Interpreting Coefficient of Determination

Chapter 1: Introduction

• Focus Topic: Interpreting the coefficient of determination, known as $r^2$, and understanding computer output for regression analysis.
• Objectives:
  • Determine $r^2$.
  • Interpret $r^2$ values.
  • Analyze computer output from linear regression.
• Example Context: Study involves data from 11 students in Texas, correlating:
  • Independent Variable (x): Percent of school days attended.
  • Dependent Variable (y): Number of correct answers on the Algebra I test.
• Key Question: How well does attendance predict test scores?

Chapter 2: A Terrible Model

• Model Without Attendance: This model predicts the mean score, which is an average of 41.3 correct answers.
• Prediction Error: Examines residuals (differences between actual and predicted scores) and uses squared residuals for evaluating fit.
  • Poor Fit: High sum of squared residuals indicates a terrible model.

Chapter 3: Better Fitting Model

• Using Attendance as a Variable: Introduces the least squares regression line to reduce residuals and improve prediction.
• Sum of Squared Residuals: Comparison between models illustrates better fit with attendance included.
• Reduction in Errors: When attendance is used, there’s a significant decrease in sum of squared residuals, improving model accuracy.

Chapter 4: Squared Regression Line

• $r^2$ Calculation: Describes the calculation of the coefficient of determination, which quantifies the reduction in error.
  • Result: A 90.3% reduction in residuals using attendance as a predictor. The interpretation of $r^2$:
  • 90.3% of the variance in test scores can be explained by attendance.
• Relation to Correlation: $r^2$ is also the square of the correlation coefficient (r).
• Template for Interpretation: "$r^2$ indicates the percentage of variance explained by the independent variable."

Chapter 5: Squared Values

• Understanding $r^2$:
  • $r^2$ ranges from 0 to 1 (or 0% to 100%).
  • Closer to 1: stronger relationship; closer to 0: weaker relationship.
  • Computation of $r^2$: Squaring the correlation coefficient gives $r^2$.

Chapter 6: Squared Value in Outputs

• Interpreting Computer Output:
  • Components: Output shows constant (y-intercept) and slope (coefficient of x).
  • Example values: y-intercept = -7.69 and slope = 0.57, leading to the regression model:
    ext{y hat} = -7.69 + 0.57 x.
  • $r^2$ value presented directly: 90.3%.
• Correlation Calculation:
  • To find correlation: $r = \pm \sqrt{r^2}$.
  • With $r^2 = 0.903$, $r$ becomes approximately 0.95 indicating a strong positive relationship due to the positive slope value.

Chapter 7: Conclusion

• Key Takeaways:
  • The coefficient of determination, $r^2$, is crucial for understanding the variance explained by an independent variable on a dependent variable.
  • It reflects model strength through the correlation squared.
  • When analyzing regression outputs, identify the slope as the coefficient of the x variable and the y-intercept as the constant.
• Caution in Statistical Analysis: Always approach data critically and compassionately, avoiding misleading statistics.