chapter 8 Relationships Among Variables

Chapter 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

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 Correlation and causation (continued)

Correlation (continued) Pearson product moment correlation, r What the coefficient of correlation means –Reliability (significance) of r –Interpreting the meaningfulness of r (r 2 )

Types of Relationships Between Variables

Using Correlation for Prediction Regression equations Y = a + bX B = r(s Y s X ) A = M Y – bM X Line of best fit Calculating Residual scores Standard error of the estimate: S Y X  S Y 1–r 2

Fitting a Regression Line

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

Procedures for Multiple Regression Multiple regression: correlating more than one predictor with a criterion How to select predictors –Forward selection and backward selection –Maximum R 2 –Stepwise –Hierarchical

Prediction Equations for Multiple Regression Y = a + b 1 X 1 + b 2 X b i X i Problems associated with multiple regression Shrinkage or population specificity Sample size Cross-validation

Reviewing the General Linear Model (GLM) Y = a + bX, this is correlation r = 1X and 1Y, both continuous Y = a + b 1 X 1 + b 2 X b i X i, this is multiple correlation R = 2 or more Xs and 1Y, all continuous

Extending the GLM to Multivariate Case X 1 b 1 + X 2 b X i b i = Y 1 b 1 + Y 2 b Y k b k Relationship among variables, Xs and Ys are continuous Canonical correlation Factor analysis Structural modeling

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

Canonical Correlation What is the relationship between multiple Xs and multiple Ys? Independent variables (predictors) –Arousal –Depression –Mood –Trait anxiety –State anxiety –Attention 6 predictor and 4 criterion variables R c and its test of significance, F Dependent variables (criteria) –Performance –Attitude –Subjectively perceived stress –Anger observed by others

Factor Analysis Can a set of variables be reduced to its underlying constructs? Characteristics important in a sport orientation questionnaire: competitiveness, goals, winning Several measures of each characteristic Administer to group of participants to see whether measures separate into underlying constructs (factors). Correlation is main procedure.

Important Characteristics of Factor Analysis Having a preliminary model Intercorrelations among variables How many factors? Interpreting loading Naming factors

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

Structural Modeling One variable does not always influence another directly. Models explain the complexity of these relationships. Structural modeling is a correlation technique that allows testing of a model. (continued)

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