Interpreting Coefficient of Determination

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?

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.

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.

$r^2$ 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."

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 $r^2$.

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.

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.