Analysis of Nonorthogonal $n$-Way Classifications

Abstract

Four problems associated with the use of Zelen's calculus of factorials in the statistical analysis of nonorthogonal $n$-way classification data are solved. These are for the situations for which (i) some effect parameters are equated to zero, (ii) some combinations (subclasses) contain no observations, (iii) expected values of mean squares under fixed, mixed, and random models are desired, and (iv) expected values of single degree of freedom sums of squares are wanted. A unified approach to these problems was developed. Relationships to previous work, to blocked experiments, to fractional replication, and to "messy data" situations are discussed. The various analyses are first described for a nonorthogonal two-way classification and then generalized to an $n$-way classification in the final section of the paper. Numerical examples are presented to illustrate the various procedures.