Module 9-2(2023)

ANOVA Hypothesis Testing Steps

1. State the Null and Alternative Hypotheses

• Null Hypothesis (H0): All group means are the same. Example: Mu1 = Mu2 = Mu3.

• Alternative Hypothesis (H1): At least one group mean is different. This indicates that while at least one mean differs, it does not necessarily mean all means are different. Possible scenarios include:

  • One mean differs while others are equal.

  • Two means are equal, and the third differs.

  • All means could be different, but not all scenarios need to be true.

2. Check the Assumptions of ANOVA

• Independence: Ensure that you have k independent populations. Each treatment group should be independent of the others.

• Random Sampling: Data within each treatment group should be randomly selected.

• Normal Distribution: Data should follow a normal distribution. Methods to check:

  • Histogram should be unimodal and symmetrical.

  • Q-Q plot should show dots closely following a linear trend.

• Equal Variability: The standard deviation across all groups should be similar.

  • Rule of thumb: Ratio of largest to smallest standard deviation should be less than 2.

  • Another check: Compare IQR from box plots, ensuring the ratio is also less than 2.

3. Calculate the Test Statistic (F-test Statistic)

• F-test Statistic Formula: F = MST / MSE

  • MST (Mean Square Treatment): Derived from SST (Sum of Squares Treatment) over degrees of freedom (df1 = k - 1).

  • MSE (Mean Square Error): Derived from SSE (Sum of Squares Error) over degrees of freedom (df2 = n - k).

• Degrees of Freedom Calculation:

  • df1 = k - 1 (where k is the number of groups).

  • df2 = n - k (where n is the total number of observations).

• F Distribution: The F-distribution is right-skewed, having no area below 0. The peak typically occurs around 1.

4. Finding the p-value

• The p-value represents the area to the right of the calculated F-statistic. It estimates the probability of observing the test statistic as extreme as F0 under the null hypothesis.

• Comparison with Alpha Level:

  • If p-value ≤ alpha: Reject null hypothesis.

  • If p-value > alpha: Do not reject null hypothesis.

5. Conclusion

• After performing the test, draw a conclusion based on the hypothesis test. Relate this back to the context of the problem being studied.

Example Application

• Example Scenario: Comparing mileage between two brands of gasoline.

  • Null Hypothesis: No difference in mileage (Mu1 = Mu2).

  • Alternative Hypothesis: At least one mean is different.

• Conduct ANOVA and check assumptions to ensure they hold. Calculate test statistics (F-statistic).

• Example calculations yield:

  • MST = 8, MSE = 0.313 results in F = 25.53.

Relationship with t-test

• The F test statistic can be derived from the t test statistic: F = (t^2 when comparing 2 groups).

• The shared p-values between ANOVA and the t-test lead to consistent conclusions regarding mean differences.

Practical Notes

• In practice, software aids in computations, but understanding the underlying calculations is essential.

• Review of incomplete ANOVA tables necessary for deriving missing values can enhance comprehension.

• Students should recognize that ANOVA is primarily used for comparing means across multiple groups while ensuring all assumptions are checked.