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Suppose there is a $2\times2$ matrix $O$ of observed values: $O = \begin{bmatrix}o_{11}&o_{12}\\o_{21}&o_{22}\end{bmatrix}$ and two matrices $E_1$ and $E_2$ of expected values: $E_1 = \begin{bmatrix}e'_{11}&e'_{12}\\e'_{21}&e'_{22}\end{bmatrix}\text{ and } E_2 = \begin{bmatrix}e''_{11}&e''_{12}\\e''_{21}&e''_{22}\end{bmatrix}.$

The total misallocation supposing $O$ is distributed according to $E_1$ is $L_1=\frac{1}{2}\left(\left|o_{11} - e'_{11}\right| + \left|o_{12} - e'_{12}\right| + \left|o_{21} - e'_{21}\right| + \left|o_{22} - e'_{22}\right|\right)$ and the total misallocation supposing O is distributed according to $E_2$ is $L_2=\frac{1}{2}\left(\left|o_{11} - e''_{11}\right| + \left|o_{12} - e''_{12}\right| + \left|o_{21} - e''_{21}\right| + \left|o_{22} - e''_{22}\right|\right).$

My question is, how can I measure how much better $E_1$ or $E_2$ is at representing $O$. Initially, I thought I could use a Chi-Squared or F distribution type test, but I don't know the distributions of $L_1$ and $L_2$.

Any help would be appreciated.

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    Under $e''$ the probability of o_{21} > 0 or o_{12} > 0 is zero, so if you have more than zero in either of those cells you can immediately say that $e'$ is better. So you should probably rethink your $e''$ hypothesis. I have two other comments: (1) If the marginals are fixed, then you might want to take a look at the Fisher Exact Test; (2) do you really need a statistical test? Your hypothesis are well separated, and it seems like you should be able to say just by looking at the data which is more likely to be true. Or you're somewhere in the middle, and both are probably false.2012-12-03

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