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suppose we have a table like this:

\begin{array}{ |l | c | r |} \hline & \mathrm{win} & \mathrm{lose} \\ \hline \mathrm{male} & X_{11} & X_{12} \\ \hline \mathrm{female} & X_{21} & X_{22} \\ \hline \end{array}

$%![enter image description here][1]$

and we want to know if winning or losing depends on sex (male or female) apart from chi-square test for Independence, which non-parametric tests we can use? Thanks

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There are a host of possibilities, though it depends on what exactly you intend by nonparametric; arguably all of these tests, including the chi-square are 'parametric'.

Some examples: You could use a two-sample proportions test (basically, normal approximation to binomial). You could do a two sample binomial test (the same thing, but based off the fact that the data are actually binomial). You could do a Fisher exact test (conditions on both margins, giving a hypergeometric).

Two sample proportions test:
http://www.statisticslectures.com/topics/ztestproportions/
http://stattrek.com/hypothesis-test/difference-in-proportions.aspx

Fisher exact test:
http://en.wikipedia.org/wiki/Fisher%27s_exact_test

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