1. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Chapter 18 Multivariate Statistics

2. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Question Multivariate statistical procedures are increasingly being used in nursing research to untangle complex relationships among how many variables? A.1 or more B.2 or more C.3 or more D.4 or more

3. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Answer C Multivariate statistical procedures are increasingly being used in nursing research to untangle complex relationships among three or more variables.

4. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Multivariate Statistics Statistical procedures for analyzing relationships among three or more variables Most commonly used procedures: –Multiple regression –Analysis of covariance –Multivariate analysis of variance –Factor analysis –Logistic regression

5. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Question Tell whether the following statement is true or false: Multiple regression makes predictions about the values of one variable based on values of a second variable.

6. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Answer False Simple linear regression makes predictions about the values of one variable based on values of a second variable. Multiple regression is a method of predicting a continuous dependent variable on the basis of two or more independent (predictor) variables.

7. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Simple Linear Regression Makes predictions about the values of one variable based on values of a second variable Estimates a straight-line fit to the data that minimizes deviations from the line

8. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Basic Linear Regression Equation Y = a + bX Where Y = predicted value of variable Y (dependent variable) a = intercept constant b = regression coefficient (slope of the line) X = actual value of variable X (independent variable) This equation solves for a and b such that sums of squares of prediction errors are minimized (least squares criterion)

9. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Multiple Linear Regression Used to predict a dependent variable based on two or more independent (predictor) variables Dependent variable is continuous (interval or ratio-level data) Predictor variables are continuous or dichotomous (dummy variables)

10. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Equation for Multiple Regression With Two Predictor Variables Y = a + b 1 X 1 + b 2 X 2 Where Y = predicted value of variable Y (dependent variable) a = intercept constant b 1 = regression coefficient for variable X 1 X 1 = actual value of variable X 1 b 2 = regression coefficient for variable X 2 X 2 = actual value of variable X 2

11. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Multiple Correlation Coefficient (R) Is the correlation index for a dependent variable and two or more independent variables Does not have negative values: it shows strength of relationships, but not direction

12. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Multiple Correlation Coefficient (R) (cont’d) Can be squared (R 2 ) to estimate the proportion of variability in the dependent variable accounted for by the independent variables Cannot be less than the highest bivariate correlation between the dependent variable and an independent variable

13. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Strategies for Handling Predictors in Multiple Regression Simultaneous multiple regression Enters all predictor variables into the regression equation at the same time

14. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Strategies for Handling Predictors in Multiple Regression (cont’d) Hierarchical multiple regression Enters predictors into the equation in a series of steps, controlled by researcher

15. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Strategies for Handling Predictors in Multiple Regression (cont’d) Stepwise multiple regression Enters predictors in a series of empirically determined steps, in the order that produces the greatest increment to R 2

16. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Beta Weights (s) Are standardized regression coefficients (all in same metric) Are sometimes used to estimate the relative importance of independent variables in the regression equation Are used with standard scores (z X ) rather than raw scores (X) –Standard scores transform raw scores to have a mean = 0 and an SD = 1

17. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Analysis of Covariance (ANCOVA) Extends ANOVA by removing the effect of extraneous variables (covariates) before testing whether mean group differences are statistically significant

18. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Analysis of Covariance (cont’d) Levels of measurement of variables: Dependent variable is continuous Independent variable is categorical (group) Covariates are continuous or dichotomous

19. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins MANOVA Multivariate analysis of variance Extension of ANOVA to more than one dependent variable Used to test the significance of differences in group means for multiple dependent variables, considered simultaneously Can be used with covariates: multivariate analysis of covariance (MANCOVA)

20. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Discriminant Analysis Used to predict categorical dependent variables (e.g., compliant/noncompliant) based on two or more predictor variables Accommodates predictors that are continuous or dichotomous Produces an index indicating the proportion of variance in the dependent variable unaccounted for by predictor variables (Wilks’ lambda—)

21. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Canonical Correlation Analyzes the relationship between two or more independent variables and two or more dependent variables Relationships are expressed by the canonical correlation coefficient (R C )

22. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Causal Modeling Tests a hypothesized multivariable causal explanation of a phenomenon Includes: –Path analysis –Linear structural relations analysis (LISREL)

23. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Path Analysis Relies on multiple regression Is applied to a prespecified model based on prior knowledge and theory Tests recursive models—ones in which causation is assumed to be unidirectional Results are often displayed in a path diagram

24. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Path Analysis (cont’d) Distinguishes two types of variable:  Exogenous variable  Endogenous variable Yields path coefficients—weights representing the effect of one variable on another; indicates relative importance of predictors

25. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Structural Equations Modeling (SEM) Another approach to causal modeling Not as many assumptions and restrictions as path analysis Can accommodate measurement errors, nonrecursive models that allow for reciprocal causal paths, and correlated errors

26. Copyright © 2012 Wolters Kluwer Health | Lippincott Williams & Wilkins Phases of SEM 1.Measurement model phase 2.Structural equations modeling phase