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Measures of the relationship between 2 variables: Correlation

Measures of the relationship between 2 variables: Correlation. Chapter 16. Correlation Coefficient. Descriptive statistic degree of relationship between 2 variables 2 dependent variables if we know value of 1 variable… how well can we predict value of other

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Measures of the relationship between 2 variables: Correlation

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  1. Measures of the relationship between 2 variables: Correlation Chapter 16

  2. Correlation Coefficient • Descriptive statistic • degree of relationship between 2 variables • 2 dependent variables • if we know value of 1 variable… how well can we predict value of other • Values of correlation coefficient • between -1 and +1 • 0 = no relationship ~

  3. Correlation Coefficient

  4. Scatter Diagrams • Also called scatter plots • 1 variable: Y axis; other X axis • makes no difference which way • plot point at intersection of values • look for trends • e.g., height vs shoe size ~

  5. 6 7 8 9 10 11 12 Scatter Diagrams 84 78 Height 72 66 60 Shoe size

  6. Correlation Coefficient Values • Sample statistic: r • population parameter: r (rho) • values: -1 <r < +1 • Scatter diagram characteristics • slope & width • determines value of r ~

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

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

  9. Width & value of r • Magnitude of r • draw imaginary ellipse around most points • Narrow: r near -1 or +1 • strong relationship between variables • Wide: r near 0 • little or no relationship between variables ~

  10. 300 300 250 250 Weight Weight 200 200 150 150 3 3 6 6 9 9 12 12 15 15 18 18 21 21 100 100 Chin ups Chin ups Width & value of r Weak relationship Strong negative relationship r near 0 r near -1

  11. Measures of Correlation • Several different measures • depends on level of measurement • Pearson’s r • interval/ratio • Spearman’s rs • ordinal and interval/ratio • Others for nominal and different combinations of levels of measurement ~

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

  13. 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 ~

  14. Correlation and Causation • Causation requires correlation, but... • Correlation does not imply causation! • Does not mean 1 variable causes changes in the other • e.g. # of household appliances negatively correlated with family size • appliances as effective birth control? • Changes may be caused by a third unknown variable ~

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