# Are there useful distributions for ternary variables (e.g. $-1,0,1$ data)?

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.

• There are discrete uniform & categorical 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
• 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
• 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
• 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
• 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 1 '14 at 20:13