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The title says it.

If one wishes to analyze ternary outcomes—that is categorical outcomes with specifically three values (-1,0,1)—are $\chi^{2}$ / contingency table tests and multinomial-logistic regression type models the only choices for making inferences, or are there well-articulated and useful ternary distributions and related inferential methods that we might consider as an alternative?

Such data result in the differences of two 1/0-coded nominal variables, in the qualitative (sign) representation of causal effects (e.g. in loop analysis), in ternary logic, and perhaps elsewhere.

kjetil b halvorsen
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Alexis
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  • There are [discrete uniform](http://en.wikipedia.org/wiki/Uniform_distribution_%28discrete%29) & [categorical](http://en.wikipedia.org/wiki/Categorical_distribution) distributions. Can you say more about what you are looking for? Also, chi-squared tests for contingency tables & multinomial LR are *analyses*, whereas the Bernoulli & binomial are *distributions*, so prima facie there seems to be some mismatch. – gung - Reinstate Monica Apr 29 '14 at 15:34
  • I do not have an application at hand. However, some data are naturally coded $(-1,0,1)$. For example, the difference in nominal variables, or the qualitative coding of causal effects in Levins' *loop analysis*. I am aware of categorical analyses, just wondering specifically about the ternary case. – Alexis Apr 29 '14 at 15:40
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    This is a potentially interesting question (I upvoted it), but it would really help if you can specify something more concrete. At present I worry that it is near the threshold for `unclear what you are asking` / `too broad`. – gung - Reinstate Monica Apr 29 '14 at 15:43
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    Reasoning from first principles, a symmetric distribution for such a variable $X$ will be completely determined by $\Pr(X=-1)=\Pr(X=1)=p,$ with $0\lt 2p\lt 1.$ *All* possible distributions are determined by two parameters $p\gt 0,q\gt 0$ with $p+q\lt 1$, where $\Pr(X=-1)=p$ and $\Pr(X=1)=q$. These simple observations are so evident that I wonder what you are really trying to get at in this question: are you looking for information about choosing a family from among these distributions in order to model some phenomenon? If so, then please share information about that phenomenon. – whuber Apr 29 '14 at 16:02
  • Closely related: http://stats.stackexchange.com/questions/18603, which concerns an application of the most general ternary distribution on $\{-1,0,1\}$. – whuber Apr 29 '14 at 16:04
  • @whuber: I normally proceed from understanding the nature of my measures/how and let that inform my approach to analysis (as gung rightly pointed out, and I hope I clarified). It's an intentionally general question I asked because I have encountered such data before, and wondered. I will remove the question if it's no good. – Alexis Apr 29 '14 at 16:09
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    Few questions are no good and yours, Alexis, is not among them. I am in sympathy with your normal procedure: it tends to be realistic and defensible. In the present case I share @gung's concern about the question being either so broad (or so trivial) as to be unlikely to garner good answers, and so I am probing (by means of these comments) for ways to focus it or help it inspire really good answers. That's all; I do not mean to suggest your question is a bad one and I'm actually hoping it will become a very good one if you could edit it to reflect a more definite problem you are facing. – whuber Apr 29 '14 at 16:41
  • @whuber I think it already has a very good answer, along the lines of an answer I'd have given if I were to answer it, only a little clearer and more succinct. It's languishing in comments at present, but it seems quite good. – Glen_b Apr 30 '14 at 00:51
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    I agree with Glen. I think you motivated multinomial/polytomous and also ordered logistic models for me. If someone wants to repost comment as an answer, I will mark it answered. – Alexis May 01 '14 at 20:13

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