# Semi-Latin squares

Meeting on Semi-Latin Squares
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

### Rosemary Bailey

Definitions and results were given for (n×n)/k semi-Latin squares and in particular, those where $\lambda$ij (the number of blocks in which treatments i and j occur) takes only two values (e.g. 0 and 1). (5×5)/2 designs with this property exist which are both Trojan and non-Trojan.

Optimality results were given. Trojan squares are optimal (for a range of criteria) where they exist, even for the n2/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.

### Steve Gilmour (and Steve Ferris)

Examples of factorial treatment structures for the (4×4)/2 Trojan design were given. For 2 treatments there are two types of allocation of treatments to letters, one having a better pattern of efficiency factors, but perhaps making the analysis more difficult. It is possible to get a more efficient block design by sacrificing orthogonality with columns. 4×2 treatments allow five types of allocation. Spreading the confounding'' with alphabets between as many contrasts of interest as possible, with the worst confounding for the most important contrasts, gives higher efficiency factors.

### Julie Leeming

The effect of considering interblock information for the (3×3)/4 design was considered. Which design is best depends on the ratio of variances in different strata. For data from an experiment on lettuce mildew, the optimal design for only intra-block information is still optimal. but it would not be if $\sigma$2R×C were smaller.

### Rodney Edmonson

Darby and Gilbert's data from a (5×5)/4 Trojan were studied. Residual analysis indicates an edge effect, so the design used was not successful.

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.

### Hervé Monod

An example of a (4×4)/5 design used in a field trial was given. The design was poor, but the analysis recommended was even worse.

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.

### Future Actions

RP proposed putting together a Genstat catalogue of designs. He would seek advice on which designs to put in it.

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