The University of Reading, 10 June 1997

Brief Summary of Contributions

Present: R. A. Bailey, R. N. Edmondson, S. J. Ferris, S. G. Gilmour, J. Jones, J. A. Leeming. A. Mead, H. Monod, R. W. Payne, D. A. Preece.

Apologies: M. G. Kenward, S. J. Welham

Optimality results were given. Trojan squares are optimal (for a range
of criteria) where they exist, even for the
*n*^{2}/*k* block designs. Where no Trojan design
exists optimal designs are known for many special cases with
*k*<*n*. With *k*<=*n*, methods of obtaining
optimal designs include inflation of Trojan design and superposition
of mutually orthogonal Latin squares.

An incomplete (4×3)/2 Trojan design with 4×2 treatments used for a
tomato trial was given, the omitted pairs of plots containing the same
level of the 4-level factor. Design has balanced confounding, but
combination of information from strata was necessary. Generally, the
problem of combining information is that stratum variances are poorly
estimated. Incomplete designs can be obtained from Youden-Freeman
rectangles and the efficiency factors carry over. (**Donald Preece** gave
an example for 14 treatments in (4×7)/2 units). A (6×5)/2 incomplete
Trojan was shown, but there may be alternatives. A (4×4)/5 ``Latinized
alpha block design'' was compared with an incomplete (5×4)/4 Trojan
design.

A question was raised about the effective degrees of freedom in
Genstat anova. **Roger Payne** explained breifly what Genstat was
doing and would send RNE more details.

A summary of known optimal semi-Latin squares was given, as were
combinations of *n* and *k* for which no optimal design was
known (especially *n*=6,10,12). ``Quasi''-inflated Trojan designs
were conjectured to be optimal.

Alpha designs and Latinized versions were reviewed. These are produced
by the Alpha+ package which finds Trojan, inflated Trojan and
quasi-inflated Trojan designs when *n* is prime. It finds
reasonably efficient, but clearly non-optimal, designs in other situaitons.

Other algorithmic methods were discussed. These could find optimal block designs, especially if simulated annealing was used, but introducing the Latin square constraints (or penalties) would cause difficulties.

RAB raised the question of putting together a book to pass on what is known to a wider audience of applied statisticians. With a title like ``Trojan Designs and their Applications'' or ``Latinized Block Designs in Practice'', various people would contribute one or more chapters, e.g.

History | DAP |

Randomization and Analysis | RAB |

Practical Situations | RNE |

Construction | HM |

Optimality | RAB |

Treatment structures | SGG/SJF |

Combining strata | JL |

Incomplete designs | RNE/DAP |

Those not working in Universities would need permission from their employers. Ideas for chapters should be sent to RAB. The RSS Lecture Notes series was suggested as a possible outlet. RAB would get in touch with the editors. A special issue of a journal was an alternative possibility.

A successor meeting may be organised, especially if we go ahead with the book, for early next year.

**Report by Steve Gilmour**

Page maintained by R. A. Bailey