Fisher’s Exact Test Calculator (2×3)

Enter the cell counts for your 2×3 contingency table:
\( \begin{array}{ccc} a & b & c \\ d & e & f \end{array} \)

* 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 Table Counts

Table probability: $$ P = \frac{R_1! \, R_2! \, C_1! \, C_2! \, C_3!}{n! \, a! \, b! \, c! \, d! \, e! \, f!}, $$ where the margins are computed from the observed table.

Fisher’s Exact Test Calculator (2×3) – Educational Guide

Fisher’s Exact Test Calculator (2×3)

Welcome to our Fisher’s Exact Test Calculator (2×3)! This tool calculates the exact p‑value for a 2×3 contingency table using Fisher’s Exact Test. It is particularly useful for analyzing categorical data when sample sizes are small and the assumptions of the chi‑square test are not met.

Table of Contents

What is Fisher’s Exact Test?

Fisher’s Exact Test is a statistical method used to determine whether there is a nonrandom association between two categorical variables. While it is commonly applied to 2×2 tables, it can also be extended to 2×3 tables, providing an exact p‑value without relying on large-sample approximations.

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2×3 Table Structure

A 2×3 contingency table has 2 rows and 3 columns. An example of such a table is:

$$\begin{array}{ccc} a & b & c \\ d & e & f \end{array}$$

Here, each cell (a, b, c, d, e, f) represents the frequency count for a specific combination of the two categorical variables.

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Calculation Overview

Fisher’s Exact Test for a 2×3 table calculates the probability of the observed table using a hypergeometric formula, and then sums the probabilities of all tables with the same marginal totals that are as extreme or more extreme than the observed one.

The general formula for the probability of a given 2×3 table with cell counts \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\), row sums \(R_1 = a+b+c\) and \(R_2 = d+e+f\), column sums \(C_1 = a+d\), \(C_2 = b+e\), \(C_3 = c+f\), and total \(n = a+b+c+d+e+f\), is:

$$P = \frac{R_1! \, R_2! \, C_1! \, C_2! \, C_3!}{n! \, a! \, b! \, c! \, d! \, e! \, f!}$$

This formula serves as the foundation for calculating the exact p‑value in Fisher’s Exact Test.

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Key Concepts

  • Contingency Table: A table that displays the frequency distribution of categorical variables.
  • Exact p‑Value: The probability of observing the data under the null hypothesis, calculated exactly without approximations.
  • Marginal Totals: The sums of the rows and columns, which are fixed during the test.
  • Extreme Tables: All possible tables with the same margins that are as extreme or more extreme than the observed table.
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Step-by-Step Calculation Process

  1. Enter the Data:

    Input the observed frequencies for each cell in your 2×3 contingency table (values for \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\)).

  2. Calculate the Observed Table Probability:

    Use the hypergeometric formula to compute the exact probability of the observed table, based on the fixed marginal totals.

  3. Enumerate Extreme Tables:

    Identify all possible 2×3 tables with the same marginal totals that are as extreme or more extreme than the observed table.

  4. Sum the Probabilities:

    Add the probabilities of all extreme tables to obtain the overall p‑value.

  5. Review the Result:

    The final p‑value indicates whether the association between the row and column variables is statistically significant.

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Practical Examples

Example: 2×3 Table Analysis

Scenario: Consider a study with a 2×3 contingency table with the following cell counts:

$$\begin{array}{ccc} 12 & 7 & 5 \\ 8 & 10 & 15 \end{array}$$

  1. Input the Data:

    Enter \(a = 12\), \(b = 7\), \(c = 5\), \(d = 8\), \(e = 10\), \(f = 15\).

  2. Compute the Observed Table Probability:

    Use the formula:

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

    where \( n = a+b+c+d+e+f \).

  3. Sum Extreme Probabilities:

    Enumerate and sum the probabilities of all tables with the same margins that are as extreme or more extreme than the observed table.

  4. Interpret the p‑Value:

    A p‑value below the significance threshold (e.g., 0.05) indicates a significant association between the variables.

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Interpreting the Results

The Fisher’s Exact Test Calculator (2×3) outputs an exact p‑value for your 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 for 2×3 tables is used in various fields, including:

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

  • Exact p‑Value: Provides a precise measure without relying on approximations.
  • Robust: Effective even with small sample sizes or low cell counts.
  • User-Friendly: Simple interface for entering data into a 2×3 table.
  • Educational: Enhances understanding of categorical data analysis and exact statistical tests.
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Conclusion

Our Fisher’s Exact Test Calculator (2×3) is an essential tool for researchers and practitioners analyzing categorical data. By calculating the exact p‑value for a 2×3 contingency table, it provides a reliable assessment of the association between variables, even when sample sizes are small. For further assistance or additional statistical resources, please explore our other calculators or contact our support team.

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