Bivariate Linear Correlation

1. Bivariate Linear Correlation

2. Linear Function Y = a + bX

3. Fixed and Random Variables A FIXED variable is one for which you have every possible value of interest in your sample. Example: Subject sex, female or male. A RANDOM variable is one where the sample values are randomly obtained from the population of values. Example: Height of subject.

4. Correlation & Regression If Y is random and X is fixed, the model is a regression model. If both Y and X are random, the model is a correlation model. Psychologists generally do not know this They think Correlation = compute the corr coeff, r Regression = find an equation to predict Y from X

5. Scatter Plot

10. Burgers (X) and Beer (Y)

11. Burger (X)-Beer (Y) Correlation

15. Burger (X)-Beer (Y) Correlation

16. Hø: ? = 0 df = n – 2 = 3 Now construct a confidence interval

17. http://glass.ed.asu.edu/stats/analysis/rci.html r =.8, n = 5. This program has problems when limit is near or above 1.

18. Try r = .8, n = 10 Clearly our estimate of ? is consistent, since the CI narrowed when N increased. The CI now excludes 0, that is, the correlation is significantly different from 0.

19. Get Exact p Value COMPUTE p=2*CDF.T(t,df).

20. Presenting the Results The correlation between my friends’ burger consumption and their beer consumption fell short of statistical significance, r(n = 5) = .8, p = .10,95% CI [-.28, .99]. Among my friends, beer consumption was positively, significantly related to burger consumption, r(n = 10) = .8, p = .006,95% CI [.35, .96].

21. Assumptions Homoscedasticity across Y|X Normality of Y|X Normality of Y ignoring X Homoscedasticity across X|Y Normality of X|Y Normality of X ignoring Y The first three should look familiar, we made them with the pooled variances t.

22. Bivariate Normal

23. When Do Assumptions Apply? Only when employing t or F. That is, obtaining a p value or constructing a confidence interval.

24. Summary Statement “The correlation between my friends’ burger consumption and their beer consumption fell short of statistical significance, r(n = 5) = .8, p = .10, 95% CI [-.28, .99].” “Among my friends, burger consumption was significantly positively related to beer consumption, ..........”

25. Shrunken r2 This reduces the bias in estimation of ? As sample size increases (n-1)/(n-2) approaches 1, and the amount of correction is reduced.

26. Spearman Rho

27. Pearson vs. Spearman

28. Uses of Correlation Analysis Measure the degree of linear association Correlation does imply causation Necessary but not sufficient Third variable problems Reliability Validity Independent Samples t – point biserial r Y = a + b? Group (Group is 0 or 1)

29. Uses of Correlation Analysis Contingency tables -- ? Rows = a + b?Columns Multiple correlation/regression

30. Uses of Correlation Analysis Analysis of variance (ANOVA) PolitConserv = a + b1 Republican? + b2 Democrat? k = 3, the third group is all others Canonical correlation/regression

31. Uses of Correlation Analysis Canonical correlation/regression (homophobia, homo-aggression) =  (psychopathic deviance, masculinity, hypomania, clinical defensiveness) High homonegativity = hypomanic, unusually frank, stereotypically masculine, psychopathically deviant (antisocial)

32. Factors Affecting Size of r Range restrictions Without variance there can’t be covariance Extraneous variance The more things affecting Y (other then X), the smaller the r. Interactions – the relationship between X and Y is modified by Z If not included in the model, reduces the r.

33. Power Analysis

34. Cohen’s Guidelines .10 – small but not trivial .30 – medium .50 – large

35. PSYC 6430 Addendum The remaining slides cover material I do not typically cover in the undergraduate course.

36. Correcting for Measurement Error If reliability is not 1, the r will underestimate the correlation between the latent variables. We can estimate the correlation between the true scores this way: rxx and rYY are reliabilities

37. Example r between misanthropy and support for animal rights = .36 among persons with an idealistic ethical ideology

38. H?: ?1 = ?2 Is the correlation between X and Y the same in one population as in another? The correlation between misanthropy and support for animal rights was significantly greater in nonidealists (r = .36) than in idealists (r = .02)

39. H?: ?WX = ?WY We have data on three variables. Does the correlation between X and W differ from that between Y and W. W is GPA, X is SATverbal, Y is SATmath. See Williams’ procedure in our text. See other procedures referenced in my handout.

40. H?: ?WX = ?YZ Raghunathan, T. E, Rosenthal, R, & and Rubin, D. B. (1996). Comparing correlated but nonoverlapping correlations, Psychological Methods, 1, 178-183. Example: is the correlation between verbal aptitiude and math aptitude the same at 10 years of age as at twenty years of age (longitudional data)

41. H?: ? = nonzero value A meta-analysis shows that the correlation between X and Y averages .39. You suspect it is not .39 in the population in which you are interested. H?: ? = .39.