1. 15: Linear Regression Expected change in Y per unit X

2. Introduction (p. 15.1) X = independent (explanatory) variable Y = dependent (response) variable Use instead of correlation when distribution of X is fixed by researcher (i.e., set number at each level of X) studying functional dependency between X and Y

3. Illustrative data (bicycle.sav) (p. 15.1) Same as prior chapter X = percent receiving reduce or free meal (RFM) Y = percent using helmets (HELM) n = 12 (outlier removed to study linear relation)

4. Regression Model (Equation) (p. 15.2) “y hat”

5. How formulas determine best line (p. 15.2) Distance of points from line = residuals (dotted) Minimizes sum of square residuals Least squares regression line

6. Formulas for Least Squares Coefficients with Illustrative Data (p. 15.2 – 15.3) SPSS output:

7. Alternative formula for slope

8. Interpretation of Slope (b) (p. 15.3) b = expected change in Y per unit X Keep track of units! –Y = helmet users per 100 –X = % receiving free lunch e.g., b of –0.54 predicts decrease of 0.54 units of Y for each unit X

9. Predicting Average Y ŷ = a + bx Predicted Y = intercept + (slope)(x) HELM = 47.49 + (–0.54)(RFM) What is predicted HELM when RFM = 50? ŷ = 47.49 + (–0.54)(50) = 20.5 Average HELM predicted to be 20.5 in neighborhood where 50% of children receive reduced or free meal What is average Y when x = 20? ŷ = 47.49 +(–0.54)(20) = 36.7

10. Confidence Interval for Slope Parameter (p. 15.4) 95% confidence Interval for ß = where b = point estimate for slope t n-2,.975 = 97.5 th percentile (from t table or StaTable) se b = standard error of slope estimate (formula 5) standard error of regression

11. Illustrative Example (bicycle.sav) 95% confidence interval for  = –0.54 ± (t 10,.975 )(0.1058) = –0.54 ± (2.23)(0.1058) = –0.54 ± 0.24 = (–0.78, –0.30)

12. Interpret 95% confidence interval Model: Point estimate for slope (b) = –0.54 Standard error of slope (se b ) = 0.24 95% confidence interval for  = (–0.78, –0.30) Interpretation: slope estimate = –0.54 ± 0.24 We are 95% confident the slope parameter falls between –0.78 and –0.30

13. Significance Test (p. 15.5) H 0 : ß = 0 t stat (formula 7) with df = n – 2 Convert t stat to p value df =12 – 2 = 10 p = 2×area beyond t stat on t 10 Use t table and StaTable

14. Regression ANOVA (not in Reader & NR) SPSS also does an analysis of variance on regression model Sum of squares of fitted values around grand mean =  (ŷ i – ÿ)² Sum of squares of residuals around line =  (y i – ŷ i )² F stat provides same p value as t stat Want to learn more about relation between ANOVA and regression? (Take regression course)

15. Distributional Assumptions (p. 15.5) Linearity Independence Normality Equal variance

16. Validity Assumptions (p. 15.6) Data = farr1852.sav X = mean elevation above sea level Y = cholera mortality per 10,000 Scatterplot (right) shows negative correlation Correlation and regression computations reveal: r = -0.88 ŷ = 129.9 + (-1.33)x p =.009 Farr used these results to support miasma theory and refute contagion theory –But data not valid (confounded by “polluted water source”)