1. 1 Review of Correlation A correlation coefficient measures the strength of a linear relation between two measurement variables. The measure is based on how close the points are to the regression line. The measures varies between -1 (perfect negative relation), 0 (no relation), and 1 (perfect positive relation).

2. 2 Correlation Formula

3. 3 R 2, the coefficient of determination The square of the correlation coefficient measure the proportion of the variance in y (the dependent variable) is “explained” or accounted for by x (the independent variable)

4. 4 Confidence Intervals Distinguish among the different things about which we may construct an confidence interval The slope, b The intercept, a A particular y for a given x (general, individ.)

5. 5 Confidence band

6. 6 Confidence Intervals The basis of all these confidence intervals is the standard deviation of the residuals, e To calculate standard deviation of residuals we must assume “normal i.i.d.” or normal, independent, identically distributed errors (e)

7. 7 Normal, i.i.d. X Y

8. 8 Std. Dev. of residuals df for model is # indep. vars. Residuals

9. 9 Standard error of slope, b

10. 10 Standard error of intercept

11. 11 Standard error of prediction, general case X o is the particular value of x for which we are predicting the conditional mean of y.

12. 12 Standard error of prediction for individual case X o is the particular value of x for which we are predicting the value of y for a case.

13. 13 Hypothesis Testing t-test for regression coefficient (slope)

14. 14 Hypothesis Testing t-test for regression intercept (a)

15. 15 Hypothesis Testing F-test for all coefficients in regression

16. 16 Output from Stata. reg homic poor Source | SS df MS Number of obs = 20 ---------+------------------------------ F( 1, 18) = 6.14 Model | 181.370325 1 181.370325 Prob > F = 0.0233 Residual | 531.573154 18 29.5318419 R-squared = 0.2544 ---------+------------------------------ Adj R-squared = 0.2130 Total | 712.943479 19 37.523341 Root MSE = 5.4343 ------------------------------------------------------------------------ homic | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------- poor |.9438495.3808596 2.478 0.023.1436932 1.744006 _cons | -.8151891 3.344025 -0.244 0.810 -7.840726 6.210348 ------------------------------------------------------------------------

17. 17 Cautions Simple regression assumes a straight line Outliers can control results Regression summarizes only over range of observed data Regression cannot prove causality Assumption of iid may be wrong Sample may not be random