Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series B, Pt. 2, 181-199

DESIGNING TWO-FACTOR EXPERMENTS FOR SELECTION THAT ALLOW RANDOMIZATION RESTRCTIONS

By

GUOHUA PAN, *Oakland University, Rochester*

And

THOMAS J. SANTNER, *Ohio State University, Columbus
*

SUMMARY. It is often convenient, or necessary, to use a randomization restricted experimental design in an investigation involving two or more factors.One important objective in such an investigation can be to select the treatment combination associated with the largest mean subject to some specified design criterion. One widely used design requirement is the so called indifference-zone probability formulation, that states that the probability of a (correct) selection of the best treatment combination should be at least equal to a specified confidence level when the difference between the largest and second largest means is at least some threshold value, usually the minimal difference of practical interest. This paper determines the number of replications or blocks required by certain statistical procedures to select the optimal treatment combination using an indifference-zone design requirement. The details are provided for the split-plot design and a short description is given of the adaptations required for general multi-factor experiments. The latter is illustrated for the strip-plot design. The analysis of the split-plot design assumes that the block effects, the potential confounding effects (the ``whole-plot error'' in agricultural experiments) and the measurement errors are normally distributed. Procedures are given for both additive and non-additive experiments and for two different states of knowledge about the confounding effect and measurement error variances. In particular, procedures are provided when there is known confounding effect and known measurement error variances, {\em or}known confounding effect but unknown measurement error variance. None of the procedures requires knowledge of the magnitude of the block effects. The constants required to implement the procedures are shown to be computable from available FORTRAN programs and tables..

*AMS (1991) subject classification.* 62F07, 62C10.

*Key words and phrases*. Indifference zone, ranking and selection, split-plot design, stripe-plot design, two-way layout, optimal design, least favourable configuration, screening, restricted randomization.