Abstract

Rosaria Lombardo, Eric J. Beh, Gianmarco Alberti

Three-way correspondence analysis (CA) provides a quick and simple graphical summary of how categories and variables are related to one another (Carlier and Kroonenberg, 1996; Lombardo, Beh, and Kroonenberg, 2021; Lombardo, van de Velden, and Beh, 2023). Here, we present some variants of three-way CA based on a generalisation of the Cressie–Read family of divergence statistics (Cressie and Read, 1984; Pardo, 1996; Beh and Lombardo, 2024).
This family relies on a power parameter, denoted by δ; changes in δ lead to special cases within the family of divergences. For example, when δ = 1, the statistic reduces to the three-way chi-squared statistic; when δ = −1/2, it yields the Freeman–Tukey statistic; and when δ = 2/3, it gives the Cressie–Read statistic.
In particular, our focus is on an ANOVA-like partitioning of these statistics and on comparing Pearson’s three-way chi-squared statistic with the generalisations of the Freeman–Tukey and Cressie–Read statistics for three-way contingency tables.
Using archaeological data from Bellanger et al. (2021), these three-way CA variants are used to explore the distribution of artefacts in the Middle Loire Valley (France). We apply them to investigate a concrete archaeological question: did ceramic innovations diffuse at different rates across sub-regions of the middle Loire Valley, or did certain areas maintain distinct traditions while others converged over time?

References
Beh, E.J., Lombardo, R.,: Correspondence analysis and the Cressie-Read family of divergence statistics. International Statistical Review 92, 17—42 (2024)

Bellanger, L., Coulon, A., Husi, P.: Determination of cultural areas based on medieval pottery using an original divisive hierarchical clustering method with geographical constraint (MapClust). Journal of Archaeological Science 132, 105431 (2021).

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Cressie, N. A. C., Read, T. R. C.: Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B 46, 440–464 (1984)

Lombardo, R, Beh, E.J., Kroonenberg, P.: Symmetrical and non-symmetrical variants of three-way correspondence analysis for ordered variables. Statistical Science, 36 (4), 542 - 561 (2021)

Lombardo, R.; van de Velden, M.; Beh, E.J. Three-way correspondence analysis in R. R Journal 15, 237–262, (2023)

Pardo, M. C.: An empirical investigation of Cressie and Read tests for the hypothesis of independence in three-way contingency tables. Kybernetika, 32, 175–183 (1996)