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Learn Biostatistics with Ria

**Come on this incredible journey and help enhance your capability for Biomedical Research**

This is a variation of the Chi-Square test of association where we calculate the significance or p value from the exact probabilities of the categories. So no matter what the frequency is inside the contingency table (even 5 or less), we can calculate the exact p value and say whether the association between the two categorical variables are significant or not.

In case you are a little rusty about probability, here are some simple reminders:

(1) ! = factorial A factorial is the product of all of the whole numbers, except zero, that are less than or equal to that number.

(2) Whenever there are n number of things, and you are told to calculate the number of ways of choosing k number of items, then you can calculate like:

(^{n}C_{k})= n! / k!(n-k)!

(3) Fisher's exact test is based on the hypergeometric distribution.

**Let us understand this with an example:** A medical clinic has 30 patients, 20 women and 10 men. A random sample of 5 patients is drawn. What is the probability that there will be 2 men?

A sample of 5 patients out of 30 can be chosen in (^{30}C_{5}) ways = 142,506 ways.

A sample of 2 men and 3 women can be drawn in (^{10}C_{2})*(^{20}C_{3}) = 51,300 ways.

Therefore the Probability of choosing 2 men and 3 women are:

[ (^{10}C_{2})*(^{20}C_{3}) ] / (^{30}C_{5}) = 51,300/142,506 = 0.359985.

Now that we have brushed up on the question of probability, let us move on to the Contingency table and see how Fisher’s Exact Test is related to this. **Sir Ronald Aylmer Fisher, FRS** (17 February 1890 – 29 July 1962) was a British polymath and biologist who was active as a mathematician, statistician, geneticist, and academic. He is said to single-handedly created the foundations for modern statistical science. This test is so called because the significance of the deviation from a Null hypothesis i.e. p-value can be calculated exactly, rather than relying on an approximation that becomes exact in the limit as the sample size grows to infinity, as with many statistical tests.

**Fisher’s exact test** can be used on the following conditions:

• when expected frequencies are < 1 in a 2 x 2 chi square table

• when the number of observations is ≤ 20 in a 2 x 2 chi square table

Although in practice it is employed when sample sizes are small, it is valid for all sample sizes.

| Habit of smoking | TOTAL | |

Gender | Smoker | Non-smoker | |

Male | a | b | a + b |

Female | c | d | c + d |

TOTAL | a + c | b + d | n |

In a scenario usually, we are given with (a + b) number of males and (c + d) number of females, in n number of people. Now I want to know, if a random sample of (a + c) is drawn, how many of them would be males. Following the previous example, we can write like this:

The Fisher’s exact p is**: p = [(a + b)! (c + d)! (a + c)! (b + d)!] / [n!a!b!c!d!]**

** Conclusion:** During reporting of the results, the usual Chi-square value should be reported along with degree of freedom, followed by the Fisher’s Exact p value, instead of the approximated p-value. We would conclude that the distribution in the observed 2 x 2 table at the .05 level is statistically significant and different from chance. Thus, we can reject Null Hypothesis and say that there is a statistically significant association between the two categorical variables.

**BIBLIOGRAPHY:**

1. Anders Hald. A history of mathematical statistics from 1750 to 1930. the University of Michigan (1988)

2. Julien I.E. Hoffman. Hypergeometric Distribution. Biostatistics for Medical and Biomedical Practitioners. Academic Press. (2015). p:179-182. Link: doi.org/10.1016/B978-0-12-802387-7.00013-5.

*Independent t-test*

Written by:

Dr. Ria Roy

Senior Resident

Department of Community and Family Medicine, AIIMS Patna

Interests: Adolescent Health, Nutrition, Biostatistics, Epidemiology, NCDs