Least Squares Regression

Introduction to Least Squares Regression Line (LSRL)

• The LSRL is determined by minimizing the sum of squared residuals.
• Key learning objectives:
  • How to determine the LSRL.
  • Important properties of the LSRL.
  • Calculation of the slope of the LSRL using correlation.

Context and Relevance

• Focused on the disparity in educational outcomes between middle/upper income and lower-income students.
• Investigating if higher school attendance can improve test scores as a potential solution to educational inequities.
• Example dataset: 11 students in Texas with data on attendance and exam performance.

Dataset and Correlation

• Identified a strong positive correlation between attendance rates and performance in algebra exams.
• A strong positive linear relationship indicates consistency in the trend of data.

Linear Models and Residuals

• Challenge: Determining the best linear model (Model A vs Model B).
• Residuals represent prediction errors; smaller residuals indicate a better fit:
  • Model A: Residual of 2
  • Model B: Residual of 1
• Just minimizing the sum of residuals may not represent the best fit.

Issues with Summing Residuals

• Example with three collinear points shows a case where sum of zero residuals does not equal a good fit.
  • A misleading model can have a sum of residuals that cancels simply because it has both positive and negative errors.
• Squaring residuals mitigates cancellation by ensuring all values contribute positively to the total error.

Least Squares Regression Line Derivation

• The LSRL is defined as the line that minimizes the sum of squared residuals across the data set.
• This is often assessed visually using technology or statistical software to derive an exact equation.

Properties of the LSRL

• The line will contain the mean point (x̄, ȳ) of the dataset:
  • For the sample: Mean attendance is 86.4% and mean correct answers is 41.3.
• The equation of the line can be expressed as: Slope = \frac{r \cdot sy}{sx}
  • Where:
    • $r$ is the correlation coefficient.
    • $s_y$ is the sample standard deviation of y values (exam scores).
    • $s_x$ is the sample standard deviation of x values (attendance).

Example Calculation

• Given:
  • Correlation $r = 0.95$ (strong positive correlation).
  • Standard deviation of test scores $s_y = 6.08$.
  • Standard deviation of attendance $s_x = 10.2$.
• Plug values into the formula to determine the slope:
  Slope = \frac{0.95 \cdot 6.08}{10.2} \approx 0.57

Conclusion

• The LSRL is crucial in statistical modeling as it minimizes squared residuals, thus ensuring a more accurate linear fit for the data.
• Key takeaways include:
  • LSRL contains the means of both variables.
  • Slope is derived from correlation and standard deviations.
• Always approach statistical analysis with critical thinking to ensure accuracy and relevancy in interpretations of data and results.