other block designs

[John P. Morgan]

Dear All,

This note outlines some important examples of block designs that we cannot
currently accomodate.  There are other, equivalent combinatorial objects for
some of the four examples listed which are not mentioned.  A common theme for
the four examples is given.

Example 1.  Our concept of block design requires that blocks are multisets, and
consequently not ordered.  As we all know, there are many block design concepts
that require order on the treatments in a block (terraces are one such example).
Such "ordered" designs could potentially be reported in our current format as
follows.  Create "point_labels" which are ordered pairs (i,j) where i in
{1,...,max_blocksize} identifies position in a block, and j is an actual
treatment value.  These pairs are lex-ordered for the correspondence with
integers 1,2,...,v reported in the list of blocks, but now v is not the actual
number of treatments; all of 1,...,v may not occur; and in terms of 1,...,v many
of the statistical and other properties make no sense.  So if this labelling
were to work, the software would have to be told to refer to the 2nd entry in
the ordered pair label for all the customary calculations.  This is all quite
messy - better would be to take all blocks ordered as given, dropping the
lex-order requirement.  Better, that is, IF we wanted to accomodate ordered
blocks at all.  I am not suggesting this.  I think this example is interesting
because the identification of treatments with ordered sets is a theme that works
for other types of designs we also cannot currently accomodate.  The next
example is an important class of designs with ordered blocks.  The third and
fourth are not about ordering within blocks, but the treatments themselves are
ordered sets (inducing similar problems for our current calculations).


Example 2.  Row-column designs are designs with two blocking factors.  Think of
the blocks of the simple block designs we now accomodate as the first factor,
then think of position within the block as the second blocking factor.  You see
that row-column designs are combinatorially nothing but our ordinary block
designs with ordering in the blocks.

Example 3.  Nested block designs such as nested BIBDs that Donald Preece has
studied extensively are neat generalizations of resolvable designs.  Resolvable
designs have blocks partitioned into resolution classes, each class being a
complete block design.  Nested designs have blocks partitioned into classes
which, when the classes are considered as blocks, make another block design with
some desirable property.  These can also be reported via blocks with ordered
pairs, but now the first member of the pair identifies class rather than
position.  One would want all of our properties calculated for the original
blcok design and again for the design defined by the classes (we do not do the
latter for the resolvable designs, but it is all trivially known in that case).

Example 4.  There are block designs known as confounded factorials where the
treatment set is all ordered t-tuples (i_1,...,i_t) for i_1\in V_1,..., i_t \in
V_t.  This would at first appear to be no problemn for us; if |V_j|=v_j, create
treatment labels 1,2,...,v where v=v_1*v_2*...*v_t and report the design in
terms of 1,...,v along with the identification with the underlying treatment set
in "point_labels". BUT, there are many such designs which are disconnected in
terms of 1,...,v.  More generally the statistical properties as now formulated
will usually be irrelevant because interest is on specific classes of
comparisons among the underlying treatment combinations, NOT in all comparisons
of 1,...,v.  As in Example 1, this would require a different implementation of
most of what we report in a way that properly dealt with the "true" treatment
structure as reported by the labels.  One more twist on this is as follows.  Any
block design where each treatment is an ordered t-tuple can be separated into t
separate block designs, one for each position in the t-tuple.  Researchers such
as Donald have constructed many of these "superimposed" designs by starting with
a separate design d_i for each set V_i and then finding ways to combine them to
achieve desirable properties.

SUMMARY.  We cannot accomodate any of these common, immediate generalizations of
our simple block designs:  (i) ordering within blocks; (ii) a second, orthogonal
blocking factor; (iii) partitions of blocks to form "super" designs, except in
the most trivial case; (iv) treatments with underlying factorial structure.  All
can be represented as simple block designs with appropriate treatment labels,
but the meaning of the labels must be properly identified and used in each case.
This may be food for future thought.

JP