1. 1 Relationships We have examined how to measure relationships between two categorical variables (chi-square) one categorical variable and one measurement variable (t-test, F-test) Now we look at relationships between two measurement variables

2. 2 Interval variable relations We want to describe the relationship in terms of form strength We want to make inferences to the population

3. 3 Our Tools Correlation to measure strength of relationship Regression to measure form of relationship

4. 4 Regression Begin with a scatterplot of two measurement variables, X and Y Let X be the independent variable Let Y be the dependent variable Plot each case as we have done before at the beginning of the course.

5. 5 Scatterplot Note:

6. 6 Note the outlier: Dallas

7. 7 Relationships Each city is represented by an X score (percent poor) and a Y score (homicide rate) We are asking about the relationship between poverty and homicide Does homicide change as percent poor changes? If so, in what way and how much?

8. 8 Looking at the scatterplot We see that as percent poor (poverty) increases (from left to right on the graph), the homicide rate increases (from low to high on the graph

9. 9 Scatterplot

10. 10 Representing relationships We represent the relationship with a straight line that goes through the middle of the points on the graph This line is the regression line It shows the average homicide rate for every level of poverty.

11. 11 Regression Line 0.005.0010.0015.00 0.00 10.00 20.00 30.00

12. 12 Regression Line Every line is represented by a formula The regression line has the following general formula ‘a’ represents the intercept of the line ‘b’ represents the slope of the line y-hat is the predicted value of y for a given x value

13. 13 Regression of homicide on poverty a = -.815b =.944 x is percent poor y is homicide rate

14. 14 Slope, the value of b The slope of the regression line is positive, it goes from the lower left to the upper right. The slope measures the amount of change in the dependent variable for every unit change in the independent variable b =.944. There is an increase of.944 units in y for every increase of 1.0 in x

15. 15 Regression Line, slope Percent families below poverty 0.005.0010.00 0.00 10.00 20.00 5 units 5 x.944 units Regression Line “rise” “run”

16. 16 Intercept, the value of a The intercept is the point where the regression line crosses the Y axis This point is the value of Y when X is zero a = -.815. The predicted rate of homicide is -.815 when there is zero poverty

17. 17 Calculate b

18. 18 Calculate a First calculate b, then

19. 19 Calculate predicted y After calculating a and b, one can use the regression line formula to calculate predicted values of y for every actual value of x

20. 20 Prediction errors Prediction errors are the difference between the predicted value of y and the actual value of y

21. 21 Prediction errors Errors (actual minus predicted) Regression Line Predicted Actual

22. 22 Ordinary Least Squares: OLS The regression line is the “best fitting” line through the data points in the graph It is the line that minimizes the sum of the squared error terms -- hence “least squares” Minimize:

23. 23 Sums of Squared Errors

24. 24 Sum of Squared Errors -0.9-0.8-0.7-0.6 0.7638.7630.2622.0614.3607.0 0.8572.9567.6562.8558.3554.3 0.9537.9535.9 534.3533.2532.4 1.0 533.7 535.0 536.7538.8541.3 1.1560.4565.0569.9575.3581.1 Minimum is 531.57 when a=-.815, b=.944 b a