Indicators

[John P. Morgan]

Date: Sun, 05 Oct 2003 19:13:06 +0100
R A Bailey <r.a.bailey at qmul.ac.uk> wrote

>JP said:
>
><begin>
>I am going to argue that the suggestion is not at all a
>recommendation for proceeding in ignorance. My proposal as made a few
>days ago is that we implement a concept that is already used, and
>that we use the terminology that is already given to that concept. We
>could talk about "pairwise partial balance" and "efficiency partial
>balance" (which can be defined as the weighted information matrix -
>see the ext rep doc - as falling in the Bose-Mesner algebra) too, but
>as far as I known this is truly striking into new territory. Is there
>any mention anywhere in the literature of either of these concepts?
><end>
>
>
>Yes.  Wilson (I believe) uses `group-divisible' to mean binary
>block designs which may have unequal block sizes but whose
>concurrence matrix is in the BMA of a group divisible association
>scheme.  In the 1980s Puri and Nigam defined `partial efficiency
>balance'.  Their definition makes it an empty concept, but it is not
>true to say that


The Wilson usage is good to learn. I don't have Puri and Nigam on hand, but 
isn't their defn of PEB same as Kageyama and Calinski, *not* requiring an 
association scheme?

>
>``Forget about any
>concept of partial efficiency balance, which will not be of interest
>in our lifetime (efficiency balance in itself is a marginal concept)''
>
>If you think this then read Volume II of Cali\'nski and Kageyama.
>

My usage of the term here was as I had defined it (with an association scheme), 
not as C&K defined. So I do not think the criticism is on target, but on the 
other hand I should know better than to cavalierly anticipate the future. 


>I think that each concept (concurrence matriz in the BMA,
>information matrix in the BMA, weighted concurrence matrix
>in the BMA) is valuable to some people.
>
>RAB
>

RAB's post with a Sunday time stamp did not arrive at my mailer until Monday 
19:53 - clearly my mailer and the list do not play well together, which can 
muddy discussions. In the meantime I had already posted a bit about the PEB 
term, as follows:

>...term is in 
>usage, most notably of late in the two volume set "Block Designs: A 
>Randomization Approach" by Calinski and Kageyama. Their use of the term 
>"partial efficiency balance" does not require an association scheme.
>

There is also now a post from PeterC with an extensive discussion of partial 
balance. Not to ignore that, but this note will touch on the narrower line of 
PEB as defined by Calinski and Kageyama and the relevance of an association 
scheme to what they are doing.

Henceforth I will use the term "partial efficiency balance" (PEB) as defined by 
C&K. Then *any* block design is PEB. The notion of an association scheme on the 
weighted info matrix is one special case; for the purposes of this post let's 
denote this property by PEB(A). And for the purposes of this post, I will 
henceforth focus only on the unequally replicated case, since the weighting is 
trivial with equirep. Our question is, does PEB(A) have particular interest? To 
talk about PEB at all you are going to get the spectral decomposition whether 
you have an association scheme or not. PEB(A) can ease the factorization effort, 
if that is an issue. 

C&K take advantage of PBIBDs in building up collections of designs with equal 
reps. They also have some constructions for unequally replicated designs that 
start from PBIBDs, but I take this as more a focus on what equirep designs are 
cataloged/available as input for their procedures than as any interest in PEB(A) 
itself. Moreover, those methods do not guarantee PEB(A), as in some 
constructions the weighted info matrices they produce may not be diagonal.

RAB above suggests something that may be different: the weighted concurrence 
matrix is in BMA. I take this to mean weighting by replication, having not first 
weighted blockwise-concurrences by blocksize; is that correct? If correct, then 
the notion will differ from PEB(A) when the blocksizes are unequal. In that 
case, the weighted concurrence matrix will not allow calculation of the 
efficiency factors, which seems to lose one of the key ideas.

Whatever notion(s) of "partially" we adopt with respect to efficiency balance or 
the weighted info matrix, if any, we may want to include in our documentation 
some explanation of their relevance/importance. I do not currently see that 
either of the notions briefly discussed above can be given much justification, 
but it may be yet to come out.

By the way, one goal of C&K is to encompass all block designs, without 
restricting to equiblocksize, equireplication, or even binarity. We are taking 
the same approach in the current external representation, though we do not yet 
handle nonbinarity. The current external representation does report the number 
of distinct efficiency factors (other than the structural zero), and thus 
reports the fundamental identifier in C&K's classification of PEB designs. This 
is some, not near to full, coverage of PEB.

JP