1. Introduction to Regression Analysis Straight lines, fitted values, residual values, sums of squares, relation to the analysis of variance

2. Population characteristics Expected value Is a conditional mean: it is dependent on The conditional mean is called the regression When this mean is a straight line, we write

3. Alternately

4. The Errors Are independent This assumption is important Excluded: Longitudinal data; repeated measures; split units; cross overs; clustering

5. The sample gives the estimates of the population characteristics For example: Some books write: Your choice (but be clear!)

6. A simple example X Y 1 2 2 3 3 4 7 5 10

9. Residual Sum of Squares Without X: With X: The difference is the sum of squares ‘attributable’ to X: The Regression SS

10. Analysis of variance table Source SS df MS Regression 40 1 40 Residual 6 3 2 Total 46 4 F = 40/2 = 20 cf F(1,3) p-value = P(F > 20) = 0.0208

11. Regression analysis. regr y x Source | SS df MS Number of obs = 5 -------------+------------------------------ F( 1, 3) = 20.00 Model | 40 1 40 Prob > F = 0.0208 Residual | 6 3 2 R-squared = 0.8696 -------------+------------------------------ Adj R-squared = 0.8261 Total | 46 4 11.5 Root MSE = 1.4142 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | 2.4472136 4.47 0.021.5767667 3.423233 _cons | -1 1.48324 -0.67 0.548 -5.720331 3.720331 ------------------------------------------------------------------------------

12. Centering. gen xc=x-3. regr y xc Source | SS df MS Number of obs = 5 -------------+------------------------------ F( 1, 3) = 20.00 Model | 40 1 40 Prob > F = 0.0208 Residual | 6 3 2 R-squared = 0.8696 -------------+------------------------------ Adj R-squared = 0.8261 Total | 46 4 11.5 Root MSE = 1.4142 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- xc | 2.4472136 4.47 0.021.5767667 3.423233 _cons | 5.6324555 7.91 0.004 2.987244 7.012756 ------------------------------------------------------------------------------