Chapter 9

1. Chapter 9 Analyzing Data Multiple Variables

2. Basic Directions • Review page 180 for basic directions on which way to proceed with your analysis • Provides statistical decision steps based upon the level of measurement for your independent and dependent variables

3. Elaboration ‘Models’ • An association has been found to be statistically significant • Consider controlling for variables that would serve as plausible explanations • Run chi-square or other comparable tests

4. Partialling • When a control variable is introduced, that is deemed first-order partialling • Should you add a second, 2nd order, and so on • The original bivariate relationship is called the zero-order relationship • Good for replicating patterns • Can use minitab stat > tables and put in multiple variables of interest • Don’t use too many – keep it clean

5. Spurious Relationships • If you introduce a third variable (a control) and the relationship that existed in the bivariate setting is now non-significant or even less strong… then, the original relationship is spurious • Consider the ice cream and murder example

6. Specification • Specification: when the control variable leads to only ‘some’ of the values of the test variable to become non-significant or weakened • It is called specification because there is a determination of which relationship holds

7. Suppressing Relationships • If there is no relationship or a very weak one, introduce control variable to see if the ‘weak’ relationship continues • Could be that the variables involved are suppressor variables • Within this structure you can also identify the intervening variables: the one that was keeping the original relationship weak

8. Partial Correlations • When a correlation exists between two variables, X and Y, the correlation may be explained by a third variable that is correlated with both X and Y. • A partial correlation is used to control for the effect of a third variable when examining the correlation between X and Y. • If the correlation between X and Y is reduced, the third variable is responsible for the effect.

9. Two-Way ANOVA • ANOVA can be used for factorial designs: ones that employ more than one IV (or factor). • The factorial design is very popular in the social sciences. The big advantage over single variable designs is that it can provide some unique and relevant information about how variables interact or combine in the effect they have on the DV. • A two way factorial design tells us about two main effects and the interaction.

10. Two-Way ANOVA • The effects • Treatment Effect: a difference in population means • Main Effect: a difference in population means for a factor collapsed over the levels of all other factors in the design • Interaction: occurs when the effect on one factor is not the same at the levels of another • Select: Stat > ANOVA > Two-Way ANOVA

11. Multiple R • Multiple correlation finds the correlation coefficient (r) for every pair of variables •  The multiple correlation coefficient, R, is the correlation coefficient between the observed values of Y and the predicted values of Y. • The value of R will always be positive and will take on a value between zero and one. • The direction of the multivariate relationship between the independent and dependent variables can be observed in the sign, positive or negative, of the regression weights.

12. Multiple R • The interpretation of R is similar to the interpretation of the correlation coefficient, the closer the value of R to one, the greater the linear relationship between the independent variables and the dependent variable.

13. Multiple Regression • Multiple regression finds the linear equation that best predicts the value of one of the variables (the dependent variable) from the others.

14. Multiple Regression • Y = a + bX + cZ + e  • The coefficients (a, b, and c) are chosen so that the sum of squared errors is minimized. • The estimation technique is then called least squares or ordinary least squares (OLS).

15. Multiple Regression • The predictors in a regression equation have no order and one cannot be said to enter before the other.  • Generally in interpreting a regression equation, it makes no scientific sense to speak of the variance due to a given predictor. • Measures of variance depend on the order of entry in step-wise regression and on the correlation between the predictors. • The semi-partial correlation or unique variance has little interpretative utility.

16. Multiple Regression • The standard test of a specified regression coefficient is to determine if the multiple correlation significantly declines when the predictor variable is removed from the equation and the other predictor variables remain. • Test is given by the t or F next to the coefficient.