Steps to Check Assumptions of Simple Linear Regression for University Statistics

📚 Understanding Simple Linear Regression Assumptions

Simple linear regression is a powerful tool, but its results are only reliable if certain assumptions hold true. These assumptions ensure that the model accurately represents the relationship between the independent and dependent variables. Let's explore these assumptions and how to check them.

📜 History and Background

Linear regression evolved from the method of least squares, developed by Carl Friedrich Gauss in the early 19th century. Over time, statisticians refined the method, identifying key assumptions necessary for valid inference. These assumptions are crucial for ensuring that the regression coefficients are unbiased and that hypothesis tests are reliable.

🔑 Key Principles and Assumptions

The validity of simple linear regression hinges on four key assumptions:

• 📏 Linearity: The relationship between the independent variable (X) and the dependent variable (Y) must be linear. This means that the change in Y for a one-unit change in X is constant.
• 🎯 Independence: The errors (residuals) must be independent of each other. This means that the error for one observation should not predict the error for another observation.
• 🌡️ Homoscedasticity: The variance of the errors must be constant across all levels of the independent variable. In other words, the spread of residuals should be roughly the same for all values of X.
• 📊 Normality: The errors must be normally distributed. This assumption is particularly important for hypothesis testing and confidence intervals.

✅ Steps to Check the Assumptions

Here's a breakdown of how to check each assumption:

1. 📈 Linearity

   • 📊 Scatter Plot: Create a scatter plot of Y versus X. Look for a linear pattern. If the relationship is non-linear (e.g., curved), linear regression may not be appropriate.
   • 📉 Residual Plot: Plot the residuals (the differences between the observed and predicted values) against the predicted values. A random scatter of points suggests linearity. A curved pattern indicates non-linearity.

2. 🤝 Independence

   • 🕰️ Time Series Plot (if applicable): If the data is collected over time, plot the residuals against time. Look for patterns or trends. Autocorrelation (correlation between consecutive residuals) violates the independence assumption.
   • 🔢 Durbin-Watson Test: This test quantifies autocorrelation. A value close to 2 suggests independence. Values significantly below 2 indicate positive autocorrelation, while values above 2 indicate negative autocorrelation.

3. ⚖️ Homoscedasticity

   • 📊 Residual Plot: Examine the residual plot (residuals vs. predicted values). Look for a consistent spread of residuals across all predicted values. Funnel shapes or patterns indicate heteroscedasticity (non-constant variance).
   • 🧪 Breusch-Pagan Test or White's Test: These statistical tests formally assess homoscedasticity. A significant p-value suggests heteroscedasticity.

4. 🔔 Normality

   • 📊 Histogram or Q-Q Plot: Create a histogram of the residuals or a Q-Q plot (quantile-quantile plot). If the residuals are normally distributed, the histogram should resemble a bell curve, and the points on the Q-Q plot should fall close to a straight line.
   • 🧪 Shapiro-Wilk Test or Kolmogorov-Smirnov Test: These tests formally assess normality. A non-significant p-value (typically > 0.05) suggests that the residuals are normally distributed.

💡 Real-world Examples

• 🏠 Real Estate: Predicting house prices based on square footage. Linearity, independence, homoscedasticity, and normality of residuals must be checked to ensure the model's reliability.
• 🍎 Agriculture: Modeling crop yield based on fertilizer application. Checking assumptions is crucial for accurate predictions and informed decision-making.

🛠️ Addressing Violations

If the assumptions are violated, several remedies can be applied:

• ⚙️ Non-linearity: Transform the independent or dependent variable (e.g., using logarithms or polynomials).
• 🔗 Non-independence: Use time series models that account for autocorrelation.
• 📉 Heteroscedasticity: Transform the dependent variable or use weighted least squares regression.
• 📊 Non-normality: Transform the dependent variable or consider non-parametric regression methods.

🔑 Conclusion

Checking the assumptions of simple linear regression is essential for ensuring the validity and reliability of the model. By systematically assessing linearity, independence, homoscedasticity, and normality, you can identify potential problems and take appropriate corrective actions. This leads to more accurate predictions and better-informed decisions. Remember to use visual diagnostics like scatter plots and residual plots, along with formal statistical tests, to thoroughly evaluate the assumptions.