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The $\chi^2$ statistic has been frequently employed in the hypothesis testing of the multinomial distributions. But after looking at the derivation of such a process (where they use Taylor expansion and discard all the remaining terms after the power of 3), I become suspicious and wonder if there's any counterexample to the statement (in which a contradiction arises after directly applying $\chi^2$ statistic to the multinomial distribution). Is there any way to strictly prove the validity of such an application of $\chi^{2}$ statistic (with mathematical rigor), or is there any counterexample to the argument?

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In statistical genetics, "empirical p-value testing" is often used to get around distributional assumptions. For example, consider a multiway table with $r$ rows and $c$ columns. For each row, the multinomial probability of the row total $\{ n_{i.} \}$ is

\begin{equation} P(\{n_{i.}\})= \frac{n_{i.}!}{\prod_j n_{ij}!} \prod_j \pi_{j|i}^{n_{ij}}, \end{equation}

where $\pi_{j|i}$ is the conditional probability in column $j$ ($j=1,2,...,c$) given row $i$.

The probability of the table $\{n_{ij}\}$ is the product of all of the multinomial row probabilities

\begin{equation} P(\{n_{ij}\})= \prod_i \left( \frac{n_{i.}!}{\prod_j n_{ij}!} \prod_j \pi_{j|i}^{n_{ij}}\right). \end{equation}

Under independence, we assume that in each column the conditional cell probabilities are equal (including the row total), shown as $\pi_{j|1}=\pi_{j|2}= \cdots = \pi_{j|r}=\pi_{.j}$. Thus, let's substitute $\pi_{.j}$ for $\pi_{j|i}$ and $n_{.j}$ for $n_{ij}$ in the above equation and obtain \begin{equation} \begin{split} P(\{n_{ij}\})&= \prod_i \left( \frac{n_{i.}!}{\prod_j n_{ij}!} \prod_j \pi_{.j}^{n_{.j}} \right)\\ &= \frac{\prod_i n_{i.}! \prod_j \pi_{.j}^{n_{.j}} } {\prod_i \prod_j n_{ij}!}. \\ \end{split} \end{equation}

The distribution of the cell counts in each row, however, also depend also on $\pi_{.j}$. Fisher assumed these to be sufficient nuisance parameters, and conditioned on them so that the resulting conditional probability does not depend on them. The contribution of $\pi_{.j}$ to the product multinomial distribution depends on the data through $n_{.j}$, which are the sufficient statistics. Each column total $n_{.j}$ has the multinomial distribution \begin{equation} P(\{n_{.j}\})= \frac{n!}{\prod_j n_{.j}!} \prod_j \pi_{.j}^{n_{.j}}. \end{equation}

The probability of $\{n_{ij}\}$ conditional on $\{n_{.j}\}$ equals the probability of $\{n_{ij}\}$ divided by the probability of $\{n_{.j}\}$, so we invert the above ratio and multiply as \begin{equation} \begin{split} P(\{n_{ij}\}|\{n_{.j}\}) &= \left( \frac{\prod_i n_{i.}! \prod_j \pi_{.j}^{n_{.j}} } {\prod_i \prod_j n_{ij}!} \right) \left( \frac{\prod_j n_{.j}!}{ \prod_j \pi_{.j}^{n_{.j}}{n!}} \right)\\ &= \frac{\prod_i n_{i.}! \prod_j n_{.j}} {n!\prod_i \prod_j n_{ij}!}. \\ \end{split} \end{equation}

$P$ above is first calculated from the observed data. Permutations are then used by re-shuffling categories for one of the two categorical variables being considered, each time recalculating $P^{(b)}$ $(b=1,2,...,B)$ repeatedly in order to simulate the null distribution. After $B$ iterations, the exact p-value is \begin{equation} P=\frac{\# \{b: P^{(b)}< P\}}{B} . \end{equation}

Randomization Test for $\chi^2$. You will notice that even for small tables, e.g. 3x4, 4x4, 4x5, 5x5, the number of iterations required for fully implementing the exact form of the test can exceed than $10^4$ configurations. Therefore, one can use the following randomization test:

  1. Calculate the raw chi-squared value for the table, call this $\chi^2$.

  2. Randomly shuffle (permute) one of the grouping variable's values.

  3. Calculate chi-squared using the grouping variables (with one of them shuffled), call this $\chi^2_{(b)}$.

  4. Repeat steps 2-3 $B=10,000$ times (default).

The p-value for this randomization test is

\begin{equation} P=\frac{\#\{b: \chi^2_{(b)} \geq \chi^2\}}{B}, \end{equation}

that is, the number of times the chi-squared value for the permuted configurations exceeds or is equal to the chi-squared value for the raw data configuration (table), divided by the number of iterations.

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