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# Algorithm for Constructing Statistical Analysis of Variance Models

IP.com Disclosure Number: IPCOM000076840D
Original Publication Date: 1972-May-01
Included in the Prior Art Database: 2005-Feb-24
Document File: 3 page(s) / 22K

IBM

## Related People

Stacy, EW: AUTHOR

## Abstract

One step in the analysis of variance for statistically designed experiments is the formal statement of a mathematical model of the design. This facilitates, for example, the computation of expected values for mean squares which are encountered in the analysis of variance. The model expresses the "structure" of data which is collected or observed. Models are linear in terms of the main effects (factors) and their interactions. The experimental design dictates which interaction terms must appear in the model.

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Algorithm for Constructing Statistical Analysis of Variance Models

One step in the analysis of variance for statistically designed experiments is the formal statement of a mathematical model of the design. This facilitates, for example, the computation of expected values for mean squares which are encountered in the analysis of variance. The model expresses the "structure" of data which is collected or observed. Models are linear in terms of the main effects (factors) and their interactions. The experimental design dictates which interaction terms must appear in the model.

This is an algorithm for constructing a model for any of a large variety of designs, which are classified as crossed, nested, cross-nested, or split-plot.

Input to the algorithm consists of nesting information about each of the factors, here denoted by A, B, C.... Means is denoted by Omega; and error is denoted by Epsilon. The following, a model for a three-way classification cross- nested with C nested in B, is an illustration in which experimental data is indicated by Y(ijkl):

(Image Omitted)

The factors A, B, and C appear along with the two-factor interaction (AB) and (AC). If no factor were nested, the model would have to accommodate one more two-factor interaction and the three-factor interaction.

Appearance or nonappearance of the symbol for a factor in any term is indicated by 1 or 0, respectively. Thus, in reference to the above illustration, the interaction (AC) is denoted by 101 since the symbol for factor B is absent.

Steps in the algorithm are: 1. Consider the factors and all conceivable interactions (two factor, three-factor, etc. up to k-factor interactions where k is the number of factors). These are generated systematically by converting the numbers 1, 2,...,(2/k/-1) from base 10 to base 2 and modifying their format by affixing zeros on the left, if necessary, so that each number is associated with k digits. (To illustrate the modified format, the number 1(base2) written 00...01, the number of indicated zeros being k-1). The resulting 2/k/-1 numbers indicate all possible combinations of k factors taken one at a time, 2 at a time, 3 at a time, etc. 2. Consider each number and discard any which indicate the presence of a factor in the absence of any factor in which it is nested. If no factor is nested in another, none of the indicated numbers would be discarded. Or, if C is nested in B, a number is discarded whenever a 1 appears in the third position but not in the second position. 3. Consider the remaining numbers for possible changes in their nonzero elements. Change 1 to 0 if the corresponding factor is such that, in the same number, some other fact...