.
The conditions (I)(III) are some necessary condition only. The construction of such design depends on the actual arrangement of the treatments into blocks and this problem is handled in combinatorial mathematics. Tables are available, giving all the designs involving at most 20 replication and their method of construction.
Theorem:
Proof: (I)
Let the incidence matrix
Observing that the quantities are the scalars and the transpose of each other, we find their values.
Consider
Similarly,
Proof: (II)
Consider
Since or 0 as or 0,
Also
Proof: (III)
From (I), the determinant of is
because since if from (II) that k = v. This contradicts the incompleteness of the design.
Thus is a nonsingular matrix.
Thus
We know from matrix theory result
But , there being b rows in N.
Thus
Interpretation of conditions of BIBD
Interpretation of (I) bk = vr
This condition is related to the total number of plots in an experiment. In our settings, there are plots in each block and there are blocks. So the total number of plots are .
Further, there are treatments and each treatment is replicated times such that each treatment occurs atmost in one block. So total number of plots containing all the treatments is . Since both the statements counts the total number of plots, hence
Interpretation of (II)
Each block has plots. Thus the total pairs of plots in a block =
There are blocks. Thus the total pairs of plots such that each pair consists of plots within a block = .
There are treatments, thus the total number of pairs of treatment = .
Each pair of treatment is replicated times, i.e., each pair of treatment occurs in blocks.
Thus the total number of pairs of plots within blocks must be .
Hence
Using in this relation, we get
Proof of (III) was given by Fisher but quite long, so not needed here.
Balancing in designs:
There are two types of balancing – Variance balanced and efficiency balanced. We discuss the variance balancing now and the efficiency balancing later.
Balanced Design (Variance Balanced):
A connected design is said to be balanced (variance balanced) if all the elementary contrasts of the treatment effects can be estimated with the same precision. This definition does not hold for the disconnected design, as all the elementary contrasts are not estimable in this design.
Proper Design:
An incomplete block design with is called a proper design.
Symmetric BIBD:
A BIBD is called symmetrical if the number of blocks = number of treatments, i.e.,
Since so from
Thus the number of pairs of treatments common between any two blocks = .
The determinant of is
When BIBD is symmetric, b = v and then using we have Thus
Since is an integer, hence when is an even number, must be a perfect square. So
Postmultiplying both sides by , we get
Hence in the case of a symmetric BIBD, any two blocks have treatment in common.
Since BIBD is an incomplete block design. So every pair of treatment can occur at most once is a block, we must have
If then it means that each treatment occurs once in every block which occurs in case of RBD.
So in BIBD, always assume v > k .
Similarly
Resolvable design:
A block design of
· b blocks in which
· each of v treatments is replicated r times
is said to be resolvable if b blocks can be divided into r sets of blocks each, such that every treatment appears in each set precisely once. Obviously, in a resolvable design, b is a multiple of r.
Theorem: If in a BIBD is divisible by r, then
Proof: Let (where is a positive integer).
Now, if possible, let
Intrablock analysis of BIBD:
Consider the model
where
We don’t need to develop the analysis of BIBD from starting. Since BIBD is also an incomplete block design and the analysis of incomplete block design has already been presented in the earlier module, so we implement those derived expressions directly under the setup and conditions of BIBD. Using the same notations, we represent the blocks totals by , treatment totals by , adjusted treatment totals by and grand total by The normal equations are obtained by differentiating the error sum of squares. Then the block effects are eliminated from the normal equations and the normal equations are solved for the treatment effects. The resulting intrablock equations of treatment effects in matrix notations are expressible as
.
in the case of BIBD. The diagonal elements of are given by
The offdiagonal elements of are given by
The adjusted treatment totals are obtained as
where denotes the sum over those blocks containing jth treatment. Denote
The matrix is simplified as follows:
Since is not as a full rank matrix, so its unique inverse does not exist. The generalized inverse of is denoted as which is obtained as
Since
the generalized inverse of is
Thus .
Thus an estimate of is obtained from as
The null hypothesis of our interest is against the alternative hypothesis at least one pair of is different. Now we obtain the various sum of squares involved in the development of analysis of variance as follows.
The adjusted treatment sum of squares is
The unadjusted block sum of squares is
.
The total sum of squares is
The residual sum of squares is obtained by
.
A test for is then based on the statistic
If
This completes the analysis of variance test and is termed as intrablock analysis of variance. This analysis can be compiled into the intrablock analysis of variance table for testing the significance of the treatment effect given as follows.
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Degrees of
freedom
Mean squares
F
Between treatment (adjusted)
Between blocks
(unadjusted)
Intrablock error
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In case, the null hyperthesis is rejected, then we go for a pairwise comparison of the treatments. For that, we need an expression for the variance of the difference of two treatment effects.
The variance of an elementary contrast under the intrablock analysis is
This expression depends on which is unknown. So it is unfit for use in the real data applications. One solution is to estimate from the given data and use it in the place of
An unbiased estimator of is
Thus an unbiased estimator of can be obtained by substituting in it as
If is rejected, then we make pairwise comparison and use the multiple comparison test. To test a suitable statistic is
which follows a tdistribution with degrees of freedom under .
