1. chapter 8 Relationships Among Variables

2. Outline What correlational research investigates Understanding the nature of correlation What the coefficient of correlation means Using correlation for prediction Partial correlation Uses of semipartial correlation Procedures for multiple regression Multivariate forms of correlation

3. Correlation What correlational research investigates –Relationship between two or more characteristics –Whether two characteristics vary in the same way Understanding the nature of correlation –Correlation coefficient: Quantitative value –Range of correlation: –1.0 to 1.0 Positive correlation:.01 to 1.0 Negative correlation: –.01 to –1.0 * Remember…correlation ≠ causation (continued)

4. Types of Relationships Between Variables

5. Correlation (continued) Pearson product moment correlation (r) –Summarizes both the magnitude and direction of the linear relationship between two variables Assumption: Linear relationship Influences on r –Variables One criterion (dependent) variable One predictor (independent) variable –Measurement reliability –Sample size

6. Correlation (continued) What the coefficient of correlation means –Reliability (significance) of r The confidence in the likelihood of a statistic occurring again if the study were repeated Table A.3, Appendix A –The correlation needed for significance decreases with increased number of participants (df) –A higher correlation is required for significance at the.01 alpha level compared to the.05 alpha level –Interpreting the meaningfulness of r (r 2 ) Coefficient of Determination Usually expressed as percentage of variation

7. Using Correlation for Prediction Prediction is based on correlation –The higher the relationship, the more accurate the prediction Prediction (regression) equation –A formula to predict some criterion based on the relationship between the predictor (independent) variable(s) and the criterion (dependent) variable

8. Using Correlation for Prediction (continued) Regression equations –Y = a + bX –b = r(s Y s X ) –a = M Y – bM X Example: p. 136-137 –Perfect correlation –Correlation of.67

9. Using Correlation for Prediction (continued) Line of best fit –Calculated regression line that results in the smallest sum of squares of the vertical distances of every point from the line Residual scores –Difference between predicted and actual scores that represents the error of prediction Standard error of the estimate –Computation of the standard deviation of all residual scores, which represents the amount of error expected in a prediction

10. Fitting a Regression Line

11. Partial and Semipartial Correlation Partial correlation: –Taking a third variable out of the relationship between two variables Semipartial correlation: –Removing the influence of a third variable on only one of the two variables in a relationship

12. Procedures for Multiple Regression Multiple regression: –Correlating more than one predictor (independent variable) with a criterion (dependent variable) Multiple regression coefficient (R) –Indicates the relationship between the criterion and a weighted sum of the predictor variables R 2 –Represents the amount of the variance of the criterion that is explained or accounted for by the combined predictors

13. Prediction Equations for Multiple Regression Y = a + b 1 X 1 + b 2 X 2 +... + b i X i Problems associated with multiple regression –Shrinkage or population specificity Generalizability –Sample size Direct relationship between the correlation and the ratio of the number of subjects to the number of variables A participant-to-variable ratio of 10:1, or higher, is recommended

14. Reviewing the General Linear Model (GLM) Correlation –Y = a + bX –r: 1X and 1Y, both continuous Multiple Correlation –Y = a + b 1 X 1 + b 2 X 2 +... b i X i, –R: 2 or more Xs and 1Y, all continuous

15. Extending the GLM to Multivariate Case Multivariate Correlation –X 1 b 1 + X 2 b 2 +... + X i b i = Y 1 b 1 + Y 2 b 2 +... + Y k b k Relationship among multiple Xs and Ys, all are continuous

16. Multiple Independent and Dependent Variables With Overlap (GLM) Criterion Measure Dependent Variable Predictor 1 Independent Variable Predictor 2 Independent Variable

17. Canonical Correlation (R c ) What is the relationship between multiple Xs and multiple Ys? Independent variables (predictors) –Arousal –Depression –Mood –Trait anxiety –State anxiety –Attention Form two linear composites Dependent variables (criteria) –Performance –Attitude –Subjectively perceived stress –Anger observed by others

18. Factor Analysis A statistical technique used to reduce a set of data by grouping similar variables into basic components (factors) Performed on data gathered from a sample who have taken a series of measurements Goal is to discover the factors that best explain a group of measurements and describe the relation of each measure to the factor

19. Identifying the Factors CompetitivenessI am a competitive person I try hard to win GoalI set goals in competition Goals help me try hard WinWinning is important I hate to lose Loadings.71.75.58.64.57.69

20. Structural Modeling A correlation technique that allows testing of a model One variable does not always influence another directly Models illustrate the complexity of these relationships –X→Y → Z –X←Y→Z (continued)

21. Structural Modeling (continued) Reprinted with permission from Research Quarterly for Exercise and Sport, Vol. 61, p. 65, Copyright © 1990 by the American Alliance for Health, Physical Education, Recreation and Dance, 1900 Association Drive, Reston, VA 20191

22. Summary Pearson r correlation –Describes relationship between two variables Linear Regression –Used to predict one variable from another Correlation is interpreted for –Significance (reliability) –Meaningfulness (r 2 ) Variance accounted for