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Understanding Correlation: Patterns, Coefficients, and Interpretation

This chapter explains the concept of correlation, patterns of correlation, calculation of correlation coefficients and their interpretation, and issues in interpreting correlation coefficients. It includes examples and guidelines for understanding correlation in research articles.

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Understanding Correlation: Patterns, Coefficients, and Interpretation

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  1. Chapter 11 Correlation Pt 1: Nov. 12, 2013

  2. Correlation • Association between scores on two variables • e.g., age and coordination skills in children, price and quality • Use scatterplots to see the relationship • Rule of thumb – if 1 var is a “predictor”, put it on the x axis

  3. Patterns of Correlation • Linear correlation – straight line relationship (appropriate to compute corr) • Curvilinear correlation – U or S shaped curves • No correlation – no trend to points in scatterplot • Positive correlation – points move from lower left to upper right (pos slope) • Negative correlation – points move from upper left to lower right (neg slope)

  4. Degree of Linear CorrelationThe Correlation Coefficient • Figure correlation using products of deviation scores • Multiply pos x pos get positive results • Multiply negative x negative  get positive results, which we want • Multiply pos x neg get negative results 1) Find means of x variable (Mx) and y variable (My) 2) Find deviation scores for each person for x variable (x-Mx) and y variable (y-My) 3) Sum these up across the sample 4) divide by sqrt of (SSx)(SSy) • where SSx=sum of squared deviations for x variable and SSy=sum of squared deviations for y variable

  5. Formula for the correlation coefficient: r = Σ [(x – Mx)(y – My)] sqrt [(SSx)(SSy)] where SSx = Σ (x-Mx)2 where SSy = Σ (y-My)2 • Positive perfect correlation: r = +1 • No correlation: r = 0 • Negative perfect correlation: r = –1 • Example on board…

  6. Correlation and Causality • Three possible directions of causality: 1. X Y 2. X Y 3. Z X Y Can only determine causality w/longitudinal study or a true experiment (w/random assignment) to rule out 3rd variables (z) Examples of 3rd variable explainingthe correlation between x & y?

  7. Issues in Interpreting the Correlation Coefficient • Statistical significance – for correlation, test is whether true corr in pop = 0. • If corr is statistically signif, means it is highly unlikely that we’d get this corr if true pop corr = 0. • Proportionate reduction in error • r2 = proportion of variance (in y) accounted for (by x) • Used to compare correlations • r = .3, r2 = .09; • r = .6, r2 = .36 (so 4x as big)

  8. Issues in Interpreting the Correlation Coefficient • Restriction in range • With limited range, corr is different than what it would be with full range (more variability) • Correlate job perf with hiring test score • But only hire people w/high test scores, so limited range (selective of good performers)

  9. Size of r: Cohen’s Guidelines • What is a large corr? • Cohen’s guidelines: • > .5 or -.5 = large, .3 or -.3 = moderate, .1 or -.1 = small • Unusual to have corr above .5 or -.5 • Consider average r = .19 for job satisfaction & job perf… • Interpretation?

  10. Correlation in Research Articles • Scatter diagrams occasionally shown • Correlation matrix presented in table. Notice only lower triangle completed & corr of variable w/itself represented with dash. • In text: “The correlation between acculturation and assimilation was significant (r = .56, p < .05).

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