1. Correlation and Regression Quantitative Methods in HPELS 440:210

2. Agenda Introduction The Pearson Correlation Hypothesis Tests with the Pearson Correlation Regression Instat Nonparametric versions

3. Introduction Correlation: Statistical technique used to measure and describe a relationship between two variables Direction of relationship:  Positive  Negative Form of relationship:  Linear  Quadratic... Degree of relationship:  -1.0  0.0  +1.0

7. Uses of Correlations Prediction Validity Reliability

8. Agenda Introduction The Pearson Correlation Hypothesis Tests with the Pearson Correlation Regression Instat Nonparametric versions

9. The Pearson Correlation Statistical Notation  Recall for ANOVA:  r = Pearson correlation  SP = sum of products of deviations  M x = mean of x scores  SS x = sum of squares of x scores

10. Pearson Correlation Formula Considerations  Recall for ANOVA:  SP =  (X – M x )(Y – M y ) SP =  XY –  X  Y / n  SS x =  (X – M x ) 2  SS y =  (Y – M y ) 2  r = SP / √SS x SS y

11. Pearson Correlation Step 1: Calculate SP Step 2: Calculate SS for X and Y values Step 3: Calcuate r

12. Step 1  SP SP =  (X – M x )(Y – M y ) SP = (-6*-1)+(4*1)+(-2*-1)+(2*0)+(2*1) SP = 6 + 4 + 2 + 0 + 2 SP = 14 SP =  XY –  X  Y / n SP = 74 – [30(100)]/5 SP = 74 - 60 SP = 14  X=30  Y=10  XY = (0*1)+(10*3)+(4*1)+(8*2)+(8*3)  XY = 0 + 30 + 4 + 16 + 24  XY = 74

13. Step 2  SS x and SS y

14. Step 3  r r = SP / √SS x SS y r = 14 / √(64)(4) r = 14 / √256 r = 14/16 r = 0.875

15. Interpretation of r Correlation ≠ causality Restricted range  If data does not represent the full range of scores – be wary Outliers can have a dramatic effect  Figure 16.9 Correlation and variability  Coefficient of determination (r 2 )

16. Agenda Introduction The Pearson Correlation Hypothesis Tests with the Pearson Correlation Regression Instat Nonparametric versions

17. The Process Step 1: State hypotheses  Non directional: H 0 : ρ = 0 (no population correlation) H 1 : ρ ≠ 0 (population correlation exists)  Directional: H 0 : ρ ≤ 0 (no positive population correlation) H 1 : ρ < 0 (positive population correlation exists) Step 2: Set criteria   = 0.05 Step 3: Collect data and calculate statistic rr Step 4: Make decision  Accept or reject

18. Example Researchers are interested in determining if leg strength is related to jumping ability Researchers measure leg strength with 1RM squat (lbs) and vertical jump height (inches) in 5 subjects (n = 5)

19. Step 1: State Hypotheses Non-Directional H 0 : ρ = 0 H 1 : ρ ≠ 0 Step 2: Set Criteria Alpha (  ) = 0.05 Critical Value: Use Critical Values for Pearson Correlation Table Appendix B.6 (p 697) Information Needed: df = n - 2 Alpha (a) = 0.05 Directional or non-directional? Critical value = 0.878 0.878

20. Step 3: Collect Data and Calculate Statistic Data: XYXY 200255000 180223960 225276075 300278100 160254000 106512627135  Calculate SP SP =  XY –  X  Y / n SP = 27135 – [1065(126)]/5 SP = 27135 - 26838 SP = 297 Calculate SS x XX-M x (X-M x ) 2 200-13169 180-331089 22512144 300877569 160-532809 213 M 11780 

21. Calculate SS y YY-M y (Y-M y ) 2 25-0.20.04 22-3.210.24 271.83.24 271.83.24 25-0.20.04 25.2 M 16.8  XX-M x (X-M x ) 2 200-13169 180-331089 22512144 300877569 160-532809 213 M 11780  r = SP / √SS x SS y r = 297 / √11780(16.8) r = 297 / √197904 r = 297 / 444.86 r = 0.667 Step 3: Collect Data and Calculate Statistic Calculate r Step 4: Make Decision 0.667 < 0.878 Accept or reject?

22. Agenda Introduction The Pearson Correlation Hypothesis Tests with the Pearson Correlation Regression Instat Nonparametric versions

23. Regression Recall  Several uses of correlation:  Prediction  Validity  Reliability Regression attempts to predict one variable based on information about the other variable Line of best fit

24. Regression Line of best fit can be described with the following linear equation  Y = bX + a where:  Y = predicted Y value  b = slope of line  X = any X value  a = intercept

25. Y = bX + a, where: Y = cost (?) b = cost per hour ($5) X = number of hours (?) a = membership cost ($25) Y = 5X + 25 Y = 5(10) + 25 Y = 50 + 25 = 75 Y = 5X + 25 Y = 5(30) + 25 Y = 150 + 25 = 175 5 25

26. Line of best fit minimizes distances of points from line

27. Calculation of the Regression Line Regression line = line of best fit = linear equation SP =  (X – M x )(Y – M y ) SS x =  (X – M x ) 2 b = SP / SS x a = M y - bM x

28. Example 16.14, p 557 SP =  (X – M x )(Y – M y ) SP = 16 SS x =  (X – M x ) 2 SP = 10 b = SP / SS x b = 16 / 10 = 1.6 a = M y - bM x a = 6 – 1.6(5) = -2 M x =5M y =6 Y = bX + a Y = 1.6(X) - 2

30. Agenda Introduction The Pearson Correlation Hypothesis Tests with the Pearson Correlation Regression Instat Nonparametric versions

31. Instat - Correlation Type data from sample into a column.  Label column appropriately. Choose “Manage” Choose “Column Properties” Choose “Name” Choose “Statistics”  Choose “Regression” Choose “Correlation”

32. Instat – Correlation Choose the appropriate variables to be correlated Click OK Interpret the p-value

33. Instat – Regression Type data from sample into a column.  Label column appropriately. Choose “Manage” Choose “Column Properties” Choose “Name” Choose “Statistics”  Choose “Regression” Choose “Simple”

34. Instat – Regression Choose appropriate variables for:  Response (Y)  Explanatory (X) Check “significance test” Check “ANOVA table” Check “Plots” Click OK Interpret p-value

35. Reporting Correlation Results Information to include:  Value of the r statistic  Sample size  p-value Examples:  A correlation of the data revealed that strength and jumping ability were not significantly related (r = 0.667, n = 5, p > 0.05) Correlation matrices are used when interrelationships of several variables are tested (Table 1, p 541)

36. Agenda Introduction The Pearson Correlation Hypothesis Tests with the Pearson Correlation Regression Instat Nonparametric versions

37. Nonparametric Versions Spearman rho  when at least one of the data sets is ordinal Point biserial correlation  when one set of data is ratio/interval and the other is dichotomous  Male vs. female  Success vs. failure Phi coefficient  when both data sets are dichotomous

38. Violation of Assumptions Nonparametric Version  Friedman Test (Not covered) When to use the Friedman Test:  Related-samples design with three or more groups  Scale of measurement assumption violation: Ordinal data  Normality assumption violation: Regardless of scale of measurement

39. Textbook Assignment Problems: 5, 7, 10, 23 (with post hoc)