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The Chi-square test for multidimensional contingency tables

Basic assumptions:


\mathcal{H}_0: & O_{ij...}=E_{ij...} $ for all categories,$\\
\mathcal{H}_1: & O_{ij...} \neq E_{ij...} $ for at least one category,$


$O_{ij...}$ and $E_{ij...}$ $-$ observed frequencies in a contingency table and the corresponding expected frequencies.

The test statistic is defined by:


This statistic asymptotically (for large expected frequencies) has the Chi-square distribution with a number of degrees of freedom calculated using the formula: $df = (r - l)(c - 1)(l - 1) + (r- l)(c- 1) + (r- 1)(l- 1) + (c- 1)(l- 1)$ - for 3-dimensional tables.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$:

$ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ 	\mathcal{H}_1, \\
$ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\

The settings window with the Chi-square (multidimensional) test can be opened in Statistics menu → NonParametric tests (unordered categories)Chi-square (multidimensional) or in ''Wizard''.


This test can be calculated only on the basis of raw data.

Cochran W.G. (1952), The chi-square goodness-of-fit test. Annals of Mathematical Statistics, 23, 315-345
en/statpqpl/porown3grpl/nparpl/anova_chipl.txt · ostatnio zmienione: 2022/02/13 17:25 przez admin

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