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Suppose I have a two by two table:

$$ \begin{array}{c|ccc} & Y & \neg Y & \\ \hline X & a & b& &\\ \neg X & c & d& &\\ \end{array} $$

and I am interested in testing the hypothesis that $\mathcal{H}_0: P(Y|X) =P(Y| \neg X)$. I don't need an exact result, and I don't have a calculator.

Are there quick arithmetic approximations to the test-statistic or p-values for common tests of this hypothesis? Pearson Chi-Square independence test, Fisher's Exact Test, exact binomial test, etc.

What about special cases when the distributions have some reasonable assumptions

  • $Y$ is common in both $X$ and $\neg(X)$ so normal approximations to sample proportions hold
  • $Y$ is rare in both $X$ and $\neg(X)$ so Poisson approximations to $Y$
  • $X$ and $\neg X$ is balanced
  • $X$ is imbalanced by 1/10 or more.
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  • $\begingroup$ Could you be more specific about what "don't have a calculator" really means? After all, it's not difficult to compute the chi-squared statistic for this table with pencil and paper or, if you're used to this sort of exercise, in your head. That reduces the question to finding p-values for a $\chi^2(1)$ statistic. $\endgroup$ – whuber Dec 28 '18 at 16:48
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    $\begingroup$ @whuber the idea is to use an approximation to get something at least a little computationally easier than finding the exact statistic, even though I agree it can be done with pencil-and-paper. If a decent approximation can be done mentally, all the better (though even that is subject to debate depending on who's reading the question). As an illustration, the approximation $e^x \approx x$ when $|x| < \epsilon$ or $x$ is very small has been useful for both verifying computation and simplifying complex expressions. $\endgroup$ – AdamO Dec 28 '18 at 17:30
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    $\begingroup$ $\frac {ad-bc}{(a+b)(c+d)(a+c)(b+d)}$. If it >= 3.84, p< 0.05, if >= 6.63 p<0.01. $\endgroup$ – user158565 Dec 28 '18 at 18:17
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Stuart Pocock published an article on this significance test for the analysis of a 1:1 randomization design:

If A is the number of events in Arm A and B is the number of events in Arm B, then the ratio:

$$Z = \frac{A-B}{\sqrt{A+B}}$$

has approximately a Normal 0,1 distribution under the null and can be used to calculate the p-values.

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