Random Choice of Fractional Replications

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.