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I am reading about testing independence in two-way contingency tables from Mood Graybill and Boes's Introduction to the Theory of Statistics and is confused about testing independence.

We have a two-way contingency table. We assume that the cells in the table follow a multinomial distribution with parameters $n$ (known) and $p_{i,j}$ (unknown) for $1 \leqslant i \leqslant r$ and $1 \leqslant j \leqslant s$ where $p_{i,j}$ is the probability of getting cell $(i,j)$ for a single trial. Let $N_{i,j}$ be the random variable representing the number in cell $i,j$.

We want to use the generalised likelihood-ratio $\Lambda$ to test $H_0:p_{i,j} = p_{i}p_{j}$

$\Lambda$ turns out to equal $\frac{(\prod_{i} {n_{i.}}^{n_{i.}})(\prod_{j} {n_{.j}}^{n_{.j}})}{n^n \prod_{i,j} {n_{i,j}}^{n_{i,j}}}$, (the $n_{i.}$ and the $n_{.j}$ are the marginal totals) the distribution of this under $H_0$ is not unique, since $H_0$ is composite and $\Lambda$ involves the unknown parameters; this makes formulating a test with a fixed Type-I error size difficult.

What the book does is to use the marginals $N_{i.}$ and $N_{.j}$. The book computes the joint probability mass function of the $N_{i,j}$ under $H_0$ which is $\frac{n!}{\prod_{i,j} {n_{i,j}!}}(\prod_{i} {p_{i.}}^{n_{i.}})(\prod_{j} {p_{.j}}^{n_{.j}})$. Then it computes the joint probability mass function of the marginals $N_{i.}$ and $N_{.j}$ under $H_0$ which is $\frac{(n!)^2}{(\prod {n_{i.}!})(\prod {n_{.j}!})}(\prod_{i} {p_{i.}}^{n_{i.}})(\prod_{j} {p_{.j}}^{n_{.j}})$. Then it computes the conditional distribution of the $N_{i,j}$ given the marginals $N_{i.}$ and $N_{.j}$ which turns out to be $\frac{(\prod {n_i. !})(\prod {n_.j !})}{n! \prod_{i,j} {n_{i,j}!}}$

The marginals are therefore (jointly) sufficient. Write $T$ for the marginals $N_{i.}$ and $N_{.j}$. The book then explains how for each $t$ (which would be a specific set of marginals in this instance) we can find $\lambda_0 (t)$ that satisfies $\int_{0}^{\lambda_0 (t)} f_{\Lambda | T=t}(\lambda | t) = 0.05$ because $f_{\Lambda | T=t}(\lambda | t)$ does not involve any unknown parameters (because $T$ is a sufficient statistic). So our test is 'conditional', we observe $T$ then we observe $\Lambda$ and reject $H_0$ if $\Lambda$ is below $\lambda_0 (t)$.

What I don't understand is how do we find $\lambda_0 (t)$? The book is vague about this. I think we are supposed to compute $f_{\Lambda | T=t}(\lambda | t)$, but how do we do this? Are we supposed to compute it from the conditional distribution of the $N_{i,j}$ given the marginals $N_{i.}$ and $N_{.j}$ which was shown to be $\frac{(\prod {n_i. !})(\prod {n_.j !})}{n! \prod_{i,j} {n_{i,j}!}}$ ? This sounds like a computational nightmare, the book does mention 'large-sample approximation', but I am not so certain what that refers to.

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