I'm getting confused by the chi squared test of homogeneity, When performing the test, do the samples need to be distributed as multinomial?

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    $\begingroup$ Well, they could be some other things -- such as Poisson for example. You need something with a conditional variance-covariance structure of the right form or otherwise the usual denominator won't be right. $\endgroup$
    – Glen_b
    Oct 17, 2016 at 9:13
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    $\begingroup$ @Glen_b In my case I have 3 independent samples, each sample has a proportion, and I want to test proportions for equality. If the sample had a multinomial distribution then the sum of theses proportions would be 1. Not in my case. Is th chi squared test of homogeneity valid in my case? $\endgroup$ Oct 17, 2016 at 9:28
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    $\begingroup$ Within each sample the success and failure counts are (at least notionally) binomial, right? $\endgroup$
    – Glen_b
    Oct 17, 2016 at 9:45
  • $\begingroup$ @Glen_b yes they are. $\endgroup$ Oct 17, 2016 at 10:32

1 Answer 1


The mention of multinomial made in your other question would be to the distribution within samples. The values across samples would be independent.

There are cases where the samples are dependent in both directions because both margins are being treated as fixed (in the 2x2 case this would be hypergeometric).

There are also cases that could be independent in both directions (Poisson counts); if you condition on the totals in one margin these would become multinomial.

For the chi-squared distribution to work you need the asymptotic multivariate normal to have the right form of dependence such that the chi-square statistic is equivalent to a quadratic form of the right kind to correspond to a sum of squares of independent standard normals of dimension equal to the df for the chi-square.

  • $\begingroup$ Within each sample the success and failure counts are binomial but the response categories are not mutually exclusive. $\endgroup$ Oct 17, 2016 at 16:09
  • $\begingroup$ But you said "3 independent samples" in the comments under your question here. Comments is not the place to trickle out enigmatic glimpses of seemingly contradictory information. [Look at the total length of additional - but apparently still incomplete - information in comments on this issue compared to your questions. Don't keep doing that. If you can, post a question that gives the information to make it clear what you are actually asking rather than posting a very general question without asking what you need to know and then trying to ask something apparently quite different in comments] $\endgroup$
    – Glen_b
    Oct 17, 2016 at 18:01
  • $\begingroup$ Yes, I have 3 independent sample and the responses are not mutually exclusive, I don't see where the contradictory information is! $\endgroup$ Oct 17, 2016 at 18:11
  • $\begingroup$ Sorry to be less than clear. If they're independent, they can't be mutually exclusive. The fact that you emphasize it anyway after my answer suggests that there's something else going on - but something unstated - that would change the answer (i.e. contradict the three independent samples). Otherwise why raise it? If they're independent, you have an answer. If they're not, you're trying to add information about that. If I didn't answer the question you wanted answered, what did you leave out? $\endgroup$
    – Glen_b
    Oct 17, 2016 at 18:16
  • $\begingroup$ I'm sorry I had poorly explained my question, I'll edit my question and I'll add an example. Is it best to delete this question and create a new one? $\endgroup$ Oct 17, 2016 at 18:24

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