Relationships Among Variables Correlation and Regression

1. Relationships Among VariablesCorrelation and Regression KNES 510 Research Methods in Kinesiology

2. Correlation • Correlation is “a statistical technique used to determine the relationship between two or more variables” • We use two different techniques to determine score relationships: • graphing technique • mathematical technique called correlation

3. Graphs of the RelationshipBetween Variables

4. Types of Relationships • The scattergram can indicate a positiverelationship, a negativerelationship, or a zerorelationship • What are the characteristics of positive, negative, and zero relationships?

5. Mathematical Technique: The Correlation Coefficient (r) • The correlation coefficient, r,* represents the relationship between the z-scores of the subjects on two different variables (usually designated X and Y) • This can be stated mathematically as the mean of the z-score products for all subjects *A more complete name for this statistic is Pearson’s product-moment correlation coefficient

6. Formula for the Correlation Coefficient • The correlation coefficient can be calculated as follows:

7. The values of the coefficient will always range from +1.00 to -1.00 • A correlation coefficient near 0.00 indicates no relationship

8. SPSS Bivariate Correlation Output

9. Interpreting the Correlation Coefficient • Because the relationship between two sets of data is seldom perfect, the majority of correlation coefficients are fractions (0.92, -0.80, and the like) • When interpreting correlation coefficients it is sometimes difficult to determine what is high, low, and average

10. The Correlation Coefficient and Cause-and-Effect • There is a high correlation between a person's shoe size and their math skills in grades K through 6 • Is this an example of cause-and-effect? • Can we predict math skill based on shoe size in grade K through 6 students?

11. Coefficient of Determination • The coefficient of determination is the amount of variability in one measure that is explained by the other measure • The coefficient of determination is the square of the correlation coefficient (r2). • For example, if the correlation coefficient between two variables is r = 0.90, the coefficient of determination is (0.90)2 = 0.81

12. Regression • When two variables are related (correlated), it is possible to predict a person’s score on one variable (Y) by knowing their score on the second variable (X)

14. This scatterplot illustrates that there is a strong, positive relationship between fat-free body mass and daily energy expenditure

15. Regression Line (Line of Best Fit) • The regression line is a line that best describes the trend in the data • This line is as close as possible to the data points • The equation for this line is: Y' = bX = C

16. Fitting a Regression Line

17. Simple Prediction • Tests have been developed to predict VO2 max from the time it takes a person to run 1.5 miles • A person's VO2 max can thus be predicted from their 1.5 mile run time because a prediction or regression equation has been developed

18. The simple linear prediction or regression equation takes the following form: Y' = a + bX Y' = predicted value a = intercept of the regression line (Y intercept) b = slope of the regression line (change in Y with each change in X) X = score on the predictor variable

19. Determining Error in Prediction • Unless two variables are perfectly related (-1.00 or +1.00) there will always be error associated with a prediction equation • We find the standard deviation of this error, the standard error of prediction (syx), using the following formula:

20. Prediction and Residuals

21. A predicted score (Y’) ± syx yields a range of scores within which a person’s true score on the predicted variable lies • If the standard error of prediction may be interpreted as the standard deviation of residuals, what are the odds that a person’s true score lies between Y’ ± syx?

22. The standard error of prediction for percent body fat estimated using the skinfold method is ±3.5% • If a person has their percent body fat estimated at 12%, between what two values does their true body fat lie (95% probability)?

23. Which of the following will more precisely predict job performance? A: r = 0.168 B: r = 0.686

24. Sample SPSS Output • Here is the SPSS output for regressing Work Simulation Job Performance (Dependent Variable) against Supervisor Ratings (Independent Variable)

25. This information can be used to create a prediction (regression) equation for predicting work performance of future applicants from supervisor ratings Y’ = – 1.156 + 0.033 X

26. Work Simulation Job Performance may also be predicted from Arm Strength • Here is the SPSS output:

27. This information can be used to create a prediction (regression) equation for predicting work performance of future applicants from supervisor ratings Y’ = – 4.095 + 0.055 X

28. We now have two regression equations for predicting Work Simulation Job Performance • Which is the better equation for accurate prediction? • To determine this, we must examine the standard error of prediction for each equation

29. Standard error of prediction using Supervisor Ratings: • Standard error of prediction using Arm Strength: • Which is the better equation?

30. Multiple Prediction • A prediction formula using a single measure X is usually not very accurate for predicting a person's score on measure Y • Multiple correlation-regression techniques allow us to predict score Y using several X scores

31. The general form of a two predictor multiple regression equation is: Y' = a + b1X1+ b2X2

32. An example of multiple correlation-regression is the prediction of percent body fat from multiple skinfold measurements DB (g/cc) = 1.0994921 - 0.0009929 (3SKF) + 0.0000023 (3SKF)2 – 0.0001392 (age)

33. Next Class • Chapters 9 & 11 • Mock Proposals in class!