Fisher's Exact Test is a statistical significance test used in the analysis of contingency tables, particularly suited for small sample sizes. Unlike the chi-square test, which approximates the probability of observed frequencies under the null hypothesis, Fisher's Exact Test calculates the exact probability. This precision makes it especially useful in Lean Six Sigma projects for analyzing categorical data with small samples, ensuring accurate hypothesis testing even when assumptions of larger sample-based tests are not met.

Background

Developed by Ronald Fisher in the early 20th century, this test is fundamental in situations where we have two categorical variables in a 2x2 contingency table and wish to examine the independence of these variables. In Lean Six Sigma projects, such scenarios can arise during the Measure or Analyze phases, where understanding relationships between variables is crucial for identifying root causes of process inefficiencies or defects.

Application in Lean Six Sigma

Lean Six Sigma practitioners leverage Fisher's Exact Test to make informed decisions based on categorical data. This could involve comparing defect rates between two machines, analyzing the impact of a new process on quality levels, or any situation where sample sizes are too small for chi-square test assumptions to hold. For instance, if a team wants to compare the defect rates before and after implementing a process improvement with a small sample size, Fisher's Exact Test would provide a reliable measure of whether the observed changes are statistically significant.

Calculating Fisher's Exact Test

The calculation focuses on the hypergeometric distribution, which models the probabilities of various configurations of successes and failures in the contingency table. The p-value obtained from the test indicates the probability of observing the data (or something more extreme) if the null hypothesis of independence between the categories is true. A low p-value (typically <0.05) suggests that we reject the null hypothesis, indicating a significant association between the categorical variables.

Advantages in Lean Six Sigma Projects

• Accuracy for Small Samples: It provides exact p-values for small sample sizes, where other tests may not be reliable.

• Non-Parametric: It does not require assumptions about the distribution of the data, making it versatile for various data types.

• Detailed Insights: It helps in identifying specific patterns or anomalies in categorical data, facilitating targeted improvements in processes.

Implementation

Implementing Fisher's Exact Test in a Lean Six Sigma project involves:

1. Data Collection: Gather categorical data in a 2x2 contingency table format.

2. Hypothesis Formulation: Define the null hypothesis (no association between the variables) and the alternative hypothesis (a significant association exists).

3. Calculation: Use statistical software or calculators to compute the test, which involves calculating the probability of the observed arrangement of data and all more extreme arrangements under the null hypothesis.

4. Decision Making: Based on the p-value, decide whether to reject the null hypothesis and interpret the results in the context of the project.

Conclusion

Fisher's Exact Test is a powerful tool for Lean Six Sigma practitioners dealing with categorical data in small samples. Its precision and reliability support data-driven decision-making, helping teams identify and validate the effectiveness of process improvements. Incorporating this test into the analytical toolkit enhances the rigor of statistical analysis in projects, contributing to more effective and efficient outcomes.

Real-Life Based Scenario:

Fisher's Exact Test is a statistical significance test used in the analysis of contingency tables when sample sizes are small. It is particularly useful when dealing with categorical data and the assumptions of the chi-squared test are not met due to the small sample size or when expected frequencies in any of the cells of a contingency table are less than 5. Unlike the chi-squared test, which approximates probability from a distribution, Fisher's Exact Test calculates the exact probability of observing the data given the null hypothesis.

Suppose a small clinic wants to investigate the effect of a new dietary supplement on reducing cold symptoms. The clinic divides a group of 20 volunteers into two groups: 10 receive the dietary supplement (Treatment), and 10 receive a placebo (Control). After a month, the clinic observes whether each volunteer reports a reduction in cold symptoms or not.

Here's the observed data:

• Treatment Group: 7 report a reduction in symptoms, and 3 do not.

• Control Group: 2 report a reduction in symptoms, and 8 do not.

  
  

The question is: Is there a statistically significant difference between the treatment and control groups in terms of symptom reduction?

Contingency Table:

Step 1: State the Hypothesis

• Null Hypothesis (H0): There is no difference in symptom reduction between the treatment and control groups.
  

• Alternative Hypothesis (H1): There is a difference in symptom reduction between the treatment and control groups.

Step 2: Calculate the Probability of the Observed Table

The probability of observing this table, given the row and column totals are fixed, is calculated using the Fisher's Exact formula:

Where:

• a=7 (Treatment with symptom reduction)

• b=3 (Treatment without symptom reduction)

• c=2 (Control with symptom reduction)

• d=8 (Control without symptom reduction)

• n=20 (Total number of observations)

Step 3: Perform the Calculation, Using the factorial values for each:

Step 4: Interpret the Result The calculated P-value is 0.0026. If we choose a significance level of 0.05, since the P-value is less than 0.05, we reject the null hypothesis. There is statistically significant evidence at the 5% level to conclude that there is a difference in symptom reduction between the treatment and control groups.

Conclusion:

The example demonstrates how Fisher’s Exact Test can be applied to a real-life scenario involving a small sample size. The test provides a precise method for determining whether there are nonrandom associations between two categorical variables. In this case, the dietary supplement appears to have a significant effect on reducing cold symptoms compared to the placebo.