On the Null-Distribution of the $F$-Statistic in a Randomized Balanced Incomplete Block Design Under the Neyman Model

Abstract

While the analysis of variance test statistic $F$ in a balanced incomplete block design without randomization is a constant under the Fisher model, i.e., a linear model without technical errors, and it has no distribution at all, under the Neyman model, i.e., a model with technical errors, its null-distribution is a non-central $F$-distribution whose non-centrality parameter being a quadratic form of unit-errors. This is carried out in Section 1. The mean value and the variance of $\theta$ with respect to the permutation distribution due to the randomization are calculated in Section 2, and in Section 3, the null-distribution of the $F$-statistic after the randomization is shown to be approximated by the familiar central $F$-distribution under the Neyman model assuming no interaction between treatments and experimental units, if the following two conditions are satisfied: (i) the variances of unit-errors within blocks are sufficiently uniform from block to block, and (ii) the number of blocks is sufficiently large. Since the unit-errors are not directly observable, how one can group the experimental units into blocks in such a way as the above Condition (i) would be satisfied is another problem, which is left open in this paper. R. A. Fisher [2] initiated the use of the so-called "randomization procedure" in order to control the unit-errors in block designs. Mathematical treatments of the Fisher randomization in randomized block and the Latin-square designs were made by B. L. Welch [12], E. J. G. Pitman [11] and M. B. Wilk [14]. Underlying models in those works may be called the "Fisher models", i.e., containing no technical errors. J. Neyman et al. [7] and M. B. Wilk [13] pointed out that there are instances in which a model with technical errors is more adequate by the very nature of the problem under consideration, and the present author calls this sort of models the "Neyman models" for convenience. M. D. McCarthy [6] investigated the null-distribution of the analysis of variance test statistic in a randomized block design under the Neyman model, and he came out with rather pessimistic results. J. Ogawa [10] treated the same problem, and his result turned out to be supporting the usual approximation by the familiar central $F$-distribution. The purpose of this article is the treatment of the same null-distribution problem for a randomized balanced incomplete block design under the Neyman model. Since a randomized block design is a limiting case of a randomized BIBD, this article should be regarded as a generalization of the earlier work.