1. Relationships between variables Statistics for the Social Sciences Psychology 340 Spring 2010

2. PSY 340 Statistics for the Social Sciences Exam 3 results Mean = 86.9 Median = 89.0 Great job!

3. PSY 340 Statistics for the Social Sciences Final project Details posted on websiteDetails Download the Harass.sav datafile (fictional dataset of harassment within a workplace)Harass Conduct analyses to answer 8 different questions Write up the results of you analyses Worth 15% of final grade

4. PSY 340 Statistics for the Social Sciences Outline (for 2 weeks) Correlation –Scatterplot, hypothesis testing, computations, SPSS Simple bi-variate regression, least-squares fit line –The general linear model –Residual plots –Using SPSS Multiple regression –Comparing models, Delta r 2 –Using SPSS

5. PSY 340 Statistics for the Social Sciences Correlation Correlations describe relationships between two variables –Age and coordination skills in children, as kids get older their motor coordination tends to improve –Price and quality, generally the more expensive something is the higher in quality it is

6. PSY 340 Statistics for the Social Sciences Correlation Correlations describe relationships between two variables, but DO NOT explain why the variables are related Suppose that Dr. Steward finds that rates of spilled coffee and severity of plane turbulents are strongly positively correlated. One might argue that turbulents cause coffee spills One might argue that spilling coffee causes turbulents

7. PSY 340 Statistics for the Social Sciences Correlation Correlations describe relationships between two variables, but DO NOT explain why the variables are related Suppose that Dr. Cranium finds a positive correlation between head size and digit span (roughly the number of digits you can remember). One might argue that bigger your head, the larger your digit span 1 21 24 15 37 One might argue that head size and digit span both increase with age (but head size and digit span aren’t directly related)

8. PSY 340 Statistics for the Social Sciences Correlation Correlations describe relationships between two variables, but DO NOT explain why the variables are related For many years instructors have noted that the reported fatality rate of grandparents increases during midterm and final exam periods. One might argue that college exams cause grandparent death Dead Grandmother/Exam Syndrome Web Page

9. PSY 340 Statistics for the Social Sciences Relationships between variables How variables co-vary with one another –As a descriptive statistic To examine this relationship you should: –Make a scatterplot - a picture of the relationship –Compute the Correlation Coefficient - a numerical description of the relationship Properties of a correlation –Form (linear or non-linear) –Direction (positive or negative) –Strength (none, weak, strong, perfect) –As an inferential statistic – comparing an observed correlation with a correlation expected due to chance

10. PSY 340 Statistics for the Social Sciences Scatterplot: Graphing Correlations Steps for making a scatterplot 1.Draw axes and assign variables to them 2.Determine range of values for each variable and mark on axes 3.Mark a dot for each person’s pair of scores Hours studied Quiz performance A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 XY Example: What is the relationship between how much you study and exam performance?

11. PSY 340 Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 XY B 1 2 C 5 6 D 3 4 E 3 2

12. PSY 340 Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 XY C 5 6 D 3 4 E 3 2

13. PSY 340 Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 XY D 3 4 E 3 2

14. PSY 340 Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 D 3 4 XY E 3 2

15. PSY 340 Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 XY

16. PSY 340 Statistics for the Social Sciences Scatterplot Y X 1 2 3 4 5 6 123 456 Imagine a line through the data points Plots one variable against the other Each point corresponds to a different individual A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 XY Useful for “seeing” the relationship –Form, Direction, and Strength

17. PSY 340 Statistics for the Social Sciences Form Non-linearLinear

18. PSY 340 Statistics for the Social Sciences NegativePositive Direction X & Y vary in the same direction As X goes up, Y goes up Positive Pearson’s r X & Y vary in opposite directions As X goes up, Y goes down Negative Pearson’s r Y X Y X

19. PSY 340 Statistics for the Social Sciences Strength The strength of the relationship –Spread around the line (note the axis scales) –Correlation coefficient will range from -1 to +1 Zero means “no relationship” The farther the r is from zero, the stronger the relationship

20. PSY 340 Statistics for the Social Sciences Strength r = 1.0 “perfect positive corr.” r 2 = 100% r = -1.0 “perfect negative corr.” r 2 = 100% r = 0.0 “no relationship” r 2 = 0.0 0.0+1.0 The farther from zero, the stronger the relationship

21. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r Hypothesis testing –Core logic of hypothesis testing Considers the probability that the result of a study could have come about if the experimental procedure had no effect If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics Step 5: Make a decision about your null hypothesis –A five step program

22. PSY 340 Statistics for the Social Sciences –Step 1: State your hypotheses : as a research hypothesis and a null hypothesis about the populations Null hypothesis (H 0 ) Research hypothesis (H A ) Hypothesis testing with Pearson’s r There are no correlation between the variables (they are independent) Generally, the variables correlated (they are not independent)

23. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r r ≥  r <  H0:H0: HA:HA: – Our theory is that the variables are negatively correlated –Step 1: State your hypotheses One -tailed Note: sometimes the symbol ρ (rho) is used Note: sometimes the symbol ρ (rho) is used

24. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r r > 0 r < 0 H0:H0: HA:HA: – Our theory is that the variables are negatively correlated –Step 1: State your hypotheses One -tailed r = 0 r ≠ 0 H0:H0: HA:HA: – Our theory is that the variables are correlated Two -tailed

25. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r –Step 2: Set your decision criteria Your alpha (α) level will be your guide for when to reject or fail to reject the null hypothesis. –Based on the probability of making making an certain type of error

26. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r –Step 3: Collect your data Descriptive statistics (Pearson’s r) 6 1 2 5 6 3 4 3 2 X Y Common formulas for the correlation coefficient: Used this one in PSY138Z-score alternative For an example of the z- score alternative, skip to the end of the powerpoint

27. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 1: SP (Sum of the Products) mean 3.64.0 6 1 2 5 6 3 4 3 2 X Y

28. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 1: SP (Sum of the Products) mean 3.64.0 2.4 0.0 6 1 2 5 6 3 4 3 2 X Y = 6 - 3.6 -2.6 = 1 - 3.6 1.4 = 5 - 3.6 -0.6 = 3 - 3.6 -0.6= 3 - 3.6 Quick check

29. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 1: SP (Sum of the Products) mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 6 1 2 5 6 3 4 3 2 X Y 2.0= 6 - 4.0 -2.0 = 2 - 4.0 2.0= 6 - 4.0 0.0 = 4 - 4.0 -2.0 = 2 - 4.0 Quick check

30. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 1: SP (Sum of the Products) mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0SP 6 1 2 5 6 3 4 3 2 X Y 4.8 * = 5.2 * = 2.8 * = 0.0 * = 1.2 * =

31. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 2: SS X & SS Y

32. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 2: SS X & SS Y mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0 6 1 2 5 6 3 4 3 2 X Y 4.8 5.2 2.8 0.0 1.2 5.76 15.20 SS X 2 =6.76 2 =1.96 2 =0.36 2 = 2 =

33. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 2: SS X & SS Y mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0 6 1 2 5 6 3 4 3 2 X Y 4.8 5.2 2.8 0.0 1.2 5.76 6.76 1.96 0.36 15.20 2 =4.0 2 = 2 = 2 =0.0 2 = 4.0 16.0 SS Y

34. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 3: compute r

35. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using SP) Stage 3: compute r mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.014.0 6 1 2 5 6 3 4 3 2 X Y 4.8 5.2 2.8 0.0 1.2 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 SS Y SS X SP

36. PSY 340 Statistics for the Social Sciences Computing Pearson’s r Stage 3: compute r 14.015.2016.0 SS Y SS X SP

37. PSY 340 Statistics for the Social Sciences Computing Pearson’s r Stage 3: compute r 15.2016.0 SS Y SS X

38. PSY 340 Statistics for the Social Sciences Computing Pearson’s r Stage 3: compute r 15.20 SS X

39. PSY 340 Statistics for the Social Sciences Computing Pearson’s r Stage 3: compute r

40. PSY 340 Statistics for the Social Sciences Computing Pearson’s r Stage 3: compute r Y X 1 2 3 4 5 6 123 456 Appears linear Positive relationship Fairly strong relationship.89 is far from 0, near +1

41. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r –Step 4: Compute your test statistics r = 0.89 Descriptive statistics (Pearson’s r) Inferential statistics: 2 choices (really the same): –A t-test & the t-tablet-table –Use the Pearson’s r table (if available)Pearson’s r table Compute your degrees of freedom (df) df = n - 2 = 5 - 2 = 3

42. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r –Step 4: Compute your test statistics Descriptive statistics (Pearson’s r) Inferential statistics: 2 choices (really the same): –A t-test & the t-tablet-table –Use the Pearson’s r table (if available) From table, with df = n - 2 = 3: t crit = 3.18 Reject H 0 Conclude that the correlation is ≠0 –Step 5: Make a decision about your null hypothesis r = 0.89

43. PSY 340 Statistics for the Social Sciences Hypothesis testing with Pearson’s r –Step 4: Compute your test statistics From table –α-level = 0.05 –Two-tailed –df = n - 2 = 3 – r crit = 0.878 Reject H 0 Conclude that the correlation is ≠0 –Step 5: Make a decision about your null hypothesis Descriptive statistics (Pearson’s r) Inferential statistics: 2 choices (really the same): –A t-test & the t-tablet-table –Use the Pearson’s r table (if available)Pearson’s r table r = 0.89

44. PSY 340 Statistics for the Social Sciences Effect sizes with Pearson’s r Pearson’s r is considered a measure of the effect size –Smallr = 0.10 –Mediumr = 0.30 –Larger = 0.50

45. PSY 340 Statistics for the Social Sciences A few more things to consider about correlation Correlations are greatly affected by the range of scores in the data –Consider height and age relationship Extreme scores can have dramatic effects on correlations –A single extreme score can radically change r When considering "how good" a relationship is, we really should consider r 2 (coefficient of determination), not just r.

46. PSY 340 Statistics for the Social Sciences Correlation in SPSS Enter each variable in separate columns –Analyze -> Correlate -> bi-variate –Enter all variables you want to examine In options can request cross products and means –Output – given as a matrix –For the scatterplot: Graphs -> legacy dialogs-> scatter/dot -> simple scatter Enter which is your X var. and which is your Y var.

47. PSY 340 Statistics for the Social Sciences Correlation in Research Articles Correlation matrix –A display of the correlations between more than two variables Acculturation Why have a “-”? Why only half the table filled with numbers?

48. PSY 340 Statistics for the Social Sciences Next time Regression: Predicting a variable based on other variables

49. PSY 340 Statistics for the Social Sciences The Correlation Coefficient Formulas for the correlation coefficient: Used this one in PSY138Common alternative

50. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 1: compute standard deviation for X and Y (note: keep track of sample or population) 6 1 2 5 6 3 4 3 2 X Y For this example we will assume the data is from a population

51. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 1: compute standard deviation for X and Y (note: keep track of sample or population) Mean 3.6 2.4 -2.6 1.4 -0.6 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 SS X Std dev 1.74 For this example we will assume the data is from a population

52. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 1: compute standard deviation for X and Y (note: keep track of sample or population) Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 SS Y Std dev 1.741.79 For this example we will assume the data is from a population

53. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.38

54. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.38 -1.49 0.8 - 0.34 0.0 Quick check

55. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.11.38 -1.49 0.8 - 0.34

56. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 2: compute z-scores Mean 3.64.0 2.4 -2.6 1.4 -0.6 2.0 -2.0 2.0 0.0 -2.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 1.1 -1.1 0.0 -1.1 1.1 0.0 1.38 -1.49 0.8 - 0.34 Quick check

57. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 3: compute r Mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 0.0 1.1 -1.1 0.0 -1.1 1.1 0.0 1.521.38 -1.49 0.8 - 0.34 * =

58. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 3: compute r Mean 3.64.0 2.4 -2.6 1.4 -0.6 0.0 2.0 -2.0 2.0 0.0 -2.0 0.0 6 1 2 5 6 3 4 3 2 X Y 5.76 6.76 1.96 0.36 15.20 4.0 0.0 4.0 16.0 Std dev 1.741.79 0.0 1.1 -1.1 0.0 -1.1 1.1 0.0 1.52 1.64 0.88 0.0 0.37 1.38 -1.49 0.8 - 0.34 4.41

59. PSY 340 Statistics for the Social Sciences Computing Pearson’s r (using z-scores) Step 3: compute r Y X 1 2 3 4 5 6 123 456 Appears linear Positive relationship Fairly strong relationship.89 is far from 0, near +1