Analysis of Variance of a Balanced Incomplete Block Design with Missing Observations

Abstract

In experimental work the results of one or more observations may be missing. An agricultural plot may be trampled, an animal may die, a test tube may be dropped, or a machine may break down. When missing values occur, the usual method of computing the various sums of squares cannot be used unless the missing values are first estimated from the existing data. Cornish' obtained a specific formula for a single missing value and an iterative procedure for several missing values by minimising the error sum of squares. He also showed that the resulting treatment sum of squares had a positive bias and how to eliminate it. Wilkinson3 obtained a general procedure for estimating missing values which involves the solutions of matrix equations. The method of Glenn and Kramer2 may also be used by considering that a balanced incomplete block design is a randomised block design with missing observations, but this procedure would require the estimation of a large number of missing values. This paper will deal with the estimation of several missing values in a balanced incomplete block design by minimising the error sum of squares. Explicit formulae for each value missing will be derived for many special cases and a completely general solution will be given. These formulae may prove to be less tedious in application than the ones now available. A direct method of analysis of the augmented data, not requiring a correction for bias in the treatment sum of squares, will be given. It should be noted that, by estimating missing values, the symmetry in the design is restored, but the estimates serve only to expedite analysis of the data; they do not by any means restore the information lost in the missing observations.