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Correlation

Correlation. Two variables: Which test?. X. Contingency analysis. Logistic regression. Y. Correlation Regression. t-test. Two variables: Which test?. X. Contingency analysis. Logistic regression. Y. Correlation Regression. t-test. Relationship Between Two Numerical Variables.

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Correlation

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  1. Correlation

  2. Two variables: Which test? X Contingency analysis Logistic regression Y Correlation Regression t-test

  3. Two variables: Which test? X Contingency analysis Logistic regression Y Correlation Regression t-test

  4. Relationship Between Two Numerical Variables

  5. Relationship Between Two Numerical Variables

  6. Correlation • What is the tendency of two numerical variables to co-vary (change together)?

  7. Correlation • What is the tendency of two numerical variables to co-vary (change together)? • Correlation coefficient r measures the strength and direction of the linear association between two numerical variables

  8. Correlation • What is the tendency of two numerical variables to co-vary (change together)? • Correlation coefficient r measures the strength and direction of the linear association between two numerical variables • Population parameter: r (rho) • Sample estimate: r

  9. Sum of squares: X and Y

  10. Sum of products Sum of squares: X and Y

  11. Shortcuts

  12. r r r r

  13. Correlation assumes... • Random sample • X is normally distributed with equal variance for all values of Y • Y is normally distributed with equal variance for all values of X

  14. Correlation assumes... • Random sample • X is normally distributed with equal variance for all values of Y • Y is normally distributed with equal variance for all values of X Bivariate normal distribution

  15. Correlation coefficient facts • -1 < r < 1; -1 < r < 1

  16. Correlation coefficient facts • -1 < r < 1; -1 < r < 1 • Positive r: variables increase together • Negative r: when one variable increases, the other decreases, and vice-versa

  17. Correlation coefficient facts • -1 < r < 1; -1 < r < 1 • Positive r: variables increase together • Negative r: when one variable increases, the other decreases, and vice-versa negative uncorrelated positive r=0 r = -1 r = 1

  18. Correlation coefficient facts • Coefficient of determination = r2 • Describes the proportion of variation in one variable that can be predicted from the other

  19. Standard error of r

  20. Confidence Limits for r

  21. Example • Are the effects of new mutations on mating success and productivity correlated? • Data from Drosophila melanogaster • n = 31 individuals

  22. X is productivity, Y is the mating success • Sum of products = 2.796 • Sum of squares for X = 16.245 • Sum of squares for Y = 1.6289

  23. X is productivity, Y is the mating success

  24. Confidence Limits for r

  25. Confidence Limits for r

  26. Confidence Limits for r

  27. Confidence Limits for r

  28. Confidence Limits for r

  29. Confidence Limits for r

  30. Example: Why Sleep?

  31. Example: Why Sleep? • 10 experimental subjects • Measured increase in “slow-wave” activity during sleep • Measured improvement in task after sleep - hand-eye coordination activity

  32. Example: Why Sleep?

  33. Why sleep? • Sum of products: 1127.4 • Sum of squares X: 2052.4 • Sum of squares Y: 830.9 • Calculate a 95% C.I. for 

  34. Hypothesis Testing for Correlations • Can test hypotheses relating to correlations among variables • Closely related to regression - the topic for next Tuesday’s lecture

  35. Hypothesis Testing for Correlations H0: r = 0 HA: r 0

  36. If r = 0,... r is normally distributed with mean 0 with df = n -2

  37. Example • Are the effects of new mutations on mating success and productivity correlated? • Data from Drosophila melanogaster

  38. Hypotheses H0: Mating success and productivity are not related (r = 0) HA: Mating success and productivity are correlated (r 0)

  39. X is productivity, Y is the mating success • Sum of products = 2.796 • Sum of squares for X = 16.245 • Sum of squares for Y = 1.6289

  40. df= n-2=31-2=29

  41. df= n-2=31-2=29

  42. Why sleep? • Sum of products: 1127.4 • Sum of squares X: 2052.4 • Sum of squares Y: 830.9 • Test for a correlation different from zero in these data.

  43. Checking Assumptions for Correlation • Bivariate normal distribution • Relationship is linear (straight line) • Cloud of points in scatter plot is circular or elliptical • Frequency distributions of X and Y are normal

  44. Linear Relationship?

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