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Let say I have a frequency table of two variables $x$ and $y$ having or not having some property.

$$ \begin{array}{lcr} \mbox{} & x & y \\ \mbox{has property} & 20 & 2 \\ \mbox{does not have property} & 61 & 79 \end{array}\ $$

What does it really mean to have positive (right hand side p-value) or negative (left-hand side p-value) association. Can I think of it as of correlation between variables $x$ and $y$? Why then for table

$$ \begin{array}{lcr} \mbox{} & x & x \\ \mbox{has property} & 20 & 20 \\ \mbox{does not have property} & 61 & 61 \end{array}\ $$

I get large p-value not rejecting the null hypothesis?

EDIT: Null hypothesis states that variables are independent, that is the proportions do not differ among $x$,$y$ vs. has/does not have property. The right / left p-value regards associations on the diagonals of the tables like here

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    $\begingroup$ Please explain on what you mean by "right hand side" and "left-hand side" p-values. In your second question, tell us specifically what the "null hypothesis" says. $\endgroup$
    – whuber
    Commented Mar 1, 2015 at 21:00
  • $\begingroup$ @whuber Post was edited. $\endgroup$
    – Misery
    Commented Mar 2, 2015 at 7:11

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The error is to believe that you can analyze two 'variables' having only one observation of each. What you have is two levels for a qualitative variable and two levels for the other. You cannot observe variation on that situation. In this case you can try a Chi square test and after that, try to explain the level of correlation (if exists) with Cramer or any other measure of relationship.

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