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We have a contingency table with 2 rows and 4 columns, how would we test the null hypothesis that the observed count in column 1 is equal or larger than expected, as well as the observed count in columns 2,3,4 is equal or less than the expected?

Namely, we would like to use a two-tailed test instead of a classical one-tailed Fisher's exact or $\chi^2$ test.

I am thinking the possible situation like below

A random sample of 15 undergraduate students reported their sex (1=male, 2=female) and college (A=business, B=engineering, C=liberal arts, D=nursing, E=pharmacy). The results were sorted into cells where, for example, D2 would be the number of female nursing students (13). The results (with column and row totals) are displayed below. T: A B C D E total

1:  11    1    1    1    1    15
2:  13    4    1    1    2    21

For me, if I want to do a modified fishers' exact test, I may just need to calculate the following probabilities using hypogeometric distribution,

P1: A B C D E total

1:  11    1    1    1    1    15
2:  13    4    1    1    2    21

P2: A B C D E total

1:  12    1    1    1    0    15
2:  13    4    1    1    2    21

P3: A B C D E total

1:  13    1    1    0    0    15
2:  13    4    1    1    2    21

P4 A B C D E total

1:  14    1    0   0    0    15
2:  13    4    1    1    2    21

    A    B    C    D    E  total

P5

1:  15    0    0    0    0    15
2:  13    4    1    1    2    21

It seems to me that the significance of the contingency table T is the sum up probability of p=P1+P2+P3+P4+P5, if this p is very small, we can accept that the count in column 1 is unexpectedly high and the count in other column is unexpectedly low, right? Does the above calculation is the same with the fisher's exact test for 2*4 table?

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    $\begingroup$ I don't think you really are describing a two tailed test here. Instead, you have a slightly more general null hypothesis than is normally the case (where you are just interested in "different" from expected), but you will still need a statistic to compare with one tail of a distribution. $\endgroup$ – Peter Ellis Sep 9 '12 at 5:56
  • $\begingroup$ can you formulate the statistic in a detail way? how about my modified post, do you think it is reasonable? $\endgroup$ – user974270 Sep 9 '12 at 6:18
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    $\begingroup$ If you are broadening the null hypothesis to get away from equal proportions then maybe you ar elooking for something different from Fisher's test. But where did you get the idea that the Fisher test is only one-sided? It can be done one-sided or two-sided. $\endgroup$ – Michael Chernick Sep 9 '12 at 10:38

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