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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.

Vobo
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Tori
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  • Welcome to Math.SE. It is much easier to read your question if the math is formatted. You can read how to do this on this site here. – Daryl Dec 03 '12 at 04:12
  • I tried to edit it but the edit button is light grey, do you know what happened? – Ivan Lerner Dec 03 '12 at 04:17
  • Thanks for editing my post... I was wondering how everyone else got such lovely formatting... – Tori Dec 03 '12 at 04:30
  • Do you know anything about the distribution of $o_{ij}$ given the expect value $e_{ij}$? – Jonathan Christensen Dec 03 '12 at 15:57
  • @JonathanChristensen I don't know what the distribution of $o_{ij}$ is, except that it is the number of days meeting type $i$ is held in meeting room $j$ over a two year period. Then $e'{ij}=365 \forall i,j$ and $e"{ij}={730,i=j;0,i\neq j}$. – Tori Dec 03 '12 at 21:52
  • I suspect $o_{ij}$ is distrbuted as either $e'{ij}$ or $e''{ij}$, but I'm trying to determine which fits better... – Tori Dec 03 '12 at 22:04
  • 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. – Jonathan Christensen Dec 03 '12 at 22:56

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