Mathematical Statistics and its Applications to the Biosciences
Meeting organized by the Bernoulli Society and the Institute of Mathematical Statistics
Rostock, 31 August-4 September 1997

Planning Experiments

Joachim Kunert,
Universität Dortmund, Germany:
On the analysis of crossover experiments with more than two treatments

In crossover experiments each experimental unit receives more than one treatment, one after the other. There are two statistical methods to reduce the impact of of possible carryover effects.

The first step is to go on using the standard estimate for treatment differences but to use balanced designs. Here each treatment is preceded by every other treatment equally often. If carryover effects occur, then the usual estimate for treatment differences is not unbiased, but with a balanced design the bias is minimal (Azaïs and Druilhet, 1997). There are randomization procedures by Azaïs (1987) and by Bailey (1985), which keep the balance intact but justify the analysis as a simple block design or a row-column design respectively.

A second step uses an estimate of treatment differences which is corrected for carryover effects. It was shown by Kunert and Utzig (1993) that the usual estimate of the variance of this corrected estimate is biassed. However, a conservative estimate of this variance can be derived.

The paper discusses results from the literature by means of simulations. We simulate the randomization distributions of the estimates of treatment differences under different randomization schemes and in the presence of carryover effects of different sizes.

References

1. Azaïs, J. M. (1987) Design of experiments for studying intergenotypic competition. Journal of the Royal Statistical Society B 49, 334-345
2. Azaïs, J. M. and P. Druilhet (1997) Optimality of neighbour-balanced designs when neighbour effects are neglected. Journal of Statistical Planning and Inference , to appear
3. Bailey, R. A. (1985) Restricted Randomization for neighbour-balanced designs. Statistics & Decisions, Supplement 2, 237-248
4. Kunert, J. and B. Utzig (1993) Estimation of variance in crossover designs Journal of the Royal Statistical Society B 55 (1993), 919-927

Emlyn Williams,
CSIRO, Canberra, Australia:
Experimental design for use in tree breeding

Tree breeding trials can run for many years and are usually very expensive to maintain. Hence breeders are keen to extract the maximum amount of quality information from these trials. It is necessary to adequately cater for field trend that is either present at establishment or develops during the course of the trial. Good experimental design with multiple blocking factors allows seedlot means to be estimated and compared as precisely as possible.

Site considerations dictate the design and layout of any forestry trial and so an extensive range of experimental design types needs to be available in order to be able to impose blocking structures appropriate for a particular site. The software package CycDesigN has ben developed to allow the computer generation of efficient experimental designs for a wide range of parameter sets and design types. Included in this package are options for resolvable, cyclic, t-latinized, partially-latinized and factorial designs in one and two dimensional blocking structures. A number of examples will be presented.

Andre Kobilinsky,
INRA, Versailles, France:
Designing experiments in the agro-food industry

Resolution IV asymmetrical designs with a mixture of 2-, 3- and 4-level factors are very efficient to study various food-industry processes, but they are difficult to build and analyse. We point out some important questions raised by them.

• A simple way to get such designs, when only 2 and 4 level factors are involved, is to decompose 4-level factors into 2-level pseudofactors, then to use the theory for classical 2^n-k fractional design to get a suitable design. However there is no simple correspondence between the different kinds of designs that can be obtained for the asymmetrical design and those for the symmetrical associated design for pseudofactors. Specific algorithm has therefore to be developed for the asymmetrical case.
• A classical way to introduce 3-level factors is to merge together two levels of a 4-level factor. However, since the resulting levels are not equally represented, resolution 4 may be lost with the usual reparametrization and a specific reparametrization has to be used to make a proper analysis.
• In many cases, these designs do not include any replication and specific methods have to be used to estimate the residual variance and detect the active effects. We briefly tackle some of the problems that arise in this context.

R. A. Bailey

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