Understanding the Shapiro-Wilk test for residual normality in linear models.

📚 Understanding the Shapiro-Wilk Test for Residual Normality

The Shapiro-Wilk test is a powerful tool to assess if a sample comes from a normally distributed population. In the context of linear models, we apply it to the residuals to check if the assumption of normality holds.

• 📊 Purpose: Assess the normality of residuals in a linear model. Non-normal residuals can indicate issues with the model's assumptions.
• 🧪 Hypotheses:
  • Null Hypothesis ($H_0$): The residuals are normally distributed.
  • Alternative Hypothesis ($H_1$): The residuals are not normally distributed.
• 🔢 Test Statistic (W): The Shapiro-Wilk test statistic, denoted as $W$, is calculated based on the ordered sample values and their corresponding expected values from a normal distribution. The formula is complex, but statistical software handles the calculation.
• ⚖️ Interpretation:
  • A small p-value (typically less than 0.05) suggests that we reject the null hypothesis and conclude that the residuals are not normally distributed.
  • A large p-value (typically greater than 0.05) suggests that we fail to reject the null hypothesis and do not have enough evidence to conclude that the residuals are not normally distributed.
• 💡 Important Notes:
  • The Shapiro-Wilk test is sensitive to sample size; it may reject normality even with slight deviations when the sample size is large.
  • Consider using visual inspections like histograms and Q-Q plots in conjunction with the Shapiro-Wilk test.

Practice Quiz

1. Which hypothesis does the Shapiro-Wilk test evaluate for residual normality in linear models?
1. The residuals are linearly related.
2. The residuals are normally distributed.
3. The residuals are independent.
4. The residuals have constant variance.
2. What does a small p-value (e.g., p < 0.05) in the Shapiro-Wilk test suggest about the residuals?
1. The residuals are normally distributed.
2. The residuals are not normally distributed.
3. The linear model is perfectly fit.
4. The sample size is too small.
3. The Shapiro-Wilk test statistic is denoted by which letter?
1. Z
2. T
3. W
4. F
4. What is a key consideration when interpreting the Shapiro-Wilk test results, especially with large sample sizes?
1. The test becomes less accurate.
2. The test may reject normality even with slight deviations.
3. The test is only valid for small datasets.
4. The test is more robust to outliers.
5. Which of the following is NOT a method used to assess residual normality?
1. Shapiro-Wilk test
2. Histogram of residuals
3. Q-Q plot of residuals
4. T-test
6. What does it mean if the Shapiro-Wilk test yields a large p-value (e.g., p > 0.05)?
1. The residuals are definitely normally distributed.
2. There is not enough evidence to conclude the residuals are not normally distributed.
3. The linear model is invalid.
4. The variance of the residuals is too high.
7. What assumption about the residuals does the Shapiro-Wilk test help to validate?
1. Homoscedasticity
2. Linearity
3. Normality
4. Independence
Click to see Answers
1. B
2. B
3. C
4. B
5. D
6. B
7. C