indicators

[Leonard Soicher]

On Wed, Sep 24, 2003 at 10:04:47AM -0400, John P. Morgan wrote:
> 
> Dear All,
> 
> This note contains three suggestions for our list of
> indicators.
> 
> 1.  Change the "t-design" indicator to "t-wise balance":
> true if and only if every t-subset of the treatment set
> occurs with the same frequency in blocks.  

Are you suggesting an indicator for each possible t???

> This simply drops
> the requirement for equiblocksize in the current indicator.
> 

If D is a t-design then D is an s-design for all s=0,...,t.  On the other
hand, a block design may be t-wise balanced but not s-wise balanced for
some s<t. Hence, it makes sense to handle t-wise balance as we do with the
t_wise_balanced element, using one or more index flag entries to specify
for given values of t whether or not the design is t-wise balanced. The
property of whther or not D is a t-design for some t>1, and if so,
the maximum t for which D is a t-design fits well as an indicator.

> Reasoning:  We have a separate indicator for
> constant_blocksize.  Why combine information for several
> properties in a single indicator when each property has
> independent interest?  I would like to suggest in general
> that indicators better carry the intent of reporting
> fundamental information if they refer to single properties
> rather than collections of properties.
> 

I agree with this principle, but believe we are handling 
t-wise balance and the t-design property correctly in the 
current ext-rep. 

> 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.
> 

We could do this, although this "affine" indicator information is already
contained in the (pairwise) block_concurrences. 

> Reasoning:  aside from intrinsic interest, this allows other
> important properties to be deduced from the set of
> indicators.  For example,
> affine+resolvable+constant_blocksize is equivalent to an
> orthogonal array of strength 2 and index \mu.  If 2-wise
> balance also holds and \mu=1, then the design is equivalent
> to a complete sets of MOLS, etc, etc.
> 
> 3.  Change the definition of "partially_balanced" to the
> following:  true if and only if the diagonal of the
> information matrix is constant, and the off-diagonal
> elements have values that follow an association scheme.
> This is equivalent to saying that the information matrix is
> in the Bose-Mesner algebra.
> 

I would like feedback from RAB and PJC on this.

> Reasoning:  This definition of partial balance has been
> implicity floating around in the statistical literature for
> quite some time, but I don't know where or if it is formally
> stated (possibly because it is so obvious, possibly because
> originally PB was thought of only for binary, equiblocksize
> and so properties were not then described in terms of the
> info matrix).  It reduces to the classical definition by
> Bose et al when the design is binary and constant_blocksize.
> In taking this route we make concrete an idea that numerous
> researchers have used.  Importantly, we gain the ability to
> talk in complete generality about partial balance without
> being restricted to either binary or constant_blocksize.
> Moreover, it breaks out the essential association scheme
> property from other fundamental properties covered by other
> indicators.  Again, why lose information on individually
> interesting properties by combining indicators of several
> properties?  A better choice for the name of this indicator
> is probably "underlying_association_scheme" or something
> similar.  Then the classical definition for partially
> balanced design is met if underlying_association_scheme,
> binary, and constant_blocksize are all true.
> 
> JP
> 

Leonard