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Interpreting Coefficient of Determination
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.
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