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13. Correlation and Regression
Topics and Notes
- 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
- 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.
- 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.
- Lecture Note
- Correlation and linear regression
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PDF
Practice and Interactive Apps
- Practice exercises #12
WEB LINK
- [Optional]Read sections 4.1-4.2 and 13.1 and 13.2 of Navidi's textbook
- Interactive statistics learning apps:
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