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 Lancaster’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.