### Abstract

Sums-of-products models can be characterized as a variant of the analysis of variance model, where interaction terms are assumed to be products of single-factor scales and where the assumption that main effects and interaction terms have zero sums is dropped. Because of the ordinal structure inherent in sums of products, these models are appropriate when data satisfy single-factor independence (the direction of the effects of a factor is independent of other factors) but have amplificatory violations of joint independence (the direction of the joint effects of a set of factors may depend on other factors, because a given factor amplifies the effects of some factors more than the effects of others). This paper describes data analysis methods for sums-of-products models that determine which main effect and interaction terms exist. The methods are exploratory in that no error theory is provided. A key concept is that of multi-factor residual with respect to a set of factors, computed by successively "taking out the means" of these factors. The methods involve analysis of multi-factor residuals in terms of their sums of squares and their N -way matrix ranks.

Original language | English (US) |
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Pages (from-to) | 327-371 |

Number of pages | 45 |

Journal | Journal of Mathematical Psychology |

Volume | 37 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Experimental and Cognitive Psychology
- Applied Mathematics

### Cite this

**Exploring N-way Tables with Sums-of-Products Models.** / Van Santen, Jan.

Research output: Contribution to journal › Article

*Journal of Mathematical Psychology*, vol. 37, no. 3, pp. 327-371. https://doi.org/10.1006/jmps.1993.1022

}

TY - JOUR

T1 - Exploring N-way Tables with Sums-of-Products Models

AU - Van Santen, Jan

PY - 1993/9

Y1 - 1993/9

N2 - Sums-of-products models can be characterized as a variant of the analysis of variance model, where interaction terms are assumed to be products of single-factor scales and where the assumption that main effects and interaction terms have zero sums is dropped. Because of the ordinal structure inherent in sums of products, these models are appropriate when data satisfy single-factor independence (the direction of the effects of a factor is independent of other factors) but have amplificatory violations of joint independence (the direction of the joint effects of a set of factors may depend on other factors, because a given factor amplifies the effects of some factors more than the effects of others). This paper describes data analysis methods for sums-of-products models that determine which main effect and interaction terms exist. The methods are exploratory in that no error theory is provided. A key concept is that of multi-factor residual with respect to a set of factors, computed by successively "taking out the means" of these factors. The methods involve analysis of multi-factor residuals in terms of their sums of squares and their N -way matrix ranks.

AB - Sums-of-products models can be characterized as a variant of the analysis of variance model, where interaction terms are assumed to be products of single-factor scales and where the assumption that main effects and interaction terms have zero sums is dropped. Because of the ordinal structure inherent in sums of products, these models are appropriate when data satisfy single-factor independence (the direction of the effects of a factor is independent of other factors) but have amplificatory violations of joint independence (the direction of the joint effects of a set of factors may depend on other factors, because a given factor amplifies the effects of some factors more than the effects of others). This paper describes data analysis methods for sums-of-products models that determine which main effect and interaction terms exist. The methods are exploratory in that no error theory is provided. A key concept is that of multi-factor residual with respect to a set of factors, computed by successively "taking out the means" of these factors. The methods involve analysis of multi-factor residuals in terms of their sums of squares and their N -way matrix ranks.

UR - http://www.scopus.com/inward/record.url?scp=38249000711&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249000711&partnerID=8YFLogxK

U2 - 10.1006/jmps.1993.1022

DO - 10.1006/jmps.1993.1022

M3 - Article

AN - SCOPUS:38249000711

VL - 37

SP - 327

EP - 371

JO - Journal of Mathematical Psychology

JF - Journal of Mathematical Psychology

SN - 0022-2496

IS - 3

ER -