indicators

[John P. Morgan]

Sun, 28 Sep 2003 13:25:27 +0100 (BST)
Leonard Soicher <l.h.soicher at qmul.ac.uk> wrote

>
>I'm not sure why JP didn't receive my posting on indicators. Perhaps there is 
>a problem with his email address used by the "reply-to" mechanism
>(I believe I sent the reply to both him and the list; this one is 
>going only to the list).
>

This is not the first time that the list and my mailer have not communicated 
properly. 


>
>Perhaps JP missed some of my previous posting, so I will 
>repeat and clarify some points:
>
...

>
>I absolutely agree with JP's principle that we should use indicators for
>basic properties, rather than combined properties.

Diverging for a moment, that principle is one of several reasons for separating 
out the association scheme property from the equireplicate and 
constant_blocksize properties in the partially_balanced indicator. 

> However,
>t-wise balanced for various t's fits better as a list of <index_flag>s
>than as an indicator. Note that we *do* handle the important special 
>cases of 1-wise balanced (<equireplicate>) and 2-wise balanced
>(<pairwise_balanced>) as indicators. Indeed, for consistency with
>equireplicate, perhaps <pairwise_balanced> should have an attribute
>"lambda". 
>

Aha! I didn't see the t-wise in the combinatorics section. Still, is it really 
too much trouble to get t-wise balance in at the indicator level? Specifically, 
is it possible, rather than reporting a single value (the max t for t-design), 
to report a vector of t-values for which the property holds? 


>>>> 2.  Add the indicator "affine":  true if and only if any two
>>>> nonparallel blocks meet in the same number \mu of points.
>>>> Analogous to the t-wise balance indicator, include the value
>>>> of \mu.
>>>> 
>[...]
>>
>>These seem to me to be pretty strong reasons. Not only is affineness 
explicitly 
>>seen, but other indicators becomes more useful. Do you disagree?
>>
>>
>
>Not at all. I support an indicator for this property or something
>slightly more general. Should we consider indicators for the 
>following properties?
>
>(1) the dual of pairwise balanced: having just one value of pairwise
>block concurrences (just one size of pairwise block intersections), and
>giving this value.
>
>(2) having just two values of pairwise block concurrences 
>(just two sizes of pairwise block intersections), and giving these
>values (0 and \mu being a special case).
>

It is just a question of how far we go with dual properties. I think the case is 
sufficiently strong for the suggested "affine." Do you want to make the case for 
(1) and (2)? Here is what immediately occurs to me:

>From a statistician's perspective, some optimality results are known for designs 
which are duals of BIBDs. (1) tells us something on this score.

If (2)=true and resolvable=true then we know (2) has the special case (0,\mu), 
which covers the examples I cited in arguing for "affine." There is some 
interest from a robustness viewpoint for knowing that (2)=true with (\mu,\mu+1).
But is this strong enough to include (2) as one of our fundamental indicators? 
Are there other reasons for the more general (2) over and above "affine"?  

If we go with (2), then do we also include the corresponding property for 
treatment concurrence? In any case, I can't see going beyond two distinct counts 
on any of this, or we will get away from the main idea for having a collection 
of elementary indicators.

JP