Exact Probability Test
Lecture 9

Fisher Exact Test

 

  1. You have to use it if

    1. Values in a cell are below 10

    2. Or the grand total is below 100

    3. Use the Fisher Exact Test

  2. Note - Always arrange the columns and rows so Cell A has the smallest number


 Men Women Marginal Total
Dieting a b a + b
Not Dieting c d c + d
Marginal Total a + c b + d a + b + c +d

  1. Example


 Men Women Marginal Total
Dieting 2 10 12
Not Dieting 3 5 8
Marginal Total 5 15 20

    1. How many combinations can we make?

      1. Men are the smallest in the study, so we have 5 combinations

      2. Look at the marginal!

      3. The Fisher test uses a Hypergeometric Distribution

      4. Probability of a particular combination, i, is:

      5. Note: 0! = 1

Equation 1

Combination 0


 Men Women Marginal Total
Dieting 0 12 12
Not Dieting 5 3 8
Marginal Total 5 15 20


Probability of this combination occurring is

Equation 2

Combination 1


 Men Women Marginal Total
Dieting 1 11 12
Not Dieting 4 4 8
Marginal Total 5 15 20


Probability of this combination occurring is

Equation 3

Combination 2


 Men Women Marginal Total
Dieting 2 10 12
Not Dieting 3 5 8
Marginal Total 5 15 20


Probability of this combination occurring is

Equation 4

Combination 3


 Men Women Marginal Total
Dieting 3 9 12
Not Dieting 2 6 8
Marginal Total 5 15 20


Probability of this combination occurring is

Equation 5

Combination 4


 Men Women Marginal Total
Dieting 4 8 12
Not Dieting 1 7 8
Marginal Total 5 15 20


Probability of this combination occurring is

Equation 6

Combination 5


 Men Women Marginal Total
Dieting 5 7 12
Not Dieting 0 8 8
Marginal Total 5 15 20


Probability of this combination occurring is

Equation 7

We manually map out the whole probability space for men on a diet


P 0 0.0036 Include in a
P 1 0.0542 Include in a
P 2 0.2384
P 3 0.3973
P 4 0.2554
P 5 0.0511 Include in a
Total 1.0000 a = 0.1089

Probability Distribution for Fisher Exact Test


Is our particular combination significant? Are men different than women on a diet?

Our data is P 2. If we choose an alpha of 5%, the best we can do in our case is to have an alpha of 11%. We take the probability that is in the tails. Alpha is the sum of P 0, P 1, and P 5. Since our value is P 2, we fail to reject and conclude men do not differ from women on a diet.