The Present Status of Variance Component Analysis

rHE ELEMENTARY THEORY of variance component analysis has been 'discussed in recent papers, by Daniels (10), Crump (8, 9) and Eisenhart (11). Since the appearance of the last of these the theory has been extended in a number of directions. It seems worthwhile at this time then to present a unified discussion of the theory in its present state which indicates the available results in order that further work on the theory may be directed at the important unsolved problems. Although an effort will be made to avoid excessive discussion of elementary well known results, enough of these will be included to make the general content of the paper essentially self contained and coherent. It seems pertinent to begin by defining rather carefully what problems are included in the province of variance component analysis. This is most easily done by considering a particular type of multiple classification which contains all of the essential elements of the situation. It must be remembered however that the general features of the next few paragraphs are common to any set of data arranged in a multiple classification and described by a linear model. Data arranged in a two-way classification with the same number of observations in each cell will be used to illustrate. Let the classes in the two criteria of classification be A, , A2 , B e, A, and B1 , B2 , * , Rb respectively. Then if yhii denotes the j-th datum in subclass AhBi the complete linear model for these data is included in the following equations: