A Unified Theory for Missing Plots in Incomplete Block and Response Surface Designs.

Abstract

In experimental work it sometimes happens that the results of one or more observations are lost through failure to record, gross errors in recording, or accidents. When observations are absent, the correct procedure is to write down a mathematical model for all observations that are present. Because the system of equations loses its symmetry, solutions become difficult. Many people have addressed themselves to the problem of missing data. The usual procedure is to insert values for the missing data so as to obtain a set of complete data allowing the usual methods of analysis to be used. Block designs and response surface designs are playing an increasing role in biostatistical and medical research. Several examples of such experiments are given. In industry and medical research, due to the limitations on availability of suitable experimental material and cost considerations, complex experiments should be considered. Frequently, experiments in medicine are not designed with forethought but arise through actual situations. The existing literature does not give full details of the consequences of missing plots in such complex designs. Also since we deal with animals and patients in biostatistical research, missing observations are more likely to arise than in agricultural experiments. Due to the increased use of complex experiments, it is essential to examine the analysis of such designs with missing plots. The purpose of this dissertation is to provide a general, unified theory of missing plots for almost all block and response surface designs used in practice. It is assumed that the missing observations arise in a r and om manner and are not due to any treatment effect. The basic methods of estimating missing plots are presented. We use Fisher's Rule to develop the theory and provide the following quantities: (1) values to be substituted for the missing plots; (2) variances of the values; (3) variances of treatment contrasts; (4) average variance of all elementary treatment comparisons; (5) bias of treatment sum of squares; (6) relative efficiency of the design. We consider the very general cases when the design matrix is singular and when the design is arranged in blocks. The theory is applied to a r and omized block design, balanced incomplete block design, linked block design, group divisible design, partially balanced incomplete block design with m associate classes, Youden square design, and designs with property A as defined by Kurkjian and Zelen. A numerical example of a triangular design is provided. Also the theory is applied to three second order response surface designs--a central composite design with orthogonal blocking, a rotatable central composite design with nonorthogonal blocking, and an equiradial design with orthogonal blocking. The consequences of a missing plot for several different central and rotatable designs are examined.