You
have to use it if
Values
in a cell are below 10
Or
the grand total is below 100
Use
the Fisher Exact Test

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 
Example

Men 
Women 
Marginal Total 
Dieting 
2 
10 
12 
Not Dieting 
3 
5 
8 
Marginal Total 
5 
15 
20 
How
many combinations can we make?
Men
are the smallest in the study, so we have 5 combinations
Look
at the marginal!
The
Fisher test uses a Hypergeometric Distribution
Probability
of a particular combination, i, is:
Note:
0! = 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
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
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
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
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
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
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 
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.

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