# Random Choice of Fractional Replications

• Shelemyahu Zacks
Chapter
Part of the Statistics for Industry, Technology, and Engineering book series (SITE)

## Abstract

In the present chapter several papers on random choice of fractional replications, which were published by Ehrenfeld and Zacks (Ann Math Stat, 32:270–297, 1961; Ann Math Stat 38:1494–1507, 1967) and Zacks (Ann Math Stat, 35:696–704, 1964; Ann Math Stat, 39:973–982, 1968) will be discussed. The first two sections introduce classical notions and notations in order to facilitate the material in the following sections. The random choice procedures presented here were designed to eliminate the bias (aliases) in estimating parameters in the classical fractional replication designs. This bias is, as will be shown, a linear function of the aliases parameters. The variances of the estimated parameters increase as a result of the random choice of fractional replications. The question is whether the mean-squared-errors of the estimators are reduced. We discuss also optimal strategies, generalized LSE, and randomized fractional weighing designs. The approach in the present chapter is a marginal analysis compared to a conditional analysis in the classical treatment of fractional replications. The difference between the two approaches is similar to the difference between the design and the modeling approaches in sampling surveys. The design approach is to choose the units of a population at random, and the properties of estimators depend on the randomization procedure. In the modeling approach the analysis is Bayesian, conditional on the units chosen, not necessarily at random. The population units are the fractions (blocks) of the full factorial experiment.

## References

1. Ehrenfeld, S., & Zacks, S. (1961). Randomization and factorial experiments. Annals of Mathematical Statistics, 32, 270–297.
2. Ehrenfeld, S., & Zacks, S. (1963). Optimal strategies in factorial experiments. Annals of Mathematical Statistics, 34, 780–791.
3. Ehrenfeld, S., & Zacks, S. (1967). Testing hypotheses in randomized factorial experiments. Annals of Mathematical Statistics, 38, 1494–1507.
4. Kenett, R. S., & Zacks, S. (2014). Modern industrial statistics with R, minitab and JMP (2nd ed.). Chichester: Wiley.
5. Zacks, S. (1963a). On a complete class of linear unbiased estimators for randomized factorial experiments. The Annals of Mathematical Statistics, 34, 769–779.
6. Zacks, S. (1963b). Optimal strategies in factorial experiments. The Annals of Mathematical Statistics, 34, 780–791.
7. Zacks, S. (1964). Generalized least squares estimators for randomized fractional replication designs. The Annals of Mathematical Statistics, 35, 696–704.
8. Zacks, S. (1966a). Randomized fractional weighing designs. The Annals of Mathematical Statistics, 37, 1382–1395.
9. Zacks, S. (1966b). Unbiased estimation of the common mean of two normal distributions based on small samples. Journal of the American Statistical Association, 61, 467–476.
10. Zacks, S. (1968). Bayes sequential design of fractional factorial experiments for the estimation of a subgroup of pre-assigned parameters. The Annals of Mathematical Statistics, 39, 973–982.