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CORRELATION

CORRELATION. Overview of Correlation. What is a Correlation? Correlation Coefficients Coefficient of Determination Test for Significance Correlation and Causality Partial and Part Correlations. What is a Correlation?. Degree of linear relationship between variables

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CORRELATION

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  1. CORRELATION

  2. Overview of Correlation • What is a Correlation? • Correlation Coefficients • Coefficient of Determination • Test for Significance • Correlation and Causality • Partial and Part Correlations

  3. What is a Correlation? • Degree of linear relationship between variables • Each individual is measured on both variables

  4. What is a Correlation? • Comparison of the way scores deviate from their means on the two variables • Standardized covariance

  5. Cross-Product Deviation • Find the difference between each person’s scores and the mean of the variable (deviation). • For each person, multiply the two deviations together. • Do the deviations tend to go in the same direction?

  6. Covariance • Add up all the cross-product deviations and average them. • The more covariance, the more the two variables go together, or co-vary. • Covariance is not standardized, so it’s hard to interpret.

  7. Pearson r • Standardized covariance • Used for two interval/ratio variables • Varies from -1 to +1

  8. Pearson r • Absolute value indicates strength of relationship • .1 - small • .3 - medium • .5 - large

  9. Pearson r • Sign indicates direction of correlation • Positive: increases on one variable correspond to increases on the other variable • Negative: increases on one variable correspond to decreases on the other variable

  10. Other Correlation Coefficients • Ordinal variables • Spearman rho or Kendall’s tau • Dichotomous variable with interval/ratio variable • Point biserial r (discrete dichotomy) • Biserial r (continuous dichotomy)

  11. Other Correlation Coefficients • Two dichotomous variables • Phi coefficient

  12. About Dichotomous Variables • Dichotomous variables are usually at the nominal level. • Numbers are assigned to the two categories in an arbitrary way. • The way the numbers are assigned determines the sign of the correlation coefficient.

  13. Review Question! How is covariance related to the correlation coefficient?

  14. Coefficient of Determination • Measures proportion of explained variance in Y based on X • r2

  15. Testing r for Significance • Null hypothesis is usually that r is zero in the population. • One tailed vs. two-tailed

  16. Assumptions • Appropriate types of data • Independent observations • Normal distributions • Linear relationship

  17. Example APA format The Pearson r was computed between rated enjoyment of frog legs and level of neuroticism. The correlation was statistically significant, r(58) = .28, p = .03.

  18. Review Question If r = .28, then r2 = .08. What does the .08 represent?

  19. Review Question! If p = .03, what probability does the .03 represent? There is a 3% chance of…..?

  20. Correlation and Causality • A correlation by itself does not show that one variable causes the other. • A correlation may be consistent with a causal relationship.

  21. The Third Variable Problem • A correlation between X and Y could be caused by a third variable influencing both X and Y.

  22. The Directionality Problem • A correlation between X and Y could be a result of X causing Y or Y causing X.

  23. Partial Correlation • Used to “partial out” the effects of a third variable (X2) on the relationship between X1 and Y • Correlation between X1 and Y with the influence of X2 removed from both X1 and Y

  24. Partial r2 X1 Y X2

  25. Interpreting Partial Correlations • Compare the simple bivariate correlation to the partial correlation. • If the partial correlation is lower, it suggests that X2 is mediating the relationship between X1 and Y.

  26. Part Correlation • Also called: semi-partial correlation • Correlation between X1 and Y with the influence of X2 (and other predictor variables) removed from just X1 • Indicates amount of unique variance in Y explained by X1 • Used in Multiple Regression Analysis

  27. Part r2 X1 Y X2

  28. Partial r2 X1 Y X2

  29. Choosing Stats Patrons at a bar are randomly assigned to one of three information conditions. In one condition, they taste a beer without being given any information about it. In a second condition, they are told that it is an inexpensive brand of beer. In a third condition, they are told that it is an expensive brand of beer. Their ratings of the taste quality are compared across the three conditions.

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