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Bivariate Correlation

Bivariate Correlation. Lesson 10. Measuring Relationships. Correlation degree relationship b/n 2 variables linear predictive relationship Covariance If X changes, does Y change also? e.g., height ( X ) and weight ( Y ) ~. Covariance. Variance

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Bivariate Correlation

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  1. BivariateCorrelation Lesson 10

  2. Measuring Relationships • Correlation • degree relationship b/n 2 variables • linear predictive relationship • Covariance • If X changes, does Y change also? • e.g., height (X) and weight (Y) ~

  3. Covariance • Variance • How much do scores (Xi) vary from mean? • (standard deviation)2 • Covariance • How much do scores (Xi, Yi) from their means

  4. Covariance: Problem • How to interpret size • Different scales of measurement • Standardization • like in z scores • Divide by standard deviation • Gets rid of units • Correlation coefficient (r)

  5. Pearson Correlation Coefficient • Both variables quantitative (interval/ratio) • Values of r • between -1 and +1 • 0 = no relationship • Parameter = ρ (rho) • Types of correlations • Positive: change in same direction • X then Y; or X then Y • Negative: change in opposite direction • X then Y; or X then Y ~

  6. Correlation & Graphs • Scatter Diagrams • Also called scatter plots • 1 variable: Y axis; other X axis • plot point at intersection of values • look for trends • e.g., height vs shoe size ~

  7. 6 7 8 9 10 11 12 Scatter Diagrams 84 78 Height 72 66 60 Shoe size

  8. Slope & value of r • Determines sign • positive or negative • From lower left to upper right • positive ~

  9. Slope & value of r • From upper left to lower right • negative ~

  10. Width & value of r • Magnitude of r • draw imaginary ellipse around most points • Narrow: r near -1 or +1 • strong relationship between variables • straight line: perfect relationship (1 or -1) • Wide: r near 0 • weak relationship between variables ~

  11. Weak relationship Strong negative relationship r near 0 r near -1 300 300 250 250 Weight Weight 200 200 150 150 3 3 6 6 9 9 12 12 15 15 18 18 21 21 100 100 Chin ups Chin ups Width & value of r

  12. Strength of Correlation • R2 • Coefficient of Determination • Proportion of variance in X explained by relationship with Y • Example: IQ and gray matter volume • r = .25 (statisically significant) • R2 = .0625 • Approximately 6% of differences in IQ explained by relationship to gray matter volume ~

  13. Factors that affect size of r • Nonlinear relationships • Pearson’s r does not detect more complex relationships • r near 0 ~ Y X

  14. Factors that affect size of r • Range restriction • eliminate values from 1 or both variable • r is reduced • e.g. eliminate people under 72 inches ~

  15. Hypothesis Test for r • H0: ρ = 0 rho = parameter H1: ρ≠ 0 • ρCV • df = n – 2 • Table: Critical values of ρ • PASW output gives sig. • Example: n = 30; df=28; nondirectional • ρCV = + .335 • decision: r = .285 ? r = -.38 ? ~

  16. Using Pearson r • Reliability • Inter-rater reliability • Validity of a measure • ACT scores and college success? • Also GPA, dean’s list, graduation rate, dropout rate • Effect size • Alternative to Cohen’s d ~

  17. Evaluating Effect Size • Pearson’s r • r= ±.1 • r = ±.2 • r = ±.5 ~ • Cohen’s d • Small: d = 0.2 • Medium: d = 0.5 • High: d = 0.8 Note: Why no zero before decimal for r ?

  18. Correlation and Causation • Causation requires correlation, but... • Correlation does not imply causation! • The 3d variable problem • Some unkown variable affects both • e.g. # of household appliances negatively correlated with family size • Direction of causality • Like psychology  get good grades • Or vice versa ~

  19. Point-biserial Correlation • One variable dichotomous • Only two values • e.g., Sex: male & female • PASW/SPSS • Same as for Pearson’s r ~

  20. Correlation: NonParametric • Spearman’s rs • Ordinal • Non-normal interval/ratio • Kendall’s Tau • Large # tied ranks • Or small data sets • Maybe better choice than Spearman’s ~

  21. Correlation: PASW • Data entry • 1 column per variable • Menus • Analyze  Correlate  Bivariate • Dialog box • Select variables • Choose correlation type • 1- or 2-tailed test of significance ~

  22. Correlation: PASW Output Figure 6.1 – Pearson’s Correlation Output

  23. Reporting Correlation Coefficients • Guidelines • No zero before decimal point • Round to 2 decimal places • significance: 1- or 2-tailed test • Use correct symbol for correlation type • Report significance level • There was a significant relationship between the number of commercials watch and the amount of candy purchased, r = +.87, p (one-tailed) < .05. • Creativity was negatively correlated with how well people did in the World’s Biggest Liar Contest, rS = -.37, p (two-tailed) = .001.

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