Introduction to Multiple Linear Regression

• Concept: Extends simple linear regression to include multiple predictors.
• Objective: Predict the outcome variable (e.g., number of active positions) using multiple inputs.

Key Points

• Predictors: Not limited to a fixed set; can include various factors influencing the response variable.
• Statistical Inference: Used to identify significant predictors among several variables.

Example of Predictors

• Total population
• Land area
• Total personal income
• Percentage of the population over 65 years old

Multiple Regression Defined

• Overview: Involves more than one predictor variable, facilitating a comprehensive model to explore influences on outcomes.
• Application: Commonly used in housing data to predict sales prices.
• Housing Data Considerations:
  • Size of the house
  • Neighborhood crime rates
  • Number of rooms
  • House age & last remodeled time

Variables in Predicting House Prices

• Binary Variables: E.g., whether a house has a backyard (coded as 1 or 0).
• Numerical Variables: E.g., age of the house, number of garage spaces.

Correlation Between Variables

• Collinearity: Occurs when two predictors are highly correlated.
• Impact on Model: Including redundant predictors can distort model relevance and interpretation.

Building the Regression Model

• Model Structure: Formula = Intercept + Predictor1 + Predictor2 + … + PredictorN
• Visualization:
  • With multiple predictors, visualize relationships as hyperplanes instead of straight lines.

Case Study: Predicting Birth Weight

• Example Predictors:
  • Weeks of pregnancy
  • Mother's age
  • Mother's weight gain
  • Smoking habits

Model Interpretation

• Intercept Understanding: Set all predictors to zero to find baseline (may not have meaningful interpretation).
• Coefficient Interpretation: Must consider other predictors when discussing significant changes.
  • Example: A positive coefficient for 'weeks' indicates an increase in weight per each week of gestation, while also considering the influence of other factors.

R-Squared Analysis

• Definition: Measures how much variability in the outcome is explained by the model.
• Change in R-Squared: Adding or removing predictors affects the percentage of variability explained.

Conclusion

• Model Diagnostics: Always analyze goodness-of-fit and correlations among predictors.
• Importance of Variable Selection: Only include variables that contribute meaningful information to the model to avoid complexity without added benefit.