## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 33, Number 2 (1962), 600-619.

### A Calculus for Factorial Arrangements

B. Kurkjian and M. Zelen

#### Abstract

This paper introduces a special calculus for the analysis of factorial experiments. The calculus applies to the general case of asymmetric factorial experiments and is not restricted to symmetric factorials as is the current theory which relies on the theory of finite projective geometry. The concise notation and operations of this calculus point up the relationship of treatment combinations to interactions and the effect of patterns of arrangements on the distribution of relevant quantities. One aim is to carry out complex manipulations and operations with relative ease. The calculus enables many large order arithmetic operations, necessary for analyzing factorial designs, to be partly carried out by logical operations. This should be of importance in programming the analysis of factorial designs on high speed computers. The principal new results of this paper, aside from the new notation and operations, are (i) the further development of a theory of confounding for asymmetrical factorials (Section 4) and (ii) a new approach to the calculation of polynomial regression (Section 5). In particular, the use of the calculus enables one to write the inverse matrix of the normal equations for a polynomial model as a partitioned matrix. As a result it only requires inverting matrices of smaller order.

#### Article information

**Source**

Ann. Math. Statist., Volume 33, Number 2 (1962), 600-619.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177704585

**Digital Object Identifier**

doi:10.1214/aoms/1177704585

**Mathematical Reviews number (MathSciNet)**

MR139240

**Zentralblatt MATH identifier**

0109.38002

**JSTOR**

links.jstor.org

#### Citation

Kurkjian, B.; Zelen, M. A Calculus for Factorial Arrangements. Ann. Math. Statist. 33 (1962), no. 2, 600--619. doi:10.1214/aoms/1177704585. https://projecteuclid.org/euclid.aoms/1177704585

#### See also

- Part II: M. Zelen, W. Federer. Applications of the Calculus for Factorial Arrangements II: Two Way Elimination of Heterogeneity. Ann. Math. Statist., Volume 35, Number 2 (1964), 658--672.Project Euclid: euclid.aoms/1177703563