1. Chapter 24 Multivariate Statistical Analysis © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ZIKMUND BABIN CARR GRIFFIN BUSINESS MARKET RESEARCH EIGHTH EDITION

2. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–2 LEARNING OUTCOMES 1.Understand what multivariate statistical analysis involves and know the two types of multivariate analysis 2.Interpret results from multiple regression analysis 3.Interpret results from multivariate analysis of variance (MANOVA) 4.Interpret basic exploratory factor analysis results 5.Know what multiple discriminant analysis can be used to do 6.Understand how cluster analysis can identify market segments After studying this chapter, you should be able to

3. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–3 What is Multivariate Data Analysis? Research that involves three or more variables, or that is concerned with underlying dimensions among multiple variables, will involve multivariate statistical analysis.Research that involves three or more variables, or that is concerned with underlying dimensions among multiple variables, will involve multivariate statistical analysis.  Methods analyze multiple variables or even multiple sets of variables simultaneously.  Business problems involve multivariate data analysis:  most employee motivation research  customer psychographic profiles  research that seeks to identify viable market segments

4. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–4 The “Variate” in Multivariate VariateVariate  A mathematical way in which a set of variables can be represented with one equation.  A linear combination of variables, each contributing to the overall meaning of the variate based upon an empirically derived weight.  A function of the measured variables involved in an analysis: V k = f (X 1, X 2,..., X m )

5. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–5 EXHIBIT 24.1 Which Multivariate Approach Is Appropriate?

6. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–6 Classifying Multivariate Techniques Dependence TechniquesDependence Techniques  Explain or predict one or more dependent variables.  Needed when hypotheses involve distinction between independent and dependent variables.  Types:  Multiple regression analysis  Multiple discriminant analysis  Multivariate analysis of variance  Structural equations modeling

7. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–7 Classifying Multivariate Techniques (cont’d) Interdependence TechniquesInterdependence Techniques  Give meaning to a set of variables or seek to group things together.  Used when researchers examine questions that do not distinguish between independent and dependent variables.  Types:  Factor analysis  Cluster analysis  Multidimensional scaling

8. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–8 Classifying Multivariate Techniques (cont’d) Influence of Measurement ScalesInfluence of Measurement Scales  The nature of the measurement scales will determine which multivariate technique is appropriate for the data.  Selection of a multivariate technique requires consideration of the types of measures used for both independent and dependent sets of variables.  Nominal and ordinal scales are nonmetric.  Interval and ratio scales are metric.

9. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–9 EXHIBIT 24.2 Which Multivariate Dependence Technique Should I Use?

10. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–10 EXHIBIT 24.3 Which Multivariate Interdependence Technique Should I Use?

11. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–11 Analysis of Dependence General Linear Model (GLM)General Linear Model (GLM)  A way of explaining and predicting a dependent variable based on fluctuations (variation) from its mean due to changes in independent variables. μ =a constant (overall mean of the dependent variable) ∆X and ∆F =changes due to main effect independent variables (experimental variables) and blocking independent variables (covariates or grouping variables) ∆ XF =represents the change due to the combination (interaction effect) of those variables.

12. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–12 Interpreting Multiple Regression Multiple Regression AnalysisMultiple Regression Analysis  An analysis of association in which the effects of two or more independent variables on a single, interval- scaled dependent variable are investigated simultaneously. Dummy variableDummy variable  The way a dichotomous (two group) independent variable is represented in regression analysis by assigning a 0 to one group and a 1 to the other.

13. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–13 Multiple Regression Analysis A Simple ExampleA Simple Example  Assume that a toy manufacturer wishes to explain store sales (dependent variable) using a sample of stores from Canada and Europe.  Several hypotheses are offered:  H1:Competitor’s sales are related negatively to sales.  H2:Sales are higher in communities with a sales office than when no sales office is present.  H3:Grammar school enrollment in a community is related positively to sales.

14. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–14 Multiple Regression Analysis (cont’d) Statistical Results of the Multiple RegressionStatistical Results of the Multiple Regression  Regression Equation:  Coefficient of multiple determination ( R 2 ) = 0.845  F -value= 14.6, p < 0.05

15. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–15 Multiple Regression Analysis (cont’d) Regression Coefficients in Multiple RegressionRegression Coefficients in Multiple Regression  Partial correlation  The correlation between two variables after taking into account the fact that they are correlated with other variables too. R 2 in Multiple RegressionR 2 in Multiple Regression  The coefficient of multiple determination in multiple regression indicates the percentage of variation in Y explained by all independent variables.

16. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–16 Multiple Regression Analysis (cont’d) Statistical Significance in Multiple RegressionStatistical Significance in Multiple Regression  F -test  Tests statistical significance by comparing the variation explained by the regression equation to the residual error variation.  Allows for testing of the relative magnitudes of the sum of squares due to the regression (SSR) and the error sum of squares (SSE).

17. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–17 Multiple Regression Analysis (cont’d) Degrees of Freedom ( d.f. )Degrees of Freedom ( d.f. )  k = number of independent variables  n = number of observations or respondents Calculating Degrees of Freedom ( d.f. )Calculating Degrees of Freedom ( d.f. )  d.f. for the numerator = k  d.f. for the denominator = n - k - 1

18. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–18 F -test

19. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–19 EXHIBIT 24.4 Interpreting Multiple Regression Results

20. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–20 ANOVA (n-way) and MANOVA Multivariate Analysis of Variance (MANOVA)Multivariate Analysis of Variance (MANOVA)  A multivariate technique that predicts multiple continuous dependent variables with multiple categorical independent variables.

21. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–21 ANOVA (n-way) and MANOVA (cont’d) Interpreting N-way (Univariate) ANOVA 1. Examine overall model F -test result. If significant, proceed. 2. Examine individual F-tests for individual variables. 3. For each significant categorical independent variable, interpret the effect by examining the group means. 4. For each significant, continuous covariate, interpret the parameter estimate (b). 5. For each significant interaction, interpret the means for each combination.

22. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–22 Discriminant Analysis A statistical technique for predicting the probability that an object will belong in one of two or more mutually exclusive categories (dependent variable), based on several independent variables.A statistical technique for predicting the probability that an object will belong in one of two or more mutually exclusive categories (dependent variable), based on several independent variables.  To calculate discriminant scores, the linear function used is:

23. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–23 Discriminant Analysis Example

24. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–24 EXHIBIT 24.5 Multivariate Dependence Techniques Summary

25. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–25 Factor Analysis Statistically identifies a reduced number of factors from a larger number of measured variables.Statistically identifies a reduced number of factors from a larger number of measured variables. Types:Types:  Exploratory factor analysis (EFA)—performed when the researcher is uncertain about how many factors may exist among a set of variables.  Confirmatory factor analysis (CFA)—performed when the researcher has strong theoretical expectations about the factor structure before performing the analysis.

26. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–26 EXHIBIT 24.6 A Simple Illustration of Factor Analysis

27. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–27 Factor Analysis (cont’d) How Many FactorsHow Many Factors  Eigenvalues are a measure of how much variance is explained by each factor.  Common rule:  Base the number of factors on the number of eigenvalues greater than 1.0. Factor LoadingFactor Loading  Indicates how strongly a measured variable is correlated with a factor.

28. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–28 Factor Analysis (cont’d) Factor RotationFactor Rotation  A mathematical way of simplifying factor analysis results to better identify which variables “load on” which factors.  Most common procedure is varimax rotation. Data Reduction TechniqueData Reduction Technique  Approaches that summarize the information from many variables into a reduced set of variates formed as linear combinations of measured variables.  The rule of parsimony: an explanation involving fewer components is better than one involving many more.

29. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–29 Factor Analysis (cont’d) Creating Composite Scales with Factor ResultsCreating Composite Scales with Factor Results  When a clear pattern of loadings exists, the researcher may take a simpler approach by summing the variables with high loadings and creating a summated scale.  Very low loadings suggest a variable does not contribute much to the factor.  The reliability of each summated scale is tested by computing a coefficient alpha estimate.

30. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–30 Factor Analysis (cont’d) CommunalityCommunality  A measure of the percentage of a variable’s variation that is explained by the factors.  A relatively high communality indicates that a variable has much in common with the other variables taken as a group.  Communality for any variable is equal to the sum of the squared loadings for that variable.

31. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–31 Factor Analysis (cont’d) Total Variance ExplainedTotal Variance Explained  Squaring and totaling each loading factor; dividing the total by the number of factors provides an estimate of variance in a set of variables explained by a factor.  This explanation of variance is much the same as R 2 in multiple regression.

32. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–32 Cluster Analysis Cluster analysisCluster analysis  A multivariate approach for grouping observations based on similarity among measured variables.  Cluster analysis is an important tool for identifying market segments.  Cluster analysis classifies individuals or objects into a small number of mutually exclusive and exhaustive groups.  Objects or individuals are assigned to groups so that there is great similarity within groups and much less similarity between groups.  The cluster should have high internal (within-cluster) homogeneity and external (between-cluster) heterogeneity.

33. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–33 EXHIBIT 24.7 Clusters of Individuals on Two Dimensions

34. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–34 EXHIBIT 24.8 Cluster Analysis of Test-Market Cities

35. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–35 Multidimensional Scaling Multidimensional ScalingMultidimensional Scaling  Measures objects in multidimensional space on the basis of respondents’ judgments of the similarity of objects.

36. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–36 EXHIBIT 24.9 Perceptual Map of Six Graduate Business Schools: Simple Space

37. © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.24–37 EXHIBIT 24.10 Summary of Multivariate Techniques for Analysis of Interdependence