Indicators

[John P. Morgan]

On Thu, 2 Oct 2003 14:44:01 +0100
Peter Cameron <p.j.cameron at qmul.ac.uk> wrote

>
>1. Affine
>---------
>I begin with a story. In the 1970s I was at Westfield College,
>where Marion Kimberley was teaching a course on design theory.
>"Design" meant "2-design" there. In a homework exercise, she asked
>students to define an affine design, expecting them to say "a 
resolvable
>design in which any two non-parallel blocks meet in \mu points". One
>student said, "a resolvable design in which any two non-disjoint 
blocks
>meet in \mu points". Marion pointed out that this is not the same 
thing
>for designs with \lambda=1 (e.g. Kirkman systems). For \lambda>1, 
nobody
>knows yet whether the two conditions are equivalent; I was able to 
show
>that there are very strong restrictions. (The first open case is 
that of
>a 2-(70,10,6) design. It is not necessary to assume resolvability; a 
single
>parallel class is enough. Without that, there are counterexamples, 
e.g. the
>3-(22,6,1) Witt design.
>
>The point is that "affine" (as usually interpreted) is more 
restrictive than
>"resolvable and block intersections 0 and \mu", and there is strong
>possibility of confusion here.
>
>The other reason I don't like this is that "affine" is so 
well-established
>and has such a body of theory that I would be reluctant to change 
the
>usage. On the other side, designs with block intersections of size
>\alpha or \beta are well studied: they are called "quasi-symmetric" 
(at
>least if they are 2-designs, and I think the generalisation here is 
>unexceptionable), and for example Shrikhande and Sane have a book 
about
>them.
>
>On the point of principle, I view indicators as being for humans 
rather than
>for computers, and would rather have indicators for concepts for 
which a
>body of theory exists than for "atomic" concepts, whataever that 
means.

The naming problem is a real concern (not just here - more below) and 
Peter's points are well taken. The quasi-symmetry is essentially 
Leonard's suggestion (not retricted to 2-designs). The suggested 
indicator or generalization could well be given a different name, to 
avoid confusion with other usages. 

There is still the separate question of whether the concept is 
sufficiently important to warrant indicator status. A list of reasons 
for its inclusion have been offered, and these should be addressed. 
If the "atomic" concept is rejected in this case, then the next step 
is to consider what is the appropriate combination. In some way I 
think this concept should be given indicator status, for reasons 
already explained. I am less sure about the more general notion of 
(generalized) quasi-symmetry. In any case, let's now focus on the 
concept, after which we can select the appropriate term.

On the more general issue (which Peter addresses in an email that 
followed the one here) here are some thoughts, and the "atomic" issue 
plays in here in a very useful way. We are taking a very general view 
in what we are building. Ours is a database for all block designs, 
including unequal replication, unequal blocksizes, and at a later 
date, nonbinarity. Had design theory historically developed with such 
a broad view, our task would be much simpler, for all of the concepts 
that we wish to report would already be named and accepted for our 
fully general setting. But of course this is not at all the case, and 
Peter's story illustrates the point well: even the term "design" 
still means (as it did then) different things to different people.

Without bothering to strictly define the term, I do support the 
"atomic" concept. First, I do not see where any indicator proposed or 
already included is not understandable by humans. The only hitch is 
that many people are not accustomed to thinking about certain 
concepts outside of certain narrowly defined (relative to our 
endeavor) setups. Our job is to define the general setup well. 
Providing simple, atomic indicators that are natural and obvious (two 
debatable terms) extensions of the general case, that reduce to the 
established concepts in the "usual" cases, is part of what we can 
provide.

All this is saying is "Think of the set of properties you want. We 
provide an indicator of each one for a general block design. Combine 
these as you will to get designs with exactly the properties you 
want." In taking this approach, we make our product *more* user 
friendly, we create the ability to look at many more combinations of 
properties than would otherwise be possible, and consequently we 
provide a vehicle for enhancing research. Will we miss out indicators 
that people will need? Yes, and we will add them as needed. But that 
is no argument for not including obvious, simple generalizations now. 


