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.

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

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

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

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