The Analysis of Designed Experiments with Missing Observations

Abstract

SUMMARY This paper examines published methods for analysing designed experiments with missing observations. An exact method originated by Wilkinson is shown to be a viable alternative to the more commonly used iterative techniques. It has the advantage that, from a small number of standard analyses of the design using specially constructed data vectors, a full and accurate analysis of the observed data can be obtained. THIS paper examines the relationship between exact and iterative procedures for analysing designed experiments with missing observations. The various techniques are presented in a common mathematical framework. The notation used originates largely from Tocher (1952). For a more detailed survey of the literature, the reader is referred to Hoyle (1971). The occurrence of missing observations destroys the orthogonal or balanced structure usually present in a designed experiment. The incomplete experiment can be analysed by using a standard least-squares method to fit successive sets of parameters corresponding to successive terms in the model. This may prove to be a tedious and possibly inaccurate procedure, especially with large and complicated experiments. A more satisfactory method would be to insert values for the missing observations and to perform the analysis using the structure of the original experiment. Yates (1933) showed that the correct values to insert are those for which, when the formally completed experiment is analysed, the residuals in the missing plots are zero. The most common procedure is to use an iterative approach similar to that proposed by Healy and Westmacott (1956). With this method, initial guesses are inserted for the missing values and, at each iteration, the residuals in the missing plots are subtracted from the current estimates. It can be proved that the residuals in the missing plots must tend to zero and that the residual sum of squares must converge to the correct value. Once these missing-value estimates have been obtained for the full model, the complete analysis is usually performed with these values inserted, the degrees of freedom for the residual sum of squares being reduced by the number of missing values. Exact procedures for analysing such an experiment consist of a series of analyses using specially constructed data vectors. Notable among these procedures are the covariance technique of Bartlett (1937) and the method of Wilkinson (1958a, b). Recent papers on exact methods (Rubin 1972; Haseman and Gaylor, 1973; John and Prescott, 1975) deal mainly with the problem of obtaining missing-value estimates for the full model. Presumably, it is intended that the complete analysis be performed with these estimates inserted for the missing obser- vations. Any method which finds missing-value estimates only for the full model and uses these