### Abstract

Experiments usually aim to study how changes in various factors affect the response variable of interest. Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters, a considerable amount of time is often necessary to implement the procedure for estimating the effects. On the other hand, it has recently been shown that the model in experimental design can also be expressed in terms of an orthonormal system. In this case, the model contains no redundant parameters. Moreover, the theorem with respect to the sum of squares for the 2-factor interaction, needed in the analysis of variance (ANOVA) has been obtained. However, 3-factor interaction is often to be considered in real cases, but the theorem with respect to the sum of squares for the 3-factor interaction has not been obtained up to now. In this paper, we present the theorem with respect to the sum of squares for the 3-factor interaction in a model based on an orthonormal system. Furthermore, we can also obtain the theorem for interactions with 4 or more factors by the similar proof. Hence, in any real case, we can execute ANOVA in the model based on an orthonormal system.

Original language | English |
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Title of host publication | Conference Proceedings - IEEE International Conference on Systems, Man and Cybernetics |

Pages | 1990-1995 |

Number of pages | 6 |

DOIs | |

Publication status | Published - 2012 |

Event | 2012 IEEE International Conference on Systems, Man, and Cybernetics, SMC 2012 - Seoul Duration: 2012 Oct 14 → 2012 Oct 17 |

### Other

Other | 2012 IEEE International Conference on Systems, Man, and Cybernetics, SMC 2012 |
---|---|

City | Seoul |

Period | 12/10/14 → 12/10/17 |

### Fingerprint

### Keywords

- Analysis of variance
- Bandlimited function
- Design for experiments
- Estimation
- Fourier transforms
- Sampling theorem

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Control and Systems Engineering
- Human-Computer Interaction

### Cite this

*Conference Proceedings - IEEE International Conference on Systems, Man and Cybernetics*(pp. 1990-1995). [6378030] https://doi.org/10.1109/ICSMC.2012.6378030

**A note on ANOVA in an experimental design model based on an orthonormal system.** / Ukita, Yoshifumi; Matsushima, Toshiyasu; Hirasawa, Shigeichi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Conference Proceedings - IEEE International Conference on Systems, Man and Cybernetics.*, 6378030, pp. 1990-1995, 2012 IEEE International Conference on Systems, Man, and Cybernetics, SMC 2012, Seoul, 12/10/14. https://doi.org/10.1109/ICSMC.2012.6378030

}

TY - GEN

T1 - A note on ANOVA in an experimental design model based on an orthonormal system

AU - Ukita, Yoshifumi

AU - Matsushima, Toshiyasu

AU - Hirasawa, Shigeichi

PY - 2012

Y1 - 2012

N2 - Experiments usually aim to study how changes in various factors affect the response variable of interest. Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters, a considerable amount of time is often necessary to implement the procedure for estimating the effects. On the other hand, it has recently been shown that the model in experimental design can also be expressed in terms of an orthonormal system. In this case, the model contains no redundant parameters. Moreover, the theorem with respect to the sum of squares for the 2-factor interaction, needed in the analysis of variance (ANOVA) has been obtained. However, 3-factor interaction is often to be considered in real cases, but the theorem with respect to the sum of squares for the 3-factor interaction has not been obtained up to now. In this paper, we present the theorem with respect to the sum of squares for the 3-factor interaction in a model based on an orthonormal system. Furthermore, we can also obtain the theorem for interactions with 4 or more factors by the similar proof. Hence, in any real case, we can execute ANOVA in the model based on an orthonormal system.

AB - Experiments usually aim to study how changes in various factors affect the response variable of interest. Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters, a considerable amount of time is often necessary to implement the procedure for estimating the effects. On the other hand, it has recently been shown that the model in experimental design can also be expressed in terms of an orthonormal system. In this case, the model contains no redundant parameters. Moreover, the theorem with respect to the sum of squares for the 2-factor interaction, needed in the analysis of variance (ANOVA) has been obtained. However, 3-factor interaction is often to be considered in real cases, but the theorem with respect to the sum of squares for the 3-factor interaction has not been obtained up to now. In this paper, we present the theorem with respect to the sum of squares for the 3-factor interaction in a model based on an orthonormal system. Furthermore, we can also obtain the theorem for interactions with 4 or more factors by the similar proof. Hence, in any real case, we can execute ANOVA in the model based on an orthonormal system.

KW - Analysis of variance

KW - Bandlimited function

KW - Design for experiments

KW - Estimation

KW - Fourier transforms

KW - Sampling theorem

UR - http://www.scopus.com/inward/record.url?scp=84872425981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872425981&partnerID=8YFLogxK

U2 - 10.1109/ICSMC.2012.6378030

DO - 10.1109/ICSMC.2012.6378030

M3 - Conference contribution

AN - SCOPUS:84872425981

SN - 9781467317146

SP - 1990

EP - 1995

BT - Conference Proceedings - IEEE International Conference on Systems, Man and Cybernetics

ER -