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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


P0 0.0036 Include in a
P1 0.0542 Include in a
P2 0.2384
P3 0.3973
P4 0.2554
P5 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 P2. 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 P0, P1, and P5. Since our value is P2, we fail to reject and conclude men do not differ from women on a diet.

 

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