Bivariate Correlation Lesson 10

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

Covariance n Variance l How much do scores (X i ) vary from mean? l (standard deviation) 2 l n Covariance l How much do scores (X i, Y i ) from their means l

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

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

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

Scatter Diagrams Height Shoe size

Slope & value of r n Determines sign l positive or negative n From lower left to upper right l positive ~

Slope & value of r n From upper left to lower right l negative ~

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

Width & value of r Weight Chin ups Strong negative relationship r near -1 Weight Chin ups Weak relationship r near 0

Strength of Correlation nR2nR2 l Coefficient of Determination l Proportion of variance in X explained by relationship with Y n Example: IQ and gray matter volume l r =.25 (statisically significant) l R 2 =.0625 l Approximately 6% of differences in IQ explained by relationship to gray matter volume ~

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

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

Hypothesis Test for r n H 0 : ρ = 0rho = parameter H 1 : ρ ≠ 0 n ρ CV l df = n – 2 l Table: Critical values of ρ l PASW output gives sig. n Example: n = 30; df=28; nondirectional l ρ CV = l decision: r =.285 ? r = -.38 ? ~

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

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

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

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

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

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

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

n Guidelines 1. No zero before decimal point 2. Round to 2 decimal places 3. significance: 1- or 2-tailed test 4. Use correct symbol for correlation type 5. Report significance level n There was a significant relationship between the number of commercials watch and the amount of candy purchased, r = +.87, p (one-tailed) <.05. n Creativity was negatively correlated with how well people did in the World’s Biggest Liar Contest, r S = -.37, p (two-tailed) =.001. Reporting Correlation Coefficients