Bivariate Correlation Lesson 15 Measuring Relationships Correlation l

Bivariate Correlation Lesson 15

Measuring Relationships 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) ~ n

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

Covariance: Problem 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) n l

Pearson Correlation Coefficient 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 n u l X then Y; or X then Y Negative: change in opposite direction u X then Y; or X then Y ~

Correlation & Graphs 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 ~ n

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

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

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

Width & value of r 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 ~ n

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

Strength of Correlation R 2 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) 2 l R =. 0625 l Approximately 6% of differences in IQ explained by relationship to gray matter volume ~ n

Guidelines for interpreting strength of correlation Table 5. 2 Interpreting a correlation coefficient Size of Correlation (r) General coefficient interpretation . 8 to 1. 0 Very strong relationship . 6 to. 8 Strong relationship . 4 to. 6 Moderate relationship . 2 to. 4 Weak relationship . 0 to. 2 Weak to no relationship *The same guidelines apply for negative values of r *from Statistics for People Who (Think They) Hate Statistics: Excel 2007 Edition By Neil J. Salkind

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 n n H 0: ρ = 0 rho = parameter H 1: ρ ≠ 0 ρCV l df = n – 2 l Table: Critical values of ρ l PASW output gives sig. Example: n = 30; df=28; nondirectional l ρCV = +. 335 l decision: r =. 285 ? r = -. 38 ? ~

Using Pearson r 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 ~ n

Evaluating Effect Size n Pearson’s r l n Cohen’s d l r = ±. 1 l Small: l r = ±. 3 l Medium: d = 0. 5 r = ±. 5 ~ l Large: d = 0. 2 d = 0. 8 Note: Why no zero before decimal for r ?

Correlation and Causation requires correlation, but. . . l Correlation does not imply causation! n The 3 d 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 ~ n

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

Correlation: Non. Parametric Spearman’s rs 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 ~ n

Correlation: PASW 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 ~ n

Correlation: PASW Output Figure 6. 1 – Pearson’s Correlation Output

Reporting Correlation Coefficients n Guidelines 1. 2. 3. 4. 5. 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 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.