Bivariate Correlation Lesson 11 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 How much do scores (Xi) vary from mean? 2 (standard deviation) s 2 2 ( X X ) i N 1 (X i X )( X i X ) N 1 Covariance How much do scores (Xi, Yi) from their means (X cov(x, y) i X )(Yi Y ) N 1 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) cov(X , Y ) ( X i X )(Yi Y ) r s X sY ( N 1) s X sY 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 ~ 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 ~ Scatter Diagrams 84 78 Height 72 66 60 6 7 8 9 Shoe size 10 11 12 Slope & value of r Determines sign positive or Height negative From lower left to upper right positive ~ 84 78 72 66 60 6 7 8 9 Shoe size 10 11 12 Slope & value of r From upper left to lower right Weight negative ~ 300 250 200 150 100 3 6 9 12 Chin ups 15 18 21 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 ~ Width & value of r Weak relationship Strong negative relationship r near 0 r near -1 Weight 300 300 250 250 Weight 200 200 150 150 100 100 3 6 9 12 Chin ups 15 18 21 3 6 9 12 Chin ups 15 18 21 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) 2 R = .0625 Approximately 6% of differences in IQ explained by relationship to gray matter volume ~ 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 Nonlinear relationships Pearson’s r does not detect more complex relationships r near 0 ~ Peeps (Y) Stress (X) Factors that affect size of r Range restriction eliminate values from 1 or both Height variable r is reduced e.g. eliminate people under 72 inches ~ 84 78 72 66 60 6 7 8 9 Shoe size 10 11 12 Hypothesis Test for r H 0: ρ = 0 rho = parameter H 1: ρ ≠ 0 ρCV df = n – 2 Table: Critical values of ρ PASW output gives sig. Example: n = 30; df=28; nondirectional ρCV = + .361 decision: r = .285 ? r = -.38 ? ~ 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 ~ Evaluating Effect Size Pearson’s r Cohen’s d r = ± .1 Small: r = ± .3 Medium: d = 0.5 r = ±.5 ~ Large: d = 0.2 d = 0.8 Note: Why no zero before decimal for r ? 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 ~ Point-biserial Correlation One variable dichotomous Only two values e.g., Sex: male & female PASW/SPSS Same as for Pearson’s r ~ 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 ~ Correlation: SPSS Data entry 1 column per variable Menus Analyze Correlate Bivariate Dialog box Select variables Choose correlation type 1- or 2-tailed test of significance ~ Reporting Correlation Coefficients 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 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. Correlation: Example Correlations WorkHours WorkHours Pearson Correlation ExCurrHours 1 Sig. (2-tailed) N ExCurrHours Pearson Correlation Sig. (2-tailed) N -.313 .081 32 32 -.313 1 .081 32 32 Correlation: Example Analysis using the Pearson’s r correlation indicated that the there was moderately strong negative relationship between the number of work hours and the number of hours spent on extracurricular activities, but the relationship was not statistically significant, r = -.31, p (two-tailed) = .08. The R2 = .097, indicating that the relationship accounts for approximately 9.7% of the variance in the number of hours spent in each activity.