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I have some project categories $(PC_1, ... ,PC_n)$ and some institutions $(I_1, ..., I_n)$ and would like to determine how fairly projects are assigned to institutions and their project categories by some external sources.

I was thinking about using some chi square based measure for this. Basically I have a two dimensional contingency table where one dimension corresponds to the project categories and the other to the institutions. Each cell of the contingency table counts the number of projects that each institution ‘runs’ for a particular project category.

I could determine $(O – E)\cdot|O-E|/E$ for each cell. Here:

  • $O$ is the observer number of projects
  • $E$ is the expected number of projects

IMHO this would give me a value for each cell which indicates how much the observed value differs from the expected value and thus how fairly the resources were assigned for an institution and project category. In other words a negative value indicates ‘unfairness’, a positive value indicates ‘favouritism’ and a value close to zero indicates ‘fairness’. What do you think?

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I would look at the signed square root, which would be on the "count" scale rather than on the squared-count scale:

$$(O-E)/\sqrt{E}$$

You could call these residuals; see for example here.

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