Last modified: 2018-05-17

#### Abstract

Pearson's chi-squared statistic is a well-established statistic either for undertaking a test of independence between two or more categorical variables, or for goodness-of-fit purposes. Here, under the complete independence assumption studying the mutual association in multi-way contingency tables for nominal and ordinal categorical variables (Beh and Lombardo, 2014), we present a case where both kinds of tests abide together. In this special case, the goodness-of-fit statistic, usually designed for the analysis of one-way data where the hypothesized marginal probabilities are theoretically derived (Andersen, 1991; Agresti, 1990), is now planned for testing the significance of each variable in a multi-way association study.

Following Lancasterâ€™s work (1951), new ANOVA-like partitions of this statistic are derived for multi-way contingency tables, where the marginal probabilities are estimated from the data and/or are theoretically driven probabilities (Lombardo, Takane and Beh, 2018; Loisel and Takane, 2016) . For ordinal categorical variables, coding by orthogonal polynomials (Emerson, 1968) will allow further, interesting partitions (Beh and Davy, 1998). To illustrate these tests, a multi-way contingency table concerning the assessment of quality of a public service will be analysed.

#### References

**References**

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Andersen, B, E. (1991). *The Statistical Analysis of Categorical Data* (Second, Revised and Enlarged Edition). Springer-Verlag.

Beh, J, E. and Davy, J, P. (1998). Partitioning Pearson's chi-squared statistic for a completely ordered three-way contingency table. *The Australian and New Zealand Journal of Statistics*, *40*, 465--477.

Beh, J, E. and Lombardo, R. (2014). *Correspondence Analysis, Theory, Practice and New Strategies*. John Wiley & Sons, Chichester, UK.

Emerson, P. L. (1968). Numerical construction of orthogonal polynomials from general recurrence formula. *Biometrics*, *24*, 696â€“701.

Lancaster, O, H. (1951). Complex contingency tables treated by the partition of the chi-square. *Journal of Royal Statistical Society, Series B*, *13*, 242--249.

Loisel, S. and Takane, Y. (2016). Partitions of Pearson's chi-square statistic for frequency tables: A comprehensive account.* Computational Statistics*, *31*, 1429--1452.

Lombardo, R., Takane, Y. and Beh, E.J. (2018). A General Formula for partitioning Multi-way Pearson's Statistic. Submitted.