Real-World Examples of Multiple Linear Regression Assumption Violations

📚 Quick Study Guide

• 📏 Linearity: The relationship between the independent and dependent variables must be linear. Violation occurs when the relationship is non-linear (e.g., quadratic, exponential).
• 🌡️ Independence: Errors (residuals) should be independent of each other. Violation occurs when errors are correlated (e.g., time series data). This is often checked using the Durbin-Watson test.
• 📊 Homoscedasticity: The variance of errors should be constant across all levels of the independent variables. Violation occurs when the variance is non-constant (heteroscedasticity). Visual inspection of residual plots can help detect this.
• 🍎 Normality: Errors should be normally distributed. Violation occurs when errors are non-normally distributed. This can be assessed using histograms or Q-Q plots of the residuals.
• 🚫 No Multicollinearity: Independent variables should not be highly correlated with each other. Violation occurs when there is high correlation among independent variables. Variance Inflation Factor (VIF) is used to detect multicollinearity.

Practice Quiz

1. Which of the following is an example of violating the linearity assumption in multiple linear regression?
   1. A) Using a linear model when the true relationship is exponential.
   2. B) Having independent variables that are highly correlated.
   3. C) Errors having non-constant variance.
   4. D) Errors being normally distributed.
2. What test is commonly used to detect autocorrelation (violation of independence of errors) in time series data?
   1. A) Shapiro-Wilk test
   2. B) Durbin-Watson test
   3. C) Levene's test
   4. D) F-test
3. Heteroscedasticity refers to the violation of which assumption?
   1. A) Linearity
   2. B) Independence of errors
   3. C) Homoscedasticity
   4. D) Normality of errors
4. How can you visually check for heteroscedasticity?
   1. A) Histogram of residuals
   2. B) Q-Q plot of residuals
   3. C) Scatter plot of residuals against predicted values
   4. D) Time series plot of residuals
5. Which of the following is used to detect multicollinearity?
   1. A) R-squared
   2. B) Adjusted R-squared
   3. C) Variance Inflation Factor (VIF)
   4. D) Standard Error
6. What does it mean if the errors in a multiple linear regression model are not normally distributed?
   1. A) The linearity assumption is violated.
   2. B) The independence assumption is violated.
   3. C) The normality assumption is violated.
   4. D) The homoscedasticity assumption is violated.
7. Suppose you build a linear regression model to predict house prices using square footage and age of the house. If square footage and age are highly correlated, what assumption is likely violated?
   1. A) Linearity
   2. B) Independence of errors
   3. C) Homoscedasticity
   4. D) No Multicollinearity