= Solution
Apart from its multinomial coefficient, the <likelihood> is $\prod_{ij}\theta_{ij}^{n_{ij}}$. Maximizing its logarithm subject to $\sum\theta_{ij}=1$ gives $n_{ij}/\theta_{ij}$ equal to a common multiplier, which is $n$. Hence $\widehat\theta_{ij}=n_{ij}/n$.
Under <independence> the log <likelihood> separates as $\sum_i n_{i\cdot}\log\phi_i+\sum_jn_{\cdot j}\log\rho_j$. The two normalization constraints give $\widehat\phi_i=n_{i\cdot}/n$ and $\widehat\rho_j=n_{\cdot j}/n$, so the fitted probabilities are $n_{i\cdot}n_{\cdot j}/n^2$. Therefore twice the difference of the maximized log likelihoods is
$$
\boxed{\Lambda=2\sum_{i,j}n_{ij}\log\frac{n_{ij}n}{n_{i\cdot}n_{\cdot j}}}.
$$
The unrestricted model has $rc-1$ free parameters and the <independence> model has $(r-1)+(c-1)$, leaving $(r-1)(c-1)=4$ <degrees of freedom> for the table. Under the null, the <likelihood>-ratio statistic is asymptotically chi-squared with four <degrees of freedom>. Since $25.192>18.467$, \b[reject <independence> at the 0.1% significance level]. This is evidence of association in the sampled variables; it does not by itself identify a causal direction.
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