Lecture 44: From Data to Decisions

Introduction

• Focus on using Excel and R for multiple regression.
• Calculate goodness of fit to assess model performance.
• Use body fat data as an example for model building.

Measuring Body Fat

• Reliable method: measure body density using hydrostatic weighing.
  • Displacement of water determines volume.
  • Use Siri equation to calculate percentage body fat.
• Challenge: Density measurement is complex.
• Easier measures: Circumference of neck, chest, abdomen, etc.

Model Building

• Goal: Use regression to predict body fat percentage from easier measures.

Using Excel for Regression

• Example 1: Abdomen circumference as predictor for body fat.
  • Scatter plot and linear regression line.
  • Goodness of fit: R squared = 0.66.
• Example 2: Weight as predictor.
  • R squared = 0.37.
  • Explanation: Height influences weight not directly related to body fat.

Goodness of Fit Measures

• Adjusted R squared: Adjusts for number of predictors.
• Akaike Information Criterion (AIC): Penalty for added complexity.
• Bayesian Information Criterion (BIC): Also penalizes complexity.

Multiple Regression

• Combine abdomen circumference and weight.
  • AIC improved from 800 to 756.
  • Adjusted R squared increased.
• Combining chest and weight.
  • AIC worse than chest alone: weight not a significant predictor.

Multicollinearity

• Correlation between predictors affects model.
• Abdomen circumference and weight are correlated, complicating interpretation.

Using R for Regression

• Load data from CSV into a data frame.
• Plotting and single regression using lm() function.
• Confidence Intervals: Calculated using confint().

Calculating Information Criteria in R

• Manually using AIC formula.
• Built-in functions: AIC() and extractAIC().
• Differences due to additive constants.
• Calculate BIC using AIC() with k = log(n).

Adjusted R Squared in R

• Part of summary() output for regression model.

Multiple Regression with R

• Add variables using + in lm().
• Example adding abdomen and weight.
• Parameters: intercept, abdomen, weight coefficients.

Conclusion

• Next lecture: Addressing multicollinearity in multiple regression.