First, I think there must be an extensive literature on likelihood-
ratio tests of ordinal constraints on the multinomial--some of our
readers must know a great deal about this.
Second, my own first encounter with this sort of problem occurred
about 30 years ago, when Amos Tversky was working on his classic
paper, "Intransitivity of preference". Tversky and I worked out
a likelihood-ratio significance test for apparent stochastic
intransitivities--situations where the observed relative frequencies
of choosing A over B, B over C, and C over A are all > 1/2. There
is a brief explanation in his paper (Psychological Review, 1969).
Third, the strategy to use the binomial with p=.10 as the null
hypothesis obviously won't do--what would happen if 200 out of
1000 people chose 7 and the other 800 chose 3? One could safely
reject p=.10 for 7, but one could scarcely conclude that 7 was the
most popular choice.
Fourth, contrary to Jens Oehlschlaegel, I don't expect that Bayesians
will do much laughing about this problem. Formulating a sensible
hierarchical prior leading to a good Bayesian analysis for this
problem strikes me as an interesting challenge.
Dave Krantz
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