1. Roger B. Hammer Assistant Professor Department of Sociology Oregon State University Conducting Social Research Ordinary Least Squares Regression

2. Conducting Social Research Housing Sales Data We have obtained a sample of 40 housing sales that took place somewhere in some year. The data contains two variables, price (in $’s) and size (total above grade finished area in feet 2 ).

3. Conducting Social Research Housing Sales Data

4. Conducting Social Research Housing Sales Data

5. The Ordinary Least Squares (OLS) Model Expected Value Predicted Value Conducting Social Research Expected Error Notation Notation: No hat on e. Estimated Error

6. Conducting Social Research Bivariate Regression Housing Sales Data Notation Notation: Using b instead of Beta-hat.

7. Conducting Social Research Housing Sales Data

8. OLS Characteristics 1)The estimated regression line passes through the means of X and Y. 2)The sum of the residuals is zero. 3)Under specific assumptions, it is the best linear unbiased estimator (blue). Conducting Social Research

9. Housing Sales Data

10. Conducting Social Research Click to Open Browser Click to Open Browser and Go to and Go to Regression Residual Regression Residual Demonstration

11. Total Sum of the Squared Deviations Residual Sum of the Squared Deviations Explained Sum of Squared Deviations Conducting Social Research

12. OLS Minimizes Residual Sum of the Squared Deviations Conducting Social Research

13. Housing Sales Data

14. OLS selects b 0 and b 1 such that the RSS is as small as possible Conducting Social Research Notation Notation: Using b instead of Beta-hat.

15. Conducting Social Research The Coefficient of Determination The proportion of the variance of Y explained by X.

16. Conducting Social Research Pearson Correlation Coefficient A standardized measure of the strength of a bivariate relationship. Theoretically ranges from -1 (perfect negative relationship) to 1 (perfect positive relationship). Negative – above average values of Y tend to occur with below average values of X and vice-versa. Positive – above average values of Y tend to occur with above average values of X and vice-versa.

17. Conducting Social Research The Coefficient of Determination and the Correlation Coefficient The correlation coefficient represents the bivariate relationship between Y and X. In bivariate (simple) regression, the coefficient of determination represent the bivariate relationship between Y and X. In bivariate regression, the coefficient of determination is equal to the squared correlation coefficient.

18. Conducting Social Research Standardized Regression Coefficients The unstandardized coefficient of X represents the unit change in Y, resulting from a 1 unit change in X. The standardized coefficient of X represents the number of standard deviation changes in Y, resulting from a 1 standard deviation change in X. Note: See Standardized Variables in Chapter 16.

19. Conducting Social Research The Standardized Regression Coefficient and the Correlation Coefficient The correlation coefficient represents the bivariate relationship between Y and X. In bivariate (simple) regression, the standardized regression coeffiecient represents the bivariate relationship between Y and X in standard deviation units. In bivariate regression, the standardized regression coefficient is equal to the correlation coefficient.

20. Conducting Social Research Total Sum of the Squared Deviations Housing Sales Data

21. Conducting Social Research Sum of Squared Residuals Housing Sales Data

22. Conducting Social Research Explained Sum of Squared Deviations Housing Sales Data

23. Conducting Social Research The Coefficient of Determination Goodness of Fit Housing Sales Data

24. Conducting Social Research Housing Sales Data Correlation of X and Y

25. Conducting Social Research Housing Sales Data Correlation of X and Y

26. Conducting Social Research The Coefficient of Determination and the Correlation of X and Y Housing Sales Data Note: Only holds true in bivariate regression.

27. Conducting Social Research The Coefficient of Determination, R 2, is a very bad way to evaluate a model, especially relative to other models.

28. Conducting Social Research Adjusted R 2 Adjusted for the number of variables in the model degrees of freedom Note: K = Number of Parameters.

29. Simple Regression Model Conducting Social Research Multiple Regression Model

30. Simple Regression Model Conducting Social Research Multiple Regression Model

31. Experiments provide strong evidence about causality since, due to random assignment, X is uncorrelated with other effects on Y. Regression mathematically adjusts for the correlations of the X’s. Other relevant variables can only be controlled for if measured and included in the model. Conducting Social Research Partial Effects

32. Conducting Social Research Housing Sales Data We have obtained all housing sales for 2005 for Benton County. The data contains a number of variables including sale price, main living area, and garage.

33. Simple Regression Model Conducting Social Research Multiple Regression Model

34. Partial Coefficient of X 1 Conducting Social Research Partial Coefficient of X 2

35. The Y-intercept always equals zero. All variables are expressed in standardized scores, measured in standard deviations from the mean. Coefficients indicate by how many standard deviations Y-hat changes, per 1 standard deviation change of X (ceteris paribus). Conducting Social Research Standardized Regression Coefficients

36. Conducting Social Research

37. Regression and Correlation Path Analysis – Structural Equation Modeling Conducting Social Research X1X1 X2X2 Y e 0.184.502 0.330.45673.24610

38. Standardized Regression Coefficients Conducting Social Research

39. Structural Equation Modeling Conducting Social Research X1X1 X2X2 Y1Y1 e1e1 X3X3 X4X4 Y2Y2 e2e2

40. Using Regression Analysis 1)Review the literature and develop the theoretical model, including hypotheses. 2)Specify the model including the measurement of the variables and the functional form. 3)Hypothesize the expected signs of the coefficients (if not already done in step 1). 4)Collect, inspect, and clean data (if not already done in step 0). 5)Estimate and evaluate model. 6)Document the results. Conducting Social Research