Bounds on the size of the \(\chi ^ 2\)-test of independence in a contingency table.

*(English)*Zbl 0695.62063Summary: Bounds are obtained on the limiting size of the level-\(\alpha\) \(\chi^ 2\)-test of independence in a contingency table, as the sample size increases. The situations considered include (a) sampling with one or both sets of marginal totals random, (b) performing the test with or without the continuity correction and (c) with or without conditioning on the event \({\mathcal E}_ k\) that the minimum estimated expected cell count is greater than a given \(k\geq 0.\)

Bounds for both the unconditional and conditional (on \({\mathcal E}_ k)\) size are derived. It is shown, for example, that the limiting conditional size of the test is unity for all \(\alpha\) if the continuity correction is used with \(k=0\) and sampling is done with both margins random. The same conclusion holds if sampling is done with one set of margins fixed and the dimensions of the table being not too small.

Bounds for both the unconditional and conditional (on \({\mathcal E}_ k)\) size are derived. It is shown, for example, that the limiting conditional size of the test is unity for all \(\alpha\) if the continuity correction is used with \(k=0\) and sampling is done with both margins random. The same conclusion holds if sampling is done with one set of margins fixed and the dimensions of the table being not too small.