The object of the analysis of the Happiness data (Davis, 1977; originally published in Clogg, 1982) is to gain insight into the relationship between happiness as reported by participants in a survey, and the number of their siblings and the number of years of schooling they had completed. The main interest lies in the three-way interaction between the two predictors and the dependent variable Happiness. Of the two-way interaction terms, that between the two predictors is of limited interest because it bears no relation to the prediction of the criterion because it is the design interaction. Its relevance very much depends on the sampling frame.

- Clogg, C. C. (1982). Some models for the analysis of association in multiway cross-classifications
having ordered categories.
*Journal American Statistical Association*,*77*, 803--815. - Davis, J. A. (1977).
*Codebook for the 1977 general social survey*. Chicago IL: National Opinion Research Center

The three-way contingency table has as its variables: Happiness reported by participants in a survey (3 categories), Number of siblings (Siblings - 5 categories) and the Years of schooling completed (Schooling - 4 categories). It is thus a frequency table containing in the cells the number of people with a particular combination of categories.

No. | Abbreviation | Description |
---|---|---|

1. | <12 | Less than 12 years |

2. | 12 | 12 years (equivalent to primary + secondary school) |

3. | 13-16 | 13-16 year (some) college |

4. | 17+ | 17+ (more than undergraduate) |

No. | Abbreviation | Description |
---|---|---|

1. | 0-1 | |

2. | 2-3 | |

3. | 4-5 | |

3. | 6-7 | |

3. | 8+ |

No. | Abbreviation | Description |
---|---|---|

1. | Happy- | Not too happy |

2. | Happy+ | Pretty happy |

3. | Happ++ | Very happy |

A three-way data array **X** = (*x*(*i,j,k*)) has the following form

|-----|i=1 |-----| |i=2 |-----| | |.. | | | |.. | | |____|i=I=4 k=K=3 | |____| k=2 |_____| k=1 j=1,.,J=5

The actual data file has the following form:

j=1,.,J=5 |-----|i=1 | |i=2 | |.. k=1 | |.. |_____|i=I=4 |-----|i=1 | |i=2 | |.. k=2 | |.. |_____|i=I=4 |-----|i=1 | |i=2 | |.. k=3 | |.. |_____|i=I=4

Thus the first mode (*i*) is nested in the third mode (*k*) and
there are 4 (Schooling) times 3 (Happiness) rows and 5 (NumSiblings)
columns.

Categorical data in contingency tables are usually divested of the influence of the marginal proportions which
represent the model of independence. Thus the original frequencies are transformed in *chi-term*s
( = observed - expected)/sqrt(expected). These chi-terms are the basis for the correspondence analysis.
For details see Kroonenberg (2008). *Applied multiway data analysis*. Hoboken NJ: Wiley (Chapter 17).

[Download the zipped Happiness Data]

| Centre for Child and Family Studies | Institute of Education and Child Studies | The Three-Mode Company | Three-Mode Data Sets | TOP |

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E-mail: kroonenb at fsw.leidenuniv.nl

Created: 10-7-2009

Last Updated: 7-8-2015