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

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)

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 10 11 12 Shoe size

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

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

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

Strength of Correlation 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 ~

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 ~ Y X

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 ~

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 ? ~

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 r = ± .1 r = ± .3 r = ±.5 ~ Cohen’s d Small: d = 0.2 Medium: d = 0.5 Large: 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: 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 ~

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

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.