Question

4. When calculating a one-sided p-value for Fisher's exact test, the focus is on the upper left-hand cell of the table, typically denoted as ny but is denoted ask in the table below No Explanatory Level 1 Level 2 Response Yes R- ER C-k R -C: + Ry CE CT To find the p-value, for each value of lik u +1.....min(R.C) or k = 0,1,...,n for k 2 and k <ni, respectively), the proportion of possible randomizations leading to that value is RIRICIC! Pr } - THR - EMC - k)!(R-C+ ! Using the hypergeometric probability distribution, verify this equation. Hint: The hyper geometric proba bility distribution is typically written in the following way: where N is the total number in the first group, Nis the total number in the second group, N = Ni + Na, is the number of successes in the first group, and is the number of successes in the second group.

Answer 1

here N1= C1, x = k, n-x = R1-k, N = T

$\frac{\binom{C1}{k}\binom{C2}{R1-k}}{\binom{T}{R1}}$

Cl! C2! k! (C1-k)!) (R1-k)!(C2-R1+)! 31

note that C2- R1 + k = R2-C1+k

hence

C1! C2! k!#(C1-k)!) (R1-k)!(R2-C1+k)! R1! R2

C1! C2! k!#(C1-k)!) (R1-k)!(R2-C1+k)! R1! R2

Just arrange above expression to get the answer