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Suppose I've a one-way table with three categories (A, B and C), and let $p_a$, $p_b$ and $p_c$ be the true proportion of observations in each category, i.e. $p_a+p_b+p_c=1$. How can I conduct a statistical test on the hypothesis "$p_a$ is the largest among the three", i.e. $p_a > p_b$ and $p_a > p_c$?

Thanks!

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Well, "$p_a$ is the largest among the three" is not a traditional null hypothesis. You can test whether they are equal - for example, $H_0 : p_a = p_b = p_c$, and, if rejected, do post-hoc tests to see where the heterogeneity lies. – Macro May 3 '12 at 15:17
But this will affect the size of the overall test and such adjustment could be an intricate matter. I admit it's rather non-standard, but the null can be formulated like "$p_a$ is not the largest" and theoretically a p-value should exist telling us how extreme our outcome is given such $H_0$. Of course if this is overly difficult then a compromise will have to be made. – David L May 3 '12 at 17:55

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You are looking for something called a "simple tree ordering test". There are quite a few methods for this in the order restricted inference literature, though I am not sure how much of it is available in software. A quick Google search lead me to a paper that seems applicable.

Peddada, S. D., Prescott, K. E. and Conaway, M. (2001), Tests for Order Restrictions in Binary Data. Biometrics, 57: 1219–1227.

Edit

Based on the comments, it appears I misunderstood the original question. It is still a simple tree ordering, which is a topic of ordered restricted statistical inference, but the reference addresses the comparison of independent binomial probabilities. I have found another paper, Nettleton, D (2009), 'Testing for the supremacy of a multinomial cell probability', Journal of the American Statistical Association. Vol. 104, Pages 1052-1059, that describes exactly your situation. I think the author has a preprint posted on his webpage, and you might be able to get most of the info from there.

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Sorry that I may not have made myself clear in the question. I don't have $n$ treatment groups there; what I only have in hand is a vector of outcomes belonging to that category. For example in a class of 100 students, 36 get A, 33 get B and 31 get C, and I'd like to test whether the proportion of students getting A is higher than that getting B and that getting C. In other words $p_a+p_b+p_c$ is constrained to be 1. I've edited my question to make it clearer. – David L May 2 '12 at 18:04
You are right, I did misunderstand you, and this does not answer your question. – Aniko May 2 '12 at 18:15
It's an interesting question, David, but I wonder about your application: what population are you trying to make inferences about? Arguably, your 100 grades form a complete census, in which case the test is simple: because 36 exceeds both 33 and 31, a grade of A is indeed the mode. – whuber♦ May 2 '12 at 21:13
Maybe we can see it as a generalization of the two-category case. Suppose in another course the 100 students are such that 51 get A and 49 get B, then obviously the empirical proportion of A is bigger than that of B, but a test of $p_a=p_b$ against $p_a>p_b$ cannot be rejected, or equivalently $p_a$ is not significantly bigger than 0.5. I would like to generalize this to three categories, but then the problem is that testing whether $p_a>1/3$ will not do the job. Or, we can treat the 100 students a sample from an underlying distribution, so that 100 is not by itself the whole population. – David L May 3 '12 at 4:08
Thanks Aniko for your article suggested. Looks like it's the one that describes this problem and I'll read it in detail once I have the time. – David L May 3 '12 at 4:10
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I don't see why this is so complicated. Why can't you simply do the two hypothesis tests and apply a mutliplicity adjustment and conclude that $p_a$ is the largest if the adjusted $p$-value is less than say $0.05$? I don't think the fact that the tests are dependent matters.

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I'm not so sure about this approach. Perhaps it depends on how you do the multiplicity adjustment. If it's something like the Bonferroni, then it seems to me it won't work well. Consider a case where one of the probabilities is extremely small compared to the other two. The decision then comes down to a single comparison, because the comparison with the low-count outcome tells us almost nothing. The multiplicity adjustment ought to have (almost) no effect in such a situation. – whuber♦ May 3 '12 at 15:21
Yes, calculating that factor could be a problem. The level of dependence of those tests will affect how far the adjustment should be made, and I'm quite clueless on that. – David L May 3 '12 at 17:50

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