Chapter 7 Stats

Chapter 7: Linear Regression

7.1 Least Squares: The Line of "Best Fit"

• Objective: Identify the optimal line that fits the data points in a scatterplot.

• Key Concepts:

  • Correlation (Example: Fat and Protein at Burger King): Strong correlation of 0.76 indicates a close linear relationship.

  • Residual: Difference between observed and predicted values, defined as:

    • Residual = Observed value - Predicted value

7.2 The Linear Model

• Line of Best Fit: A line minimizing the sum of squared residuals; also known as the least squares line.

• Equation Format: yˆ = b0 + b1x

  • b1: slope, indicates the rate of change in y with respect to x.

  • b0: y-intercept, the expected value of y when x is 0.

7.3 Finding the Least Squares Line

• Slope and Correlation:

  • Both have correlated signs; slope includes units of y/x.

• Finding y-intercept: Based on the means of x and y, and the slope can be calculated.

7.4 Regression to the Mean

• Concept: Children of tall parents are generally shorter than the parents themselves.

• Regression to the Mean: Refers to the tendency of extreme values to move towards the average in subsequent observations.

7.5 Examining the Residuals

• Residual Definition: Difference between actual data point and model prediction.

• Good Model Indicators: Scatterplot of residuals should display no patterns, direction, shape, or outliers.

7.6 R²: The Variation Accounted for by the Model

• Interpretation: Indicates how much variation in the dependent variable (y) is explained by the model.

• Example Understanding: An R² of 0.58 means 58% of variability in one variable is accounted for by another.

7.7 Regression Assumptions and Conditions

• Key Conditions:

  • Quantitative Variable Condition: Both variables must be quantitative.

  • Straight Enough Condition: Scatterplots should show linearity.

  • Outlier Condition: Outliers can skew results and should be managed.

  • Does the Plot Thicken? Condition: The spread should remain consistent; no increasing variability.

General Notes

• Model Validation: Always check the conditions and the residuals before using the regression results.

• Causation: Correlation does not imply causation; a scientific explanation is needed to draw such conclusions.