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I need to see if the proportions in a rare disease for four groups of 500 patients differ, each group of 500 receiving a different treatment (1-4).

The observed frequencies are as follows:

G1: 16, G2:12, G3: 2, G4: 4.

As far as I understand the chi-square test requires a count of at least 5 in each cell and I cannot group the patients G3-G4. Is there an alternative to chi-square in this case.

As a further step ib case the null hypothesis is rejected how could I prove that treatment G4 has lower incidence than G1 and G2?

Thanks in advance for answering.

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    $\begingroup$ It is the expected frequencies which matter, not the observed. So if (16 + 12 + 2 + 4) / 4 is greater than 5 you would be OK. $\endgroup$
    – mdewey
    Commented Aug 21, 2016 at 14:51

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Within the chi-square type of framework you could use an exact test.

However, since you think disease depends on treatment, I suggest logistic regression. Not only does this let you estimate the effect of each treatment, it lets you adjust for covariates that might be related to the disease.

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