Fisher’s Exact Test Calculator (2×2)

Fisher’s Exact Test Calculator (2×2)

Enter the counts for the 2×2 table:
\( \begin{array}{cc} a & b \\ c & d \end{array} \)

* The p‑value is computed by summing the probabilities of all tables (with fixed margins) whose probability is ≤ that of the observed table.

Step 1: Enter 2×2 Table Counts

Fisher’s Exact Test p‑value is computed as: $$ P = \frac{\binom{a+b}{a}\binom{c+d}{c}}{\binom{n}{a+c}}, $$ summing over all tables with probability ≤ that of the observed table.

Fisher’s Exact Test Calculator (2×2) – Educational Guide

Fisher’s Exact Test Calculator (2×2)

Welcome to our Fisher’s Exact Test Calculator (2×2)! This tool allows you to compute the exact p‑value for a 2×2 contingency table using Fisher’s Exact Test. It is an essential resource for analyzing categorical data, especially when sample sizes are small.

Table of Contents

What is Fisher’s Exact Test?

Fisher’s Exact Test is a statistical significance test used for analyzing 2×2 contingency tables. It is especially useful when sample sizes are small and the assumptions of the chi-square test may not hold. The test calculates the exact probability of observing the data assuming that the null hypothesis of no association is true.

  • 2×2 Contingency Table: A table with two rows and two columns representing two categorical variables.
  • Exact p‑Value: The precise probability of obtaining the observed distribution under the null hypothesis.
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Test Overview and Formula

For a 2×2 table with cell counts:

$$\begin{array}{cc} a & b \\ c & d \end{array}$$

The exact probability of this table is given by:

$$P = \frac{(a+b)! (c+d)! (a+c)! (b+d)!}{a! \, b! \, c! \, d! \, n!}$$

where \( n = a + b + c + d \). Fisher’s Exact Test sums the probabilities of all tables as or more extreme than the observed table.

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Key Concepts

  • Contingency Table: A table showing the frequency distribution of variables.
  • Exact Test: A statistical test that does not rely on approximations, providing an exact p‑value.
  • Null Hypothesis: Assumes no association between the categorical variables.
  • One‑Tail vs Two‑Tail: Depending on the research question, the test can be one‑tailed or two‑tailed.
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Step-by-Step Calculation Process

  1. Enter the Cell Counts:

    Input the counts for each cell in the 2×2 contingency table (a, b, c, d).

  2. Compute the Factorials:

    Calculate the factorial for each cell count and the sums of the rows and columns.

  3. Apply the Exact Probability Formula:

    Substitute the values into the formula:

    $$P = \frac{(a+b)! (c+d)! (a+c)! (b+d)!}{a! \, b! \, c! \, d! \, n!}$$

  4. Sum the Probabilities (if needed):

    For a two‑tailed test, sum the probabilities of all tables as extreme or more extreme than the observed table.

  5. Review the p‑Value:

    The resulting p‑value indicates the likelihood of observing the data if the null hypothesis is true.

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Practical Examples

Example: 2×2 Table Analysis

Scenario: Consider a study with the following contingency table:

$$\begin{array}{cc} 8 & 2 \\ 1 & 9 \end{array}$$

  1. Enter the Values:

    a = 8, b = 2, c = 1, d = 9, with \(n = 20\).

  2. Calculate the Exact p‑Value:

    Substitute these into the formula to compute the exact p‑value.

  3. Interpret the Result:

    The p‑value indicates whether there is a statistically significant association between the variables.

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Interpreting the Results

The Fisher’s Exact Test Calculator provides an exact p‑value for the 2×2 contingency table. A low p‑value (typically less than 0.05) suggests that the association between the categorical variables is statistically significant, leading to rejection of the null hypothesis.

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Applications

Fisher’s Exact Test is widely used in:

  • Medical Research: Analyzing treatment outcomes in small samples.
  • Genetics: Testing associations between genetic markers and diseases.
  • Social Sciences: Examining survey data with small cell counts.
  • Quality Control: Evaluating categorical quality data when sample sizes are limited.
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Advantages

  • Exact p‑Value: Provides precise results without relying on approximations.
  • Robust: Suitable for small sample sizes or when expected cell counts are low.
  • User-Friendly: Simple interface for entering 2×2 table values.
  • Educational: Enhances understanding of categorical data analysis and exact tests.
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Conclusion

Our Fisher’s Exact Test Calculator (2×2) is a vital tool for researchers and practitioners analyzing categorical data. By computing the exact p‑value for a 2×2 contingency table, it provides an accurate assessment of the association between variables, even in studies with small sample sizes. For additional assistance or further statistical resources, please explore our other calculators or contact our support team.

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