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Correlation & Regression

Correlation & Regression. Association & Prediction. Measuring association. Editorial and letter to the editor, Indianapolis Star re CDC data Differing opinions regarding degree of association How to quantify the association between two variables ie Smoking deaths & tax

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Correlation & Regression

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  1. Correlation&Regression Association & Prediction

  2. Measuring association • Editorial and letter to the editor, Indianapolis Star re CDC data • Differing opinions regarding degree of association • How to quantify the association between two variables • ie Smoking deaths & tax • ie Smoking percent & tax • ie Smoking percent & smoking death

  3. Breast feeding & IQ Smoking & Criminal Behavior Abortion & Crime Lot’s of Anecdotal & Clinical Relationships

  4. Is there a relationship?

  5. Plot out the data The Scattergram

  6. Plot out the data The Scattergram Janet (756,3.8) John

  7. Plot out the data The Scattergram Each point represents a pair of scores from a single subject (case)

  8. The Scattergram

  9. Add 2 more students

  10. The Scattergram

  11. Quantifying Relationships • Pearson: developed the technique • Pearson r • Pearson correlation coefficient • Pearson product-moment correlation coefficient • r

  12. Correlation • Co rrelation: how score on one variable is related to score on another variable • More specifically • How relative performance on one variable is related to relative performance on another variable • ie How each score relates to its’ mean and variability

  13. Quantify relationship to the mean: Deviation Score • X = independent variable • Y = dependent variable • X - X (score on one variable related to its mean; deviation score of X; x) • Y - Y (score on another variable related to its mean; deviation score of Y; y)

  14. Calculation of r : deviation score method  ( (Xi - X) (Yi -Y) ) r = [(Xi - X)2 * (Yi - Y)2]

  15. Calculation of r : deviation score method ( Xi - X) Deviation score of X x Note: will be + or - for each case

  16. Calculation of r : deviation score method ( Yi - Y) Deviation score of Y y Note: will be + or - for each case

  17. Calculation of r : deviation score method (Xi - X) ( Yi - Y) Product of paired deviation scores Product of x and y xy Note: product will be + or - for each case

  18. Calculation of r : deviation score method [(Xi - X) ( Yi - Y)] Sum of product of paired deviation scores Sum of xy Covariance Note: will be + or - depending on ALL of the individual cases!!!!

  19. Calculation of r : deviation score method  ( (Xi - X) (Yi -Y) ) r = (Xi - X)2 * (Yi - Y)2

  20. Calculate r : T1&T2, T1&T3, T1&T4

  21. r by deviation score method X=8 Y=8 20 20 20

  22. r T1&T2 = 1.00Perfect Positive Relationshipsee scattergram next slide

  23. Graphical presentation of the data: perfect + relationship

  24. T1 & T2 = 1.00 • perfect positive • T1 & T3 = -1.00 • perfect negative • T1& T4 = 0.00 • no relationship

  25. Possible values of r • Range from -1.00 to +1.00 • any value in between • closer the value to -1.00, stronger the - relationship between the two variables • closer the value to +1.00, stronger the + relationship between the two variables Guess the correlation game

  26. Possible values of r • Range from -1.00 to +1.00 • any value in between • closer the value to -1.00, stronger the - relationship between the two variables • closer the value to +1.00, stronger the + relationship between the two variables Just what does r value of +0.25 mean?

  27. Factors limiting a PMCC • Homogenous group • subjects very similar on the variables • Unreliable measurement instrument/technique • measurements bounce all over the place) • Nonlinear relationship • Pearson's r is based on linear relationships • Ceiling or Floor with measurement • lots of scores clumped at the top or bottom...therefore no spread which creates a problem similar to the homogeneous group [skewed data set(s)]

  28. Assumptions of the PMCC • Measures are approximately normally distributed • Check with frequency distribution • The variance of the two measures is similar (homoscedasticity) • check with scatterplot • The relationship is linear • check with scatterplot • The sample represents the population • Variables measured on a interval or ratio scale

  29. Not Causation Only Association

  30. Correlations and causality • Correlations only describe the relationship, they do not prove cause and effect • Correlation is a necessary, but not a sufficient condition for determining causality • There are Three Requirements to Infer a Causal Relationship…

  31. Correlations and causality • A statistically significant relationship between the variables • The causal variable occurred prior to the other variable • There are no other factors that could account for the cause • Correlation studies do not meet the last requirement and may not meet the second requirement

  32. Correlations and causality • If there is a relationship between A and B it could be because • A ->B • A<-B • A<-C->B

  33. Smoking & LBP r = 0.45 Low Back Pain Smoking

  34. Smoking & LBP r = 0.45 Low Back Pain Smoking ? Low Back Pain Smoking

  35. Smoking & LBP r = 0.45 Low Back Pain ? Smoking Lifestyle factors ( ie strength)

  36. Interpreting r • r is not a proportion. • r = 0.25 does not mean one quarter similarity between the variables • r = 0.50 does not mean one half similarity between the variables • r describes the co-variability of the variables

  37. Coefficient of Determination • r2 : simply square the r value • What percentage of the variance in each variable is explained by knowledge of the variance of the other variable • what percentage of the variance within Y is predicted by the variance within X?

  38. Coefficient of Determination • (Shared Variation) • Correlation Coefficient Squared • Percentage of the variability among scores on one variable that can be attributed to differences in the scores on the other variable • The coefficient of determination is useful because it gives the proportion of the variance of one variable that is predictable from the other variable

  39. Notes about r2 • Coefficient of determination explains shared variance • therefore 1-r2 is unexplained • r = 0.70 gives about 50% explained variance (why???) • always calculate r2 to evaluate extent of the correlation

  40. Use of Correlation • Reliability of a test/measure • relate test-retest scores • relate tester1 to tester2 • Validity of a test • HR and fitness (aerobic capacity) • Relate multiple dependent variables (do all measure the same construct?)

  41. Cautions concerning r • Appropriate only for linear relationships (use Anxiety&Performance.sav) • Sensitive to range of talent • smaller range, lower r • Sensitive to sampling variation • smaller samples, more unstable • r calculated is not population r

  42. Anxiety & Skill Performance

  43. Meyer et al, 2002 MSSE, 34:7, 1065-1070

  44. Adachi et al, 2002. Mechanoreceptors in the ACL contribute to the joint position sense. Acta Orthop Scand, 73:2:330-334.

  45. Click here for a web site to review correlation concepts introduced in this lecture

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