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Additive and Multiplicative Models for Three-Way Contingency Tables: Darroch (1974) Revisited
| Content Provider | Semantic Scholar |
|---|---|
| Author | Kroonenberg, P. M. Anderson, Carolyn Jane |
| Copyright Year | 2006 |
| Abstract | In an only occasionally referenced paper, Darroch (1974) discussed the relative merits of additive and multiplicative definitions of interaction for higher-order contingency tables. In particular, he compared the following aspects: partitioning properties, closeness to independence, conditional independence as a special case, distributional equivalence, subtable invariance, and constraints on the marginal probabilities. On the basis of this investigation, he believed that multiplicative modelling is preferable over additive modelling, “but not by so wide a margin as the difference in the attention that these two definitions have received in the literature” (p. 213). One important aspect of modelling contingency tables did not figure in this comparison: interpretability. The potential systematic relationships in multivariate categorical data becomes progressively more complex as the number of variables and/or the number categories per variable increase. In turn, interpretation becomes increasingly difficult. We consider techniques for data that can be logically formatted as three-way contingency tables. This does not limit us to three variables but rather it limits us to, at most, three types or modes of variables. The focus in this chapter lies with the interpretation of the dependence present in threeway tables and how insight can be gained into complex patterns of different types of dependence. Since the major aim in most empirical sciences is to apply (statistical) models to data and to obtain a deeper insight into the subject matter, we consider it worthwhile to take up Darroch’s comparison and extend his set of criteria by considering the interpretational possibilities (and impossibilities) of multiplicative and additive modelling of contingency tables. The investigation will primarily take place at the empirical level guided by a particular data set that consists of four variables but is best analysed as a three-way table. Similarities between additive and multiplicative modelling techniques for two-way tables have been discussed by Escoufier (1982), Goodman (1985, 1996), and Van der Heijden, Mooijaart and Takane (1994). Limited discussions of the three-way case can be found in Van der Heijden and Worsley (1988) and Green (1989). Neither of the latter two consider threeway correspondence analysis (three-way CA) nor multiple correspondence analysis (MCA). |
| Starting Page | 455 |
| Ending Page | 486 |
| Page Count | 32 |
| File Format | PDF HTM / HTML |
| DOI | 10.1201/9781420011319.ch21 |
| Alternate Webpage(s) | https://www.leidenuniv.nl/fsw/three-mode/pdf/k/kroonenberganderson2005.pdf |
| Alternate Webpage(s) | https://doi.org/10.1201/9781420011319.ch21 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
