Chapter 5 Correlation I Introduction to Correlation and Regression

1. Chapter 5 Correlation I Introduction to Correlation and Regression A. Describing the Linear Relationship Between Two Variables, X and Y 1. Pearson product-moment correlation coefficient (r)

2. 2. Bivariate frequency distributions (scatterplots) for various correlation coefficients (r)

4. 3. Upper and lower limits for r: +1 to –1 B. Correlation and Regression Distinguished 1. Characteristics of regression situations One dependent variable, Y, and one or more independent variables, X Levels of independent variables are selected in advance  The value of the dependent variable for a given level of the independent variable is free to vary

5.  The researcher is primarily interested in predicting Y from a knowledge of X 2. Characteristics of correlation situation  Neither variable is considered the independent variable  The researcher is primarily interested in assessing the strength of the relationship between X and Y  X and Y are both free to vary

6. II Correlation A.Formula for Pearson Product-Moment Correlation Coefficient

7. 1. Understanding the formula for r; what the numerator tells you  Covariance  Information in the cross products

10. 2. If the majority of the data points fall in quadrants 1 and 3, the cross product is positive and r > 0 3. If the majority of the data points fall in quadrants 2 and 4, the cross product is negative and r < 0 4. If the data points are equally dispersed over the four quadrants, the cross product equals zero and r = 0 5. The cross product is largest when the data points fall on a straight line 6. The cross product is small when the data points fall in an elongated circle (ellipse)

11. Table 1. Height and Weight of Girl’s Basketball Team 1 7.0 140 .64 289 13.6 2 6.5 130 .09 49 2.1 3 6.5 140 .09 289 5.1 4 6.5 130 .09 49 2.1 5 6.5 120 .09 9 –0.9 6 6.0 120 .04 9 0.6 7 6.0 130 .04 49 –1.4 8 6.0 110 .04 169 2.6 9 5.5 100 .49 529 16.1 10 5.5 110 .49 169 9.1

12. B. Scatterplot for Data in Table 1

13. C. Computation of r for Data in Table 1

14. III Interpretation of the Correlation Coefficient A. Coefficient of Determination, r2 , and Nondetermination, k2

15. B. Visual Representation of r2 and k2

17. IV Common Errors in Interpreting r A. Interpreting r in Direct Proportion to its Size B. Interpreting r in Terms of Arbitrary Labels    

18. 1. Typical reliability coefficients 2. Typical validity coefficients C. Inferring Causation from Correlation V Some Factors That Affect the Correlation Coefficient

19. A. Nature of the Relationship Between X and Y 1. Eta or eta squared can be used to describe the curvilinear relation between X and Y

20. B. Truncated Range

21. C. Subgroups with Different Means or Standard Deviations

23. D. Discontinuous Distribution

24. E. Non-Normal Distributions

25. F. Heterogeneous & Homogeneous Array Variances

26. VI Spearman Rank Correlation (rs) A. Strength of Monotonic Relationship Based On Ranks, RXi and RYi B. Computational Example

27. Table 2. Progress of Patients in Therapy as Ranked by Occupational Therapist, RX, and Physical Therapist, RY 1 5 7 –2 4 2 3 3 0 0 3 1 2 –1 1 4 7 6 1 1 5 4 5 –1 1 6 2 1 1 1 7 8 8 0 0 8 6 4 2 4

28. C. Computation of rs 1. Dealing with tied ranks

29. VII Other Kinds of Correlation Coefficients Coefficient Symbol Characteristics 1. Eta  X and Y quantitative, curvilinear relationship 2. Biserial rb X and Y quantitative, but one variable forced into a dichotomy 3. Cramér’s V X and Y both dichotomous correlation 4. Multiple R All X’s and Y’s quantitative,correlation linear relationships