13. Correlation and Regression

Topics and Notes

1. Correlation coefficients
• a. Linear correlation coefficient
•   ■ Three sum of squares: $$S^2_{xy}=\sum_{i=1}^n(x_i-\bar{x})(y-\bar{y}), S^2_{xx} =\sum_{i=1}^n(x_i-\bar{x})^2, S^2_{yy}=\sum_{i=1}^n(y_i-\bar{y})^2$$
•   ■ Pearson correlation coefficient: $r= \frac{S^2_{xy}}{\sqrt{S^2_{xx}}\sqrt{S^2_{yy}}}$
• b. Interpretation: strength and direction of linear correlation
2. Least square regression
• a. Least square regression y = b + mx
• b. Interpretation of m: The amount by which y changes when x increases by one unit.
• c. Estimation: m = $S^2_{xy}/S^2_{xx}$, b = ȳ -m x̄.
• d. Coefficient of determination
•   ■ Formula: R² = r², r is the Pearson correlation coefficient.
•   ■ Interpretation: Percentage of variation in y captured by the linear regression
•   ■ R² measures goodness of the regression.
3. Application and Inference on Linear Regression
• Prediction with linear regression
• Inference of slope parameter m
•   ■ Testing H₀: m = 0. p-value method.
•   ■ Confidence interval of m. If 0 is not in the interval, m is not equal to 0.
4. Lecture Note
• Correlation and linear regression HTML PDF

Practice and Interactive Apps

1. Practice exercises #12 WEB LINK
2. [Optional]Read sections 4.1-4.2 and 13.1 and 13.2 of Navidi's textbook
3. Interactive statistics learning apps:
   • a. Understanding Correlation Coefficients and Regression: Simulation INTERACTIVE APPS
   • b. Correlation Coefficients and Regression INTERACTIVE APPS

Weekly Assignments

1. This week's Assignment (Weekly quiz)
   • a. Available on D2L: 12:00 PM, Thursday
   • b. Due: 11:30 PM, Sunday