Discussion
To test the equality of \(r\) independent multinomial distributions, multinomial \((k, p_{i_{1}}, p_{i_{k}})\) \(i = 1 \ldots r\) the null hypothesis is \(H_{0} : P_{ij} = P_{j}\ \forall j = 1 \ldots k\) where \(p_{j}\) are unknown \(j = 1 \ldots k\), a \(\chi^{2}\) test statistic is used. The degrees of freedom of the \(\chi^{2}\) test are :
Answer: D
The degrees of freedom are \((r - 1)(k - 1)\).
The data form an \(r \times k\) contingency table. Each of the \(r\) rows has \(k - 1\) free probabilities, \(r(k - 1)\) parameters in all. Under \(H_{0}\) the rows share one probability vector with \(k - 1\) free parameters, estimated from the data. The degrees of freedom are the difference:
\[ r(k - 1) - (k - 1) = (r - 1)(k - 1) \]
This is the same as the chi-square test of homogeneity or independence. \(r(k - 1)\) forgets the estimated common probabilities, and \(rk - 1\) treats the whole table as one multinomial.
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