>
>2. Partially balanced
>---------------------
>For general block designs, there are (at least) three different 
kinds of
>balance: pairwise balance, variance balance, and efficiency balance. 
If
>the block size is constant these all reduce to the same thing.
>
Reduction to the same also requires equireplication and binarity.

>May something similar be true for partial balance? Thus, "pairwise 
partial
>balance" means that the concurrence matrix is in the Bose-Mesner 
algebra
>of an association scheme; "variance partial balance" means that the 
>information matrix is (do I have this right JP?) and there may be a 
kind of 
>"efficiency partial balance" as well.
>
>My proposal is that we do not in ignorance assume that only one of 
these
>three concepts is important. We could easily have three indicators 
as
>outlined above. 
>

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? 
If we do not know of any, then this may not be the right time to 
implement (they can always be added later), and they may not warrant 
indicator status.  But we have a concept that has seen considerable 
usage, it is named "partial balance", and it covers the full 
generality of what we are doing. Why are we defining it only for 
equireplicate, binary, constant blocksize when usage does not make 
any of these restrictions?

Here is a bit to clarify my earlier remarks. The typical way in which 
"partial balance" appears in the statistical literature is through 
one or more design constructions, the results of which are then shown 
to have partial balance by establishing that their information 
matrices are in the Bose-Mesner algebra. Typically not even the B/M 
term is used, it is just shown that the elements of the info matrix 
take values in accordance with some association scheme. This is done 
for many classes of designs (nested, row-column, nested row-col, 
test/control, etc). This is a large part of the reason for bringing 
it up now -  we either ignore the partial balance concept when we 
turn our attention to other classes of designs, or we come back to 
the same issue, leaving one of two possible questions
1) why did they define partial balance for block designs but not for 
class X?
2) why did they define partial balance in general for class X but 
only for a special case of block designs?

Of course, we could also remove all mention of partial balance from 
the current external rep. Probably not the move we will make, but 
tenable if we are concerned with proceeding in ignorance and if we do 
not know the relevant literature and usage of the term. I have tried 
to explain above that the suggested avenue is not a path of 
ignorance, and that there is established usage of the term.

Although it disrespects the current usage of the term "partial 
balance" as explained above, I do prefer Peter's term "partial 
variance balance" in being more descriptive. We also face the issue 
of statistical versus mathematical terminology, and although one 
field may have laid claim to a particular term, in creating a 
resource for all researchers interested in block designs 
(mathematicians, statisticians, computer scientists, etc) we bear 
some responsibility to select terms that are understandable and 
acceptable to all (in so far as this is possible). Sometimes this 
requires modification of existing terminology. 

Having thought about Peter's comments and made responses above, I 
still support changing the current "partial_balance" indicator as 
previously suggested, which adopts the "atomic" principle (removing 
any mention of blocksize or replication, addressing only the 
association scheme) but would rename it something like 
"information_matrix_association_scheme". Add a parallel indicator 
"concurrence_matrix_association_scheme." The terms themselves are 
easily understandable by humans and give us all what we want. They 
allow thinking about this concept from the combinatorial and the 
statistical perspectives, so address our main audiences and the 
differences among them. Indeed, they make more clear the ways in 
which those two audiences think differently about this concept 
(though my ignorance is that I do not know if combinatorialists 
consider it at all for unequal blocksizes, etc). 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). 
The implementation of the partial_balance_properties element should 
include an optional way to explore the concept of partial variance 
balance when the design is *not* binary, equireplicate, 
equiblocksize; this option would be omitted in the binary, 
equireplicate, equiblocksize case.
 


>I assume it will turn out to be the case that for equireplicate 
designs
>with constant block size, all three concepts will coincide and 
become the
>classical notion of partial balance.
>
>By the way, I take the "diagonal" to be one of the associate classes 
in an
>association scheme. This implies that a pairwise partially balanced 
design
>is automatically equireplicate. Is this agreed?
>

Do we really want to exclude nonbinarity? I hope not! Isn't it our 
plan to allow nonbinarity at some unspecified future time? With 
nonbinarity we need not have equireplication. The same is true for 
variance partial balance even for equal blocksizes.

JP