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I need to give a direction to the association between an ordinal (independent) and a nominal variable (dependent). This, after having performed a Chi-square test.

The ordinal variable is like a blood test score with values (high, low, average) and the nominal variable is yes/no (reflecting the possibility of being sick).

Is it OK to use the Goodman gamma to do this? If not because one variable is nominal, which is a good alternative?

Nick Cox
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    "yes/no" can be treated as ordered. – Glen_b Dec 31 '13 at 03:30
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    As Glen said, dichotomous nominal variable can be _statistically_ treated as ordinal. For non-dichotomous case, see http://stats.stackexchange.com/q/73065/3277 – ttnphns Dec 31 '13 at 08:41
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    Cramer's V would also make sense as the Chisquared test is checking the null hypothesis of a true V of 0. – Michael M Dec 31 '13 at 08:58
  • Glen this is a very good news! Is there any place where I can get a formal proof of this? I need it for a review. If I treat it as ordered, does it means that I can also use Spearman's Rho? – Forinstance Dec 31 '13 at 12:05
  • Goodman here should be Goodman-Kruskal. – Nick Cox Jan 14 '14 at 11:41

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I would argue that in your particular case, "possibility of being sick" is ordinal, statistically and otherwise. The dichotomous possibility of being sick is essentially synonymous with a statement about probability of being sick both in cases where the possibility does exist (i.e., probability $> 0$, rank $= 1$) and in cases where it doesn't (i.e., probability $= 0$, rank $< 1$). This means you can directly translate your possibility dichotomy into an ordinal, binary probability variable without any loss of information, alteration of the data itself, or additional assumptions, simply by revising your understanding of the variable.

That's not exactly a formal proof, and I might be overlooking some logical subtlety, but in addition to the comments you've received, this answer is another vote for, "Yes, it's okay!" based on different reasoning about your specific instance.

Nick Stauner
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