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I have a data set consisting of a number of different hospitals. I am trying to compare the proportion for hospital X to the overall proportion in the data set (which includes hospital X).

Should I not use a two sample proportion test for this? I am worried because the overall proportion contains the observations for hospital X, violating the assumption of independence. Please let me know what approach I should use. Also, if there are going to be sample size issues (i.g. a really large hospital would have a greater impact on the overall proportion, is this more of a problem compared to smaller hospitals?)

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I would suggest 1 of 2 approaches (there could be others as well).

If you have the counts for both hospital X and the group of all hospitals, then you can subtract hospital X's numbers from the totals to have 2 groups: hospital X and all hospitals other than X. You can then do the 2 sample proportions test comparing the 2 sets of numbers.

If for all hospitals you only have a proportion, but not the counts then you can consider the group of all hospitals to be the population of interest and do a 1 sample proportions test to see if hospital X has the proportion of the population.

Another approach, though it is really a variation on the 2nd, would be to randomly generate a bunch of samples using the population proportion and the sample size equal to that original sample from X, then compare how often the proportion in the original sample is more extreme than the randomly generated ones.

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