Regression Analysis with SPSS

1. Regression Analysiswith SPSS Robert A. Yaffee, Ph.D. Statistics, Mapping and Social Science Group Academic Computing Services Information Technology Services New York University Office: 75 Third Ave Level C3 Tel: 212.998.3402 E-mail: yaffee@nyu.edu February 04

2. Outline • Conceptualization • Schematic Diagrams of Linear Regression processes • Using SPSS, we plot and test relationships for linearity • Nonlinear relationships are transformed to linear ones • General Linear Model • Derivation of Sums of Squares and ANOVADerivation of intercept and regression coefficients • The Prediction Interval and its derivation • Model Assumptions • Explanation • Testing • Assessment • Alternatives when assumptions are unfulfilled

3. Conceptualization of Regression Analysis • Hypothesis testing • Path Analytical Decomposition of effects

4. Hypothesis Testing • For example: hypothesis 1 : X is statistically significantly related to Y. • The relationship is positive (as X increases, Y increases) or negative (as X decreases, Y increases). • The magnitude of the relationship is small, medium, or large. If the magnitude is small, then a unit change in x is associated with a small change in Y.

5. Regression AnalysisHave a clear notion of what you can and cannot do with regression analysis • Conceptualization • A Path Model of a Regression Analysis

6. In a path analysis, Yi is endogenous. It is the outcome of several paths. Direct effects on Y3: C,E, F Indirect effects on Y3: BF, BDF Total Effects= Direct + Indirect effects

7. Interaction coefficient: C X1 and X2 must be in model for interaction to be properly specified.

8. A Precursor to Modeling with Regression • Data Exploration: Run a scatterplot matrix and search for linear relationships with the dependent variable.

9. Click on graphs and then on scatter

10. When the scatterplot dialog box appears, select Matrix

11. A Matrix of Scatterplots will appear Search for distinct linear relationships

14. Decomposition of the Sums of Squares

15. Graphical Decomposition of Effects

16. Decomposition of the sum of squares

17. Decomposition of the sum of squares • Total SS = model SS + error SS and if we divide by df • This yields the Variance Decomposition: We have the total variance= model variance + error variance

18. F test for significance and R2 for magnitude of effect • R2 = Model var/total var • F test for model significance • = Model Var/Error Var

19. ANOVA tests the significance of the Regression Model

20. The Multiple Regression Equation • We proceed to the derivation of its components: • The intercept: a • The regression parameters, b1 and b2

21. Derivation of the Intercept

22. Derivation of the Regression Coefficient

23. If we recall that the formula for the correlation coefficient can be expressed as follows:

25. Extending the bivariate case To the Multiple linear regression case

26. It is also easy to extend the bivariate intercept to the multivariate case as follows.

27. Significance Tests for the Regression Coefficients • We find the significance of the parameter estimates by using the F or t test. • The R2 is the proportion of variance explained.

28. F and T tests for significance for overall model

29. Significance tests • If we are using a type II sum of squares, we are dealing with the ballantine. DV Variance explained = a + b

30. Significance tests T tests for statistical significance

31. Significance tests Standard Error of intercept Standard error of regression coefficient

32. Programming Protocol After invoking SPSS, procede to File, Open, Data

33. Select a Data Set (we choose employee.sav) and click on open

34. We open the data set

35. To inspect the variable formats, click on variable view on the lower left

36. Because gender is a string variable, we need to recode gender into a numeric format

37. We autorecode gender by clicking on transform and then autorecode

38. We select gender and move it into the variable box on the right

39. Give the variable a new name and click on add new name

40. Click on ok and the numeric variable sex is created It has values 1 for female and 2 for male and those values labels are inserted.

41. To invoke Regression analysis,Click on Analyze

42. Click on Regression and then linear

43. Select the dependent variable: Current Salary

44. Enter it in the dependent variable box

45. Entering independent variables • These variables are entered in blocks. First the potentially confounding covariates that have to entered. • We enter time on job, beginning salary, and previous experience.

46. After entering the covariates, we click on next

47. We now enter the hypotheses we wish to test • We are testing for minority or sex differences in salary after controlling for the time on job, previous experience, and beginning salary. • We enter minority and numeric gender (sex)

48. After entering these variables, click on statistics

49. We select the following statistics from the dialog box and click on continue

50. Click on plots to obtain the plots dialog box