# Simple arithmetic approximations to categorical analyses

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.
• 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. – whuber Dec 28 '18 at 16:48
• @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. – AdamO Dec 28 '18 at 17:30
• $\frac {ad-bc}{(a+b)(c+d)(a+c)(b+d)}$. If it >= 3.84, p< 0.05, if >= 6.63 p<0.01. – user158565 Dec 28 '18 at 18:17

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