Bivariate Regression

1. Bivariate Regression CJ 526 Statistical Analysis in Criminal Justice

2. Regression Towards the Mean Measure tend to “fall toward” the mean Tall parents have tall children, but not as tall as themselves Sir Francis Galton

3. Regression • Prediction: predicting a variable from one or more variables • Karl Pearson, Pearson r correlation coefficient, uses one variable to make predictions about another variable (bivariate prediction)

4. Multivariate Prediction Uses two or more variables (considered independent variables) to make predictions about another variable Y = a +b1x1+b2x2+b3x3+e

5. Criterion Variable Criterion variable: The variable whose value is predicted A = a constant, x (1, 2, etc) the independent variables, and b(1,2,) are the slopes. They are standardized and referred to as beta weights

6. Predictor Variables • The variable(s) whose values are used to make predictions • Predictions are made based on independent variables which are weighted (by the beta weights) that BEST predict the predictor variable

7. Regression Line • A straight line that an be used to predict the value of the criterion variable from the value of the predictor variable

8. Line of Best Fit 2. Graphically, the regression line is the line that minimizes the size of errors that are made when using it to make predictions

9. Predicted Value (Y’) • Values of Y that are predicted by the regression line • The regression line is the line of best fit, that makes the prediction • There will be error • Error, or e = Y –Y’

10. Least-Squares Criterion The regression line is determined such that the sum of the squared prediction errors for all observations is as small as possible

11. Regression Equation • The equation of a straight line (bivariate, one predictor and one predicted variable) • Y’ = 3 X + 2 • X = 4, Y’ = 3(4) + 2 = 14 • X = 2, Y’ = 3(2) + 2 = 8

12. Regression equation • Multiple regression equations an expansion of the equation example above to 2 or more predictor variables to predict a predicted variable

13. Standard Error of Estimate Measure of the average amount of variability of the predictive error

14. Standard Error of Estimate

15. Range of Predictive Error SYX becomes smaller as r increases

16. Multiple regression • Multiple regression can tell us how much variance in a dependent variable is explained by independent variables that are combined into a predictor equation

17. Collinearity • Very often independent variables are intercorrelated, related to one another • i.e., lung cancer can be predicted from smoking, but smoking is intercorrelated with other factors such as diet, exercise, social class, medical care, etc.

18. Multiple Regression • One purpose of multiple regression is to determine how much prediction in variability is uniquely due to each IV

19. Proportion of variance • R squared • The F test can be used to determine the statistical significance of R squared.

20. SPSS Procedure Regression • Analyze, Regression, Linear • Move DV into Dependent • Move IV into Independent • Method • Enter • Statistics • Estimate • Model fit • R squared change • Descriptives

21. SPSS Procedure Regression Output • Descriptive Statistics • Variables • Mean • Standard Deviation • N • Correlations • Pearson Correlation • Sig (1-tailed) • N

22. SPSS Procedure Regression Output -- continued • Variables Entered/Removed • Model Summary • R • R Square • Adjusted R Square • Standard Error of the Estimate

23. SPSS Procedure Regression Output -- continued • Change Statistics • R Square Change • F Change • Df1 • Df2 • Sig F Change

24. SPSS Procedure Regression Output -- continued • ANOVA • Sum of Squares • Df • Mean Squares • F • Sig

25. SPSS Procedure Regression Output -- continued • Coefficients • Model • Constant (Y-Intercept) • IV • Unstandardized Coefficients • B • Standard Error of B • Standardized Coefficients • Beta • t • sig