Research Methods & Design in Psychology

1. Research Methods & Design in Psychology Lecture 4 Correlation Lecturer: James Neill

2. Readings • Howell (Fundamentals) • Ch9 (Correlation) • Howell (Methods) • Ch6 (Categorical Data and Chi-Square) • Ch 9 (Correlation and Regression)

3. Overview • Correlational analyses • Types of answers • Types of correlation • Interpretation • Assumptions / Limitations

4. The World is Made of Covariation

5. Covariations are the Building Block of Complex Models

11. Correlational Research Questions Are two variables related? Interesting questions tend to: • test a novel relationship, e.g. • “is time spent studying for exams associated with increased incidence of brain cancer?” • avoid simply showing an expected relationship, e.g. • “is time spent studying studying for exams associated with higher exam marks?”

12. Correlational analyses 1 Correlational analyses are used to examine the extent to which two variables have a simple linear relationship. Correlations provide the building blocks for: • Factor analysis • Reliability • Regression • Etc.

13. Correlational analyses 2 Linear relationship between 2 variables: • direction and • strength • ranges from -1 to +1 • Sign indicates direction • Size indicates strength

14. Correlational analyses 3 Measures the extent to which: • differences in one variable can be predicted from differences in the other variable • one variable varies with another variable A correlation is also an effect size.

15. Types of Answers • No relationship (independence) • Linear relationship: • As one variable increases, so does the other (+ve) • As one variable increases, the other decreases (-ve) • Non-linear • Restricted range • Heterogeneous samples

16. Types of Correlation • Phi / Cramer’s V • Spearman’s rank / Kendall’s Tau b • Point bi-serial rpb • Product-moment or Pearson’s r

17. Types of Correlation

18. Tufte: Graphics reveal data.

19. Nominal by Nominal

20. Contingency Tables 1 • Bivariate frequency tables • Marginal totals • Can include %

21. Contingency Tables 2

22. Example

23. Example

24. Example

25. Clustered Bar Graph • Bar graph of frequencies or percentages with the category axis clustered by coloured bars to indicate the two variable’s categories

26. Example

27. Chi-square 1

28. Example

29. Example

30. Phi (f) & Cramer’s V Phi () • Two dichotomous variables (2x2, 2x3, 3x2) • E.g., Gender & Pass/Fail Cramer’s V • Two dichotomous variables (3x3 or greater) • E.g., Favourite Season x Favourite Sense

31. Example

32. Ordinal by Ordinal

33. Spearman’s rho (rs) • For ranked (or recoded to ordinal) data • Uses product-moment correlation, but interpretations must be adjusted to consider the underlying ranked scales • e.g. Olympic Placing vs. World Ranking

34. Kendall’s Tau-b • Kendall’s tau-b • for ordinal/ranked data • takes joint ranks into account • Ranges -1 to +1, but only for square tables

35. Dichotomous by Interval/Ratio

36. Point-biserial correlation • Point-biserial correlation (rpb) • one dichotomous & one continuous variable • calculate as for Pearson’s r, but interpretations must be adjusted to consider the underlying ranked scales • e.g., gender and self-esteem

37. Example

38. Example

39. Product-moment correlation (r) • For two interval and/or ratio variables • r = covxy sxsy

40. Interval/Ratio by Interval/Ratio

41. Scatterplots • Plot each pair of observations (X, Y) • x = predictor variable (independent) • y = criterion variable (dependent) • Check for: • outliers • linearity • ‘Line of best fit’ y = a + bx

42. The correlation between 2 variables is a measure of the degree to which: • Pairs of numbers (points) cluster together around a best-fitting straight line

43. Scatterplot showing relationship between age & cholesterol

44. Strong positive (.81)

45. Weak positive (.14)

46. Moderately strong negative (-.76)

47. Stop global warming: Become a pirate

48. Correlation Estimation Indicate level (high, med., or low) and sign of the correlation for: • number of guns in community and number firearm deaths • robberies and incidence of drug abuse • protected sex and incidence of AIDS • community education level and crime rate • solar flares and suicide

49. Covariance • Variance shared by 2 variables • Covariance reflects the direction of the relationship: +ve cov indicates + relationship -ve cov indicates - relationship. Cross products

50. -ve dev. products +ve dev. products -ve dev. products +ve dev. products Covariance – Cross-products -ve cross products