Lecture Notes: Regression Analysis and Model Evaluation

Key Topics

• Understanding different types of regression models
• Example of linearity and non-linearity in data models
• Utilizing R for model analysis

Types of Models

• Linear Model: Assumes a straight-line relationship between variables.
• Log-Linear Model: Used for skewed data where the dependent variable is the logarithm.
• Linear-Log Model: Logarithm of the independent variable is used.
• Log-Log Model: Logarithm of both dependent and independent variables, used in supply-demand scenarios.
• Quadratic/Cubic Models: Represented by parabolic/cubic equations when data isn't linear.

Example Scenarios

• Rabbit and Grass Field: Demonstrates non-linear relationships; as rabbit population increases, food availability per rabbit decreases.
• Birds and Grasshoppers: Demonstrates supply and demand relationships.

Using R for Regression Analysis

• Tools: RStudio is used for analysis.
• Dataset: Loaded and renamed for convenience.

Data Exploration

• Scatter Plot: Used to visualize potential relationships between variables.
  • Linear Appearance: Straight-line scatter indicates potential linear relationship.
  • Example: Plot of G$X1 vs G$X2 shows a straight line, indicating linearity.

Model Evaluation

• Linear Model Creation: Use lm() to create a model and summary() to evaluate it.
  • Example: Perfect model with R-squared value of 1 indicates 100% data explanation.
• Residuals Analysis: Plots to evaluate model fit.
  • Residuals vs Fitted Values: Evaluates errors; small errors indicate a good fit.
  • Normal QQ Plot: Assesses normal distribution of errors.
  • Spread-Location Plot: Checks for homoscedasticity; horizontal and random spread indicates equal variance.
  • Residuals vs Leverage: Identifies influential data points (outliers).

Further Model Testing

• Non-linear Data: When data does not fit a linear model, explore other relationships such as quadratic.
• Logarithmic Models: Test for log-linear or linear-log relationships when data appears stacked or non-linear.
  • Example: Analysis of G$C vs G$A did not fit a linear model well, suggesting a potential logarithmic relationship.

Key Takeaways

• Model Fit Indicator: R-squared value; closer to 1 indicates a better fit.
• Error Distribution: Normal distribution suggests good model assumptions.
• Homoscadastity: Equally spread residuals across predictor ranges affirm equal variance.
• Leverage: High leverage points could indicate outliers.

Conclusion

• Always verify model assumptions through residual analysis and graphical diagnostics.
• Next steps: Further practice in evaluating and selecting appropriate models based on data behavior.