Some Contributions to Anova in One or More Dimensions: I

Abstract

Two models are considered in detail which are the Models I and II of ANOVA in the terminology of Eisenhart [2]. The present paper, which deals with the one dimensional or univariate case, and its sequel, which will deal with the multidimensional or multivariate case, seek to give a unified general treatment, using matrix methods, of certain problems under the two models of ANOVA. Section 1 of each paper, which deals with Model I, is of the nature of a resume giving the main results of a general treatment discussed elsewhere [10, 11, 12] by one of the authors. Section 2 of each paper, which deals with Model II or variance components model, is self-contained, and presents a natural tie-up between the analyses under the two models for a $k$-way classification. Results in estimation, testing of hypotheses and confidence bounds are presented, although the main emphasis is on the results in confidence bounds (simultaneous and/or separate) on meaningful parametric functions which are physically natural and mathematically convenient measures of departure from customary null hypotheses. It will be seen that a mixed model, which would include both Models I and II as special cases, can be defined, and the associated problems can be studied by using methods which are, essentially, a combination of the methods given for the separate models in Sections 1 and 2, respectively, of this paper. Since nothing essentially new is involved in such a study, this paper does not explicitly discuss it. Unless otherwise stated, capital letters will denote matrices and small letters in boldface will denote column vectors. Such letters when primed denote transposes. For instance, $A(p \times q)$ denotes a matrix with $p$ rows and $q$ columns, $A'(q \times p)$ denotes the transpose of $A, \mathbf{a}(p \times 1)$ denotes a column vector with $p$ elements and $\mathbf{a}'(1 \times p)$, the transpose of $\mathbf{a}$, is a row vector. In particular, $I(p)$ will denote the identity matrix of order $p$ and $0(p \times 1)$ and $0(p \times q)$ will stand, respectively, for the null vector of order $p$ and the null matrix with $p$ rows and $q$ columns.