Paper: Design of Experiments and Sample Survey Module: Balanced Incomplete Block Design-I
Module No: DOE-10
Principal investigator:
Dr. Bhaswati Ganguli,
Professor, Department of Statistics, University of Calcutta
Paper co-ordinator:
Dr. Bikas Kumar Sinha,
Retired Professor, Indian Statistical Institute, Kolkata
Content writer: Dr. Santu Ghosh,
Lecturer, Department of Environmental Health Engineering, Sri Ramachandra University, Chennai
Content reviewer: Dr. Sugata SenRoy,
I take this opportunity to thank Dr Bhaswati Ganguly of the Department of Statistics, Calcutta University for approaching me with a specific purpose. In this UGC initiated epathsala programme, for the subject of
’Statistics’, in her capacity as the Principal Investigator, she wanted me to act as a coordinator for the topic : Design of Experiments [DoE] and Sample Survey [SS]. I gladly accepted her proposal and volunteered to prepare all the 40 Modules as asked for. I have followed a distinctive style while preparing the modules viz., as that of a dialogue between an Instructor [Professor Bikram Kanti Sahay(BKS)] and his two students [Ms.
Sagarika Ghosh(SG) and Mr. Subhra Sankar Gupta(SSG)]. I fondly hope this instructional discourse and my efforts on two of my favorite topics in Statistics will be appreciated and found useful.
In the video recordings, I will impersonate as BKS.Mr. Samopriya Basu[MSc (Statistics), Calcutta University] andMs. Moumita Chatterjee[University of Calcutta, Kolkata] will impersonate as the students [SSG and SG] respectively.
Professor Bikas k Sinha Retired Professor of Statistics Indian Statistical Institute Kolkata
July 10, 2015
BKS
I am now going to talk about a very special subclass of block designs.
To start with, these are non-orthogonal but necessarily connected.
More than that, these are binary designs and hence the incidence matrix admits of 0-values!
Such designs form a special subclass of ’binary proper equireplicate’
connected .... block .... designs.
Recall that we are in the framework of constant block size, say, k.
BKS
As usual, we start with ’v’ treatments in a block design involving ’b’ blocks, each of size ’k’.
The total number of observations ’n’ equalsbk and we assume it to be divisible by ’v’, the number of treatments.
We confine to the class of ’equireplicate’ designs so thatr =bk/v stands for the constant replication number.
Note that binary block designs with ’k =v’ are easily identified as RBDs.
To exclude this case, we assume henceforth that ’k <v’.
SG
Sir,
In that case, we necessarily have ’r <b’ since we have the identity : n=bk=vr.
This is also clear from the fact that we are dealing with incomplete block designs.
I am curious to know about the special subclass you are referring to.
Does it have anything to do with the incidences of treatment pairs across the blocks?
BKS
Yes, right you are! Note that so far we did not mention about the joint occurrence of pairs of treatments .... is it the same for all pairs ?
SG
Sir, this may not necessarily happen ... it must call for extra conditions on the design parameters ... I believe.
BKS
Let’s investigate.
If all treatment pairs occur equally often across the blocks, we will really have a special structure of the Incidence MatrixN.
Let us call this extra parameter as ’λ’.
Can you visualize this ?
SG
Let me try to say something· · · since I can see something!
Within each block there are ’k’ treatments .... all different .... so that we have k(k−1)/2 pairs of treatments formed.
All the ’b’ blocks taken together, this leads tobk(k−1)/2 total number of formations of treatment pairs.
Again, since we are talking about ’v’ treatments and each pair occurringλ times, we have a total ofλv(v−1)/2 pairs to be formed.
Therefore, we must have
bk(k−1) =λv(v−1).
SG
However, we already know .... bk =vr and so ... it amounts to
r(k−1) =λ(v−1).
Sir,
what if the design parameters [b,v,r,k] do not ensure that λ=r(k−1)/(v−1) is an integer ?
Take, for example,b= 12,v = 10,k = 5,r = 6. This yields r(k−1)/(v−1) = 24/9 = 8/3...which is not an integer.
