I don't have much experience with some of these measures of association, so I can't give much practical advise. But I think the broadest distinction is to separate those that are used for a) ordinal x ordinal tables, b) ordinal x nominal tables, and c) nominal x nominal tables.

I your case, I assume that Satisfaction should be treated as ordinal, and Sex should be treated as nominal. In that case, you would choose those from b), which include Freeman's theta and epsilon-squared (the one that would be used with a Kruskal-Wallis test). However, since Sex has only two levels, those measures in a) would work also.

Fantastic Sal Mangiafico I don't know these statistical tests but I see that they have been used to determine association between categorical variables. Thank you very much for the information, I will update the forum

P.S. In the case of one dichotomous variable and one ordinal variable, you could also use those effect size statistics that are commonly used with a Wilcoxon-Mann-Whitney test. These include Cliff's delta, Vargha and Delaney's A, and the common language effect size statistic. Your audience may be more familiar with these, and they have a simple interpretation. Vargha and Delaney's A is simply the probability that an observation in one group is larger than an observation in another group. Cliff's delta is linearly related to this, and ranges from -1 to 1. In the case of a dichotomous nominal variable, I believe the absolute value of Freeman's theta is equal to Cliff's delta.

As to the question about dropping the 0-count categories, it may depend upon what you are trying to do. I think, in general, you would leave the categories with zero counts. In the case of your example, I think a chi-square test of association would fail, but a Mann-Whitney or Cochran-Armitage test would work.

To comment on your list:

Most of the measures in your list are measures of association, or effect size statistics. They are analogous to r in correlation analysis. Some of them range from -1 to 1, or 0 to 1.

However, Cochran-Armitage is a hypothesis test. The salient result is a p-value.

It is often helpful to report both a hypothesis test and an effect size statistic. They convey different kinds of information.

Sometimes it's easy to confuse the two because modern software often reports both together. For example, the cor.test() function in R will report r, but also a p-value for the null hypothesis that this r is equal to 0.

Sal Mangiafico Totally true, I have confused the Cochran-Armitage test with the others.