Steps to perform residual analysis for checking model assumptions

📚 What is Residual Analysis?

Residual analysis is a method used to assess the appropriateness of a statistical model by examining the residuals, which are the differences between the observed and predicted values. It helps in verifying if the assumptions of the model, such as linearity, independence, homoscedasticity (constant variance), and normality of errors, are met. If these assumptions are violated, the model's predictions and inferences may not be reliable.

📜 History and Background

The concept of residuals has been around since the early days of statistical modeling, with significant developments in the 20th century. Early statisticians like Gauss and Legendre laid the groundwork for least squares estimation, which inherently involves minimizing residuals. The formalization of residual analysis as a diagnostic tool gained prominence with the increasing use of regression models in various fields.

🔑 Key Principles of Residual Analysis

• 📈 Linearity: Check if the relationship between the independent and dependent variables is linear.
• 🤝 Independence: Ensure that the residuals are independent of each other (no autocorrelation).
• 📊 Homoscedasticity: Verify that the variance of the residuals is constant across all levels of the independent variables.
• 🧪 Normality: Confirm that the residuals are normally distributed.

🪜 Steps to Perform Residual Analysis

1. Fit the Model: Start by fitting the statistical model to your data. For example, a linear regression model: $y = \beta_0 + \beta_1x + \epsilon$, where $y$ is the dependent variable, $x$ is the independent variable, $\beta_0$ and $\beta_1$ are the coefficients, and $\epsilon$ is the error term.
2. Calculate Residuals: Compute the residuals ($e_i$) for each observation as the difference between the observed value ($y_i$) and the predicted value ($\hat{y_i}$): $e_i = y_i - \hat{y_i}$.
3. Plot Residuals: Create various plots to examine the residuals:
   • 📉 Residuals vs. Fitted Values Plot: Plot the residuals against the predicted values. This helps assess linearity and homoscedasticity.
   • 📊 Normal Probability Plot (Q-Q Plot): Plot the quantiles of the residuals against the quantiles of a normal distribution. This helps assess normality.
   • ⌛ Residuals vs. Order of Data: Plot the residuals against the order in which the data was collected. This helps assess independence (especially for time series data).
   • 📦 Histogram of Residuals: Create a histogram of the residuals to visually inspect their distribution.
4. Interpret Plots:
   • 📉 Residuals vs. Fitted Values: Look for a random scatter of points around zero. Non-random patterns (e.g., a funnel shape or a curve) indicate violations of linearity or homoscedasticity.
   • 🧪 Normal Probability Plot: Check if the points fall approximately along a straight line. Deviations from the line indicate non-normality.
   • ⌛ Residuals vs. Order: Look for any patterns or trends. Patterns suggest a violation of independence (autocorrelation).
   • 📊 Histogram: Check if the histogram resembles a normal distribution.
5. Take Corrective Actions: If the assumptions are violated, consider the following:
   • 🛠️ Non-linearity: Transform the variables (e.g., using logarithmic or polynomial transformations).
   • ⚖️ Non-constant Variance: Apply variance-stabilizing transformations (e.g., Box-Cox transformation) or use weighted least squares.
   • 📈 Non-normality: Consider using robust regression techniques or transforming the dependent variable.
   • 🕰️ Autocorrelation: Use time series models that account for autocorrelation (e.g., ARIMA models).

🌍 Real-world Example

Consider a simple linear regression model predicting a student's exam score based on the number of hours studied. After fitting the model, you perform residual analysis and observe a funnel shape in the residuals vs. fitted values plot, indicating heteroscedasticity. To correct this, you might apply a logarithmic transformation to the dependent variable (exam score) and re-evaluate the residuals.

📝 Conclusion

Residual analysis is a crucial step in validating statistical models. By carefully examining residuals, you can identify violations of model assumptions and take appropriate corrective actions to improve the reliability and accuracy of your analysis. This ensures that the conclusions drawn from the model are valid and trustworthy.