Difference squares

The differences between treatments in different sets within a block (in cyclic designs) are similarly useful in OBIBDs to the differences between treatments within blocks in BIBDs. See here for further details..

Relationships with other designs

OBIBDs are closely related to many other designs - see here for more.

Geometric interpretation of OBIBDs

To some extent, designs constructed by the non-geometric methods can be interpreted as if they had been constructed as a geometry. There are four features of the design which are relevant.

  1. With each point of either set is associated r points of the other set, namely, the points of the other set with which it occurs: these sets of r points can be regarded as planes.
  2. These r associated points can be grouped into r lines of k points, namely the blocks in which they occur when associated with a given point of the other set. Each point occurs k times. These form a configuration in the sense of e.g. Carmichael or HCD IV.6.
  3. These blocks or lines can be regarded as the intersection of the planes.
  4. These lines are paired off with lines of the other set; the lines of the other set necessarily consist of the individual points with which the planes are associated.
  5. The multiple intersections of configurations may be of interest on their own account. When k = 3 and λ = 1, the 3 lines formed by taking the intersection of these 3 configurations two at a time,
    • may all coincide, as above,
    • the three may be mutually skew,
    • two may intersect with the third skew to both,
    • the three may pass through a single point
    • or one line may intersect the other two in distinct points.
    The number of points involved in these patterns is 3,9,8,7,7 respectively: as a crude measure, the number of distinct points may suffice, but in some cases it may be appropriate to distinguish between the last two cases.

A couple of formulae summarise some of this information.

  • The first counts how many configurations are present of each type, in exponential form such as aibj...
  • The second counts how many multiple configuration intersections there are of each type, giving another exponential formula.

The intersections of pairs of configurations will be of less interest, with pergolas at least, because the underlying symmetric BIBD will ensure that they will all be the same.

The examples for v=19 and v=15 may help to make this clear. Further consideration of the correlated block design underlying the pergola with v=19 is here.