## How to calculate four numbers from marginal total in cross tabulation?

When you execute Fisher’s exact test with cross tabulation, the marginal total is constant. Therefore, you could calculate the remaining three numbers if you could get ‘a’, the number of true positive. Because all four numbers are 0 or greater than 0, the range of ‘a’ is between 0 and the smaller one of either the number of ‘TRUE’ or the number of ‘positive’.

Fisher’s probability is function dependent on ‘a’. Fisher’s probability follows super geometric distribution. If ‘positive’ and ‘negative’ are separated by cut-off value in continuous variable, changing cut-off value makes a change in numbers of positive, negative and true-positive. The numbers of true and false never change regardless of change in cut-off value. Therefore, Fisher’s probability is function dependent on cut-off value.

 TRUE FALSE Marginal total POSITIVE a P – a P NEGATIVE T – a a + N – P – T N – P Marginal total T N – T N

At first you know only about numbers of ‘N’ meaning of grand total, ‘T’ meaning of true and ‘P’ meaning of positive.

 TRUE FALSE Marginal total POSITIVE P NEGATIVE Marginal total T N

Next, you can calculate numbers ‘N – P’ meaning of negative and ‘N – T’ meaning of false. Then you have got marginal total.

 TRUE FALSE Marginal total POSITIVE P NEGATIVE N – P Marginal total T N – T N

If you could get ‘a’, you would get false negative ‘T – a’ and false positive ‘P – a’.

 TRUE FALSE Marginal total POSITIVE a P – a P NEGATIVE T – a N – P Marginal total T N – T N

At last, you could get true negative ‘a + N – P – T’.

 TRUE FALSE Marginal total POSITIVE a P – a P NEGATIVE T – a a + N – P – T N – P Marginal total T N – T N

## How to calculate Fisher’s exact test with logarithm?

Chi-square test is known to compare between ratios with two-by-two table. But you couldn’t use chi-square test if total number was smaller than 20 or expected value was smaller than 5.

Even if you couldn’t use chi-square test, you could use Fisher’s exact test and calculate accurate p-value. Although the test has reliability, it requires huge amount of calculation with factorial function and software may overflow. You would easily calculate it with conversion to the logarithm first. Next, you could add or subtract the logarithm. At last, you could convert the result to the power of e, the base of natural logarithm.

 TRUE FALSE Marginal total POSITIVE a b a + b NEGATIVE c d c + d Marginal total a + c b + d N
$\displaystyle \begin{array} {rcl} P &=& \frac{(a+b)!(c+d)!(a+c)!(b+d)!}{N!a!b!c!d!}\vspace{0.2in}\\&=& \exp \left[ LN \left( \frac{(a+b)!(c+d)!(a+c)!(b+d)!}{N!a!b!c!d!} \right) \right]\vspace{0.2in}\\ &=& \exp [ LN((a+b)!) + LN((c+d)!) + LN((a+c)!) + LN((b+d)!)\vspace{0.2in}\\& & - LN(N!) - LN(a!) - LN(b!) - LN(c!) - LN(d!) ]\end{array}$