A question arises that how a BIBD compares to an RBD. Note that BIBD is an incomplete block design whereas RBD is a complete block design. This point should be kept in mind while making such restrictive comparison.
We now compare the efficiency of BIBD with a randomized block (complete) design with replicates. The variance of an elementary contrast under a randomized block design (RBD) is
where under RBD.
Thus the relative efficiency of BIBD relative to RBD is
The factor (say) is termed as the efficiency factor of BIBD and
The actual efficiency of BIBD over RBD not only depends on the efficiency factor but also on the ratio of variances . So BIBD can be more efficient than RBD as can be more than because
Efficiency balanced design:
A block design is said to be efficiency balanced if every contrast of the treatment effects is estimated through the design with the same efficiency factor.
If a block design satisfies any two of the following properties:
(i) efficiency balanced,
(ii) variance balanced and
(iii) an equal number of replications,
then the third property also holds true.
Missing observations in BIBD:
The intrablock estimate of missing (i, j)th observation is
the sum of Q value for all other treatment (but not the jth one) which are present in the
ith block.
All other procedures remain the same.
Interblock analysis and recovery of interblock information in BIBD
In the intrablock analysis of variance of an incomplete block design or BIBD, the treatment effects were estimated after eliminating the block effects from the normal equations. In a way, the block effects were assumed to be not marked enough and so they were eliminated. It is possible in many situations that the block effects are influential and marked. In such situations, the block totals may carry information about the treatment combinations also. This information can be used in estimating the treatment effects which may provide more efficient results. This is accomplished by an interblock analysis of BIBD and used further through the recovery of interblock information. So we first conduct the interblock analysis of BIBD. We do not derive the expressions a fresh but we use the assumptions and results from the interblock analysis of an incomplete block design. We additionally assume that the block effects are random with variance
After estimating the treatment effects under interblock analysis, we use the results for the pooled estimation and recovery of interblock information in a BIBD.
In case of BIBD,
The interblock estimate of can be obtained by substituting the expression on in the earlier obtained interblock estimate.
Our next objective is to use the intrablock and interblock estimates of treatment effects together to find an improved estimate of treatment effects.
In order to use the interblock and intrablock estimates of together through pooled estimate, we consider the interblock and intrablock estimates of the treatment contrast.
The intrablock estimate of treatment contrast is
The interblock estimate of treatment contrast is
The variance of is obtained as
Since
so
Similarly, the variance of is obtained as
The information on and can be used together to obtain a more efficient estimator of by considering the weighted arithmetic mean of and . This will be the minimum variance unbiased and estimator of when the weights of the corresponding estimates are chosen such that they are inversely proportional to the respective variances of the estimators. Thus the weights to be assigned to intrablock and interblock estimates are reciprocal to their variances as and respectively. Then the pooled mean of these two estimators is
where ,
Now we simplify the expression of so that it becomes more compatible in further analysis.
Since so the numerator of can be expressed as
Similarly, the denominator of can be expressed as
Let
where Using these results we have
where
Thus the pooled estimate of the contrast is
The variance of is
is called as the effective variance.
Note that the variance of any elementary contrast based on the pooled estimates of the treatment effects is
The effective variance can be approximately estimated by
where MSE is the mean square due to error obtained from the intrablock analysis as
and
.
The quantity depends upon the unknown and To obtain an estimate of , we can obtain the unbiased estimates of and and then substitute them back in place of and in . To do this, we proceed as follows.
An estimate of can be obtained by estimating from the intrablock analysis of variance as
.
The estimate of depends on . To obtain an unbiased estimator of , consider
for which
Thus an unbiased estimator of is
where
Thus
Recall that our main objective is to develop a test of hypothesis for and we now want to develop it using the information based on both interblock and intrablock analysis.
To test the hypothesis related to treatment effects based on the pooled estimate, we proceed as follows.
Consider the adjusted treatment totals based on the intrablock and the interblock estimates as
and use it as usual treatment total as in earlier cases.
The sum of squares due to is
Note that in the usual analysis of variance technique, the test statistic for such hull hypothesis is developed by taking the ratio of the sum of squares due to treatment divided by its degrees of freedom and the sum of squares due to error divided by its degrees of freedom. Following the same idea, we define the statistics
where is an estimator of . It may be noted that depends on The value of itself depends on the estimated variances . So it cannot be ascertained that the statistic necessary follow the distribution. Since the construction of is based on the earlier approaches where the statistic was found to follow the exact distribution, so based on this idea, the distribution of can be considered to be approximately distributed. Thus the approximate distribution of is considered as distribution with and degrees of freedom. Also, is an estimator of which is obtained by substituting the unbiased estimators of and .
An approximate best pooled estimator of is
and its variance is approximately estimated by
In case of the resolvable BIBD, can be obtained by using the adjusted block with replications sum of squares from the intrablock analysis of variance. If sum of squares due to such block total is and corresponding mean square is
then
and for a resolvable design. Thus
and hence
The analysis of variance table for the recovery of interblock information in BIBD is described in the following table:
Source
Sum of squares
Degrees of freedom
Mean square
Between treatment
(unadjusted)
Between blocks
(adjusted)
Intrablock error
v  1
b  1
bk – b – v + 1
Total
bk  1
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