So .... we are stuck!
BKS
Not only that...
Even if there is an integer solution toλ, we may not be in a position to visualize an incidence matrix of orderv×badmitting of such a combinatorial structure!
Take, for example,b= 8,v = 16,k = 10,r = 5 for whichλ= 3. Do we have a solution? It is not very clear right away· · · these seem to be puzzling issues.
Fortunately for us, many such situations have been studied by the researchers and, by now, we have some knowledge as to what is possible and what not. [The solution to the example here is in the negative.]
SG
Interesting.
Anyway, what is the special subclass you were referring to?
Is that the one with ’constantλ’ ?
If there are so many issues as to the existence of such designs, what is so special about it?
Is it very common in practice?
Are there catalogues which an experimenter can consult, given the other design parameters [b,v,r,k]?
BKS
SG, I am really surprised by your curiosity to know and criticize ...
this is a very good attitude ... you must ask questions .... seek clarifications ....
Let us go slow ....
all I can say is that this area of research has created immense academic interest among the design theorists for the last 90-100 years ... almost!!!
AND, for your information,
Indian Design Theorists have kept a permanent visible mark in credible research along this area.
Let me introduce this subclass of designs ....
called ’Balanced Incomplete Block Designs’ ... abbreviated as BIBDs.
SG
Sir,
So a BIBD is defined in terms of its four parameters [b,v,r,k] and the derived parameter ’λ’, right ?
May be .... not really ... since there is also a relation ’bk=vr0 ...so ...
basically ... there are three defining parameters ....
I am confused ...Oh ! now I see ...
there are three independent parameters .... two others are dependent and derivable from the given three ... right ?
BKS
You may say so, SG.
Is it clear why such a combinatorial structure is referred to as a BIBD ? We can see that it is an incomplete block design sincek <v and the design, to start with, is binary and proper and equireplicate.
But, why is it ’balanced’ ?
What is the meaning of ’balancing’ ? Does it have to do with ’variance balance’ ?
I mean .... all treatment effects elementary contrasts [of the form (τi−τj)]
are estimated with the same variance .... does that mean .... it is connected to start with?
SG
Sir, you are going fast. Let us try to assimilate. Yes ...
if we are discussing about variance .... we need to assume that the design is connected ....
Interesting ...
Does it not follow necessarily from the combinatorial definition of BIBD ? What about application of Park-Shah technique ?
Oh, I see. It is almost immediate to me.
Consider block 1.
It hask treatments - all distinct.
SG
Clearly, all treatment effect differences in this block are estimable and hence all the treatments in the block are in one single equivalence group.
Since any two treatments occur together inλ[≥1] blocks, we can find another block with at least one treatment common between the two.
Hence, through this common treatment, all treatments in the two blocks become connected etc etc ... it seems pretty obvious now that all treatments will belong to a single equivalence group and hence the design will be ultimately connected.
Right, Sir? I am now curious to study the form of the C-matrix for a BIBD!
BKS
OK · · · let’s do that.
Note that for a BIBD[b,v,r,k, λ], NN’ has all diagonal elements equal to
’r’ and all off-diagonal elements equal to ’λ’.
Hence,
I C=rδ−NN0/k
and it simplifies to a matrix of the formaI +bJ where
I a+b=r−(r/k) = [r(k−1)]/k=λ(v−1)/k and
I b=−λ/k
BKS
Therefore,
I a+vb= 0 so that
I C=aI −a/v J=a[I−J/v].
This also suggests that C-matrix has all positive eigenvalues the same as equal to
I a=vλ/k=vr(k−1)/k(v−1).
I hope .... you can see all this .... these are pretty much easy and routine deductions ... one has to use the parametric relations for a BIBD at convenient places.
SG
Yes, Sir, I can see all that. I think ....
it now follows thatC+ can be taken as [1/a]I and hence,
Var(τ\i−τi0) = σ2[2/a] =σ2[2k(v−1)/vr(k−1)]
= [2σ2/r]×[k(v−1)/v(k−1)].
(To be continued...)
