On a Complete Class of Linear Unbiased Estimators for Randomized Factorial Experiments

Abstract

Consider a factorial system of order $N = p^m$, which consists of $m$ factors each at $p$ levels. The factorial model relates the expected yield to the various treatment combinations in terms of a linear function of $N = p^m$ parameters $(\beta_0, \beta_1, \cdots, \beta_{N-1})$, which represent the main effects and interactions. A subset of $S = p^s(s < m)$ preassigned parameters is specified for estimation and testing of hypotheses. The other $N - S$ parameters are considered as nuisance ones. Unbiased estimates of the $S = p^s$ preassigned parameters may be obtained by different procedures of balanced random allocation designs (see Dempster [2], [3]). In the present paper we consider unbiased estimators with two types of randomized fractional replication, studied previously by Ehrenfeld and Zacks [4]. These procedures, designated by R.P.I. and R.P.II, are based on orthogonal fractional replication designs. As will be shown in a subsequent paper, both procedures have some optimal properties in cases where the nuisance parameters may assume arbitrary values. R.P.I. consists of choosing at random, with or without replacement, $n$ blocks of treatment combinations from the set of $M = p^{m-s}$ blocks, constructed by confounding the nuisance parameters. R.P.II. consists of choosing at random $n$ treatment combinations independently from each one of the $p^s$ blocks, constructed by confounding the pre-assigned parameters. The estimator of the pre-assigned parameters studied in the previous paper is the "least squares estimator", commonly applied in fractional replication procedures (see Kempthorne [5], Cochran and Cox [1]). When all the nuisance parameters are zero then this estimator is the least-squares estimator, and thus the best unbiased linear estimator. However, if the nuisance parameters are not zero there is no uniformly best unbiased estimator. The first question raised is whether unbiased estimates of the pre-assigned parameters can be attained by choosing a block of treatment combinations, in a similar manner to R.P.I., but with unequal probabilities, (unbalanced designs). As proven in the present paper, unbiased estimates cannot be attained with a procedure that assigns unequal probabilities to different blocks. Thus, the class of linear unbiased estimators is studied with respect to R.P.I. The same estimators can be applied to R.P.II., or to any balanced allocation design that yields a factorial model with similar properties to those of R.P.I. The statistical model adopted in the present paper is the orthogonal factorial model. Accordingly, the main effects and interactions are represented by orthogonal contrasts of the expected values of the observed random variable. This is a common practice for measuring main-effects and interactions (see Kempthorne [5]). The resulting algebra and numerical procedures are relatively simple and free of the difficult problems associated with the approach of Dempster [2], [3]. Basic notions as well as the required algebra for the factorial model are represented in Section 2. Theorems concerning the specification of the class of linear unbiased estimators, and the characterization of the subclass of conditional least squares estimators are given in Section 3. In Section 4 we prove that the subclass of conditional least squares estimators is complete. For this purpose we first derive an explicit formula for the variance-covariance matrix of any linear unbiased estimator with R.P.I.; and then we show that to any linear unbiased estimator not in the subclass of the conditional least-squares estimators one can find a conditional least-squares estimator so that the difference between the corresponding variance-covariance matrices is positive definite. For simplifying the arguments and notation, the definitions and theorems in Sections 3 and 4 are given relative to a particular choice of the pre-assigned parameters for special defining parameters, by which the classification of the treatment combinations into blocks in R.P.I. is carried out. In Section 5 the results are generalized for arbitrary sets of pre-assigned and defining parameters. In Section 6 the results are extended for R.P.II.