Basic Quantitative Methods in the Social Sciences (AKA Intro Stats)

1. Basic Quantitative Methods in the Social Sciences(AKA Intro Stats) 02-250-01 Lecture 9

2. Assignment Due and Course Evaluations • All four modules of the assignment are due in the first 5 minutes of class. NO assignment will be accepted after 4:05 PM. • Course evaluations will be completed during the first 10 minutes of class.

3. Correlation • We are often interested in knowing about the relationship between two variables. • Consider the following research questions: • Does the incidence of crime (X) vary with the outdoor temperature (Y) in Detroit? • Does pizza consumption (X) have anything to do with how much time one spends surfing the web (Y)? • Does severity of depression (X) vary as a function of Ecstacy use (Y)? • Do the occurrence of pimples (X) increase as air pollution increases (Y) in Windsor?

4. Correlation • These are all examples of relationships. • In each case, we are asking whether one variable (X) is related to another variable (Y). Stated differently: Are X and Y correlated? • More specifically: Are changes in one variable reliably accompanied by changes in the other? • “Correlation coefficients” can be calculated so that we can measure the degree to which two variables are related to each other.

5. Scatter Plot Used to Describe Correlation • We can plot the X and Y points on a Scatter plot. • We plot the Y scores on the vertical axis and the X scores on the horizontal axis. • We then can draw a straight line to try to represent or describe the points on our scatter plot.

6. Graphing Relationships • When our height and weight scores are plotted, we see some irregularity. • We can draw a straight line through these points to summarize the relationship. • The line provides an average statement about change in one variable associated with changes in the other variable. r = .770

7. Correlation AGE WEIGHT

8. Imagine if…. • All of the dots fell exactly on the line? What would that mean? • All of the dots clustered close to the line, but few fell on the line – What would that mean? • The dots were widely dispersed around the line, such that the line is only a vague representation of how the scatterplot looks. What would that mean?

9. Correlation: Positive R • Lets look at some different scatter plots. • A positive relationship.

10. Various degrees of linear correlation

11. Correlation: Negative R • Lets look at some different scatter plots. • A negative relationship.

12. Various degrees of linear correlation

13. Correlation: No Relationship • Lets look at some different scatter plots. • No Relationship:

14. What Direction Relationship Is Described in This Scatter Plot?

15. Logic Dictates… • We can measure the distance between each dot and the line. • If a perfect correlation (1.000) is represented by all of the dots falling on the line, while a line whose dots vary around it indicates a weaker correlation… • The degree to which the two variables are correlated can be thought of as the mean distance between the dots and the line. This is calculated algebraically.

16. Covariance • Conceptually, the correlation between X and Y is based on covariance – a statistic representing the degree to which two variables vary together. • Like variance, covariance is based on deviations from the mean. • r is calculated as • But wait! Just like calculating variance, there is an easier formula

17. The Pearson Product-Moment Correlation Coefficient (r) • r is a quantitative expression of the degree to which two variables are correlated in a linear relationship. • Linear relationship: This means that the scatterplot points are clustered more or less symmetrically about a straight line, such that the line is an adequate representation of the relationship. • Non-linear or curvillinear relationship: The scatterplot points do not cluster around a straight line. Example? Arousal/performance

18. Characteristics of r • r has two components: • The degree of relationship • The direction of relationship • r ranges from –1.000 to +1.000

19. Are X & Y Correlated?

20. [ ] [ ] (SC)2 (SU)2 SC2 SU2 N N The Pearson r (SC) (SU) SCU N r = Note: This formula really is the same as the one in the book, just slightly rearranged.

21. We Need: • Sum of the Xs SC • Sum of the Ys SU • Sum of the Xs squared (SC)2 • Sum of the Ys squared(SU)2 • Sum of the squared Xs SC2 • Sum of the squared Ys SU2 • Sum of Xs times the Ys SCU • Number of Subjects (N)

22. Correlation Arithmetic

23. [ ] [ ] (15)2 (17)2 55 63 5 5 The Pearson r (15) (17) 57 5 r =

24. [ ] [ ] (15)2 (17)2 55 63 5 5 The Pearson r 255 57 5 r =

25. [ ] [ ] (15)2 (17)2 55 63 5 5 The Pearson r 57 51 r =

26. [ ] [ ] (15)2 (17)2 55 63 5 5 The Pearson r 6 r =

27. [ ] [ ] 225 289 55 63 5 5 The Pearson r 6 r =

28. The Pearson r 6 r = [ ] [ ] 55 45 63 57.8

29. The Pearson r 6 r = [ ] [ ] 10 5.2

30. The Pearson r 6 r = 52

31. The Pearson r 6 r = 7.2111

32. The Pearson r .832 r =

33. Hypothesis Testing with Correlations • H0 =  = 0 ( = “rho” – population correlation coefficient) • Ha =   0 (there is a significant relationship between X and Y) • Technically, you could do a one-tailed test for correlations ( <0 or  >0), but for our purposes we will always test whether there simply is a relationship – therefore, we will always do a two-tailed test for correlations. • Find the critical value for .05 with df=n-2 (where N is the number of paired observations) in Table E.2 p. 440

34. The Pearson r .832 r = Is an r of .832 significant? See Table E.2 (p.440) for n - 2 df ( 5 - 2 = 3 df) and an alpha (a) of .05

35. The Pearson r .832 r = Is an r of .832 significant? The “Critical r” = .878 r = .832 Therefore, the correlation is NOT significant

36. Popcorn Consumption • Researcher X hypothesizes that popcorn consumption varies as a function of stress. He gives a random sample of 5 people a self-report measure of stress that produces scores ranging from 1 (little or no stress) to 10 (very stressed), and then has them watch a movie. He measures how many kernels of popcorn each of them eat. Is popcorn consumption correlated with stress?

37. Are X & Y Correlated? Stress Ratings # of Kernals

38. [ ] [ ] (SC)2 (SU)2 SC2 SU2 N N The Pearson r (SC) (SU) SCU N r =

39. We Need: • Sum of the Xs SC • Sum of the Ys SU • Sum of the Xs squared (SC)2 • Sum of the Ys squared(SU)2 • Sum of the squared Xs SC2 • Sum of the squared Ys SU2 • Sum of Xs times the Ys SCU • Number of Subjects (N)

40. Correlation Arithmetic

41. [ ] [ ] (SC)2 (SU)2 SC2 SU2 N N The Pearson r (SC) (SU) SCU N r =

42. [ ] [ ] (29)2 (40)2 189 370 5 5 The Pearson r (29) (40) 256 5 r =

43. [ ] [ ] (29)2 (40)2 189 370 5 5 The Pearson r 1160 256 5 r =

44. [ ] [ ] (29)2 (40)2 189 370 5 5 The Pearson r 256 232 r =

45. [ ] [ ] (29)2 (40)2 189 370 5 5 The Pearson r 24 r =

46. [ ] [ ] 841 1600 189 370 5 5 The Pearson r 24 r =

47. The Pearson r 24 r = [ ] [ ] 189 168.2 370 320

48. The Pearson r 24 r = [ ] [ ] 20.8 50

49. The Pearson r 24 r = 1040

50. The Pearson r 24 r = 32.2490