Intro to Business Statistics: Analysis of Variance ANOVA - Post Hoc Tests, Tukey HSD Bonferroni Correction

What This Is

Post hoc tests are used to compare the means of multiple groups after an ANOVA (Analysis of Variance) test has been conducted. A retail chain wants to know if average daily sales exceed $10,000 in different store locations. They collect data from 5 stores and want to compare the means of sales in each store. After conducting an ANOVA test, they find that the F-statistic is significant, indicating that at least one group mean is different from the others. However, they need to determine which specific groups are different from each other. This is where post hoc tests come in.

Key Formulas & Symbols

• Tukey's HSD (Honestly Significant Difference) = q * sqrt((MSE) / n) where q = critical value from the studentized range distribution, MSE = mean square error, n = sample size.
• Bonferroni Correction =-/ k where-= significance level, k = number of comparisons.
• Studentized Range Distribution (q) = q(df, ?) where df = degrees of freedom,-= significance level.
• Mean Square Error (MSE) = (SSR / (k - 1)) where SSR = sum of squares regression, k = number of groups.
• Sum of Squares Regression (SSR) =? (xi - x?)² where xi = individual data points, x? = mean of the group.
• Degrees of Freedom (df) = n - k where n = sample size, k = number of groups.
• Studentized Range Distribution (q) = q(df, ?) where df = degrees of freedom,-= significance level.

Step-by-Step Procedure

1. State hypotheses: State the null and alternative hypotheses for each comparison. For example, H?: = = and H?: at least one mean is different.
2. Choose test: Choose the post hoc test to use, either Tukey's HSD or Bonferroni correction.
3. Compute test statistic: Compute the test statistic for each comparison using the chosen post hoc test.
4. Find p-value or critical value: Find the p-value or critical value for each comparison using the test statistic and the chosen post hoc test.
5. Compare to ?: Compare the p-value or critical value to the significance level (? = 0.05).
6. Conclude: Conclude which groups are significantly different from each other based on the results of the post hoc test.

Common Mistakes

• Mistake: Using the Bonferroni correction when the sample sizes are not equal.
• Correction: Use Tukey's HSD instead, which is more robust to unequal sample sizes.
• Mistake: Not accounting for the number of comparisons when using the Bonferroni correction.
• Correction: Use the correct formula for the Bonferroni correction, which takes into account the number of comparisons (k).
• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.

Quick Practice Problems

1. A company wants to compare the average salaries of employees in different departments. They collect data from 4 departments and want to compare the means of salaries in each department. After conducting an ANOVA test, they find that the F-statistic is significant, indicating that at least one group mean is different from the others. Using Tukey's HSD, what is the critical value for a comparison between two groups?

Answer: q = 3.77 (from the studentized range distribution with df = 3 and-= 0.05).

1. A researcher wants to compare the average scores of students in different classes. They collect data from 5 classes and want to compare the means of scores in each class. After conducting an ANOVA test, they find that the F-statistic is significant, indicating that at least one group mean is different from the others. Using the Bonferroni correction, what is the critical value for a comparison between two groups?

Answer:-/ k = 0.05 / 10 = 0.005.

1. A company wants to compare the average prices of products in different categories. They collect data from 3 categories and want to compare the means of prices in each category. After conducting an ANOVA test, they find that the F-statistic is significant, indicating that at least one group mean is different from the others. Using Tukey's HSD, what is the test statistic for a comparison between two groups?

Answer: q * sqrt((MSE) / n) = 3.77 * sqrt(10 / 20) = 1.93.

Last-Minute Cram Sheet

1. Tukey's HSD: q * sqrt((MSE) / n) where q = critical value from the studentized range distribution.
2. Bonferroni Correction:-/ k where-= significance level, k = number of comparisons.
3. Studentized Range Distribution (q): q(df, ?) where df = degrees of freedom,-= significance level.
4. Mean Square Error (MSE): (SSR / (k - 1)) where SSR = sum of squares regression.
5. Sum of Squares Regression (SSR):? (xi - x?)² where xi = individual data points, x? = mean of the group.
6. Degrees of Freedom (df): n - k where n = sample size, k = number of groups.
7. p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true.
8. Use Tukey's HSD when sample sizes are unequal.
9. Use Bonferroni correction when sample sizes are equal.
10. Always account for the number of comparisons when using the Bonferroni correction.