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Chap019
- 1. Chapter 19Chapter 19 Multivariate Analysis:Multivariate Analysis: An OverviewAn Overview McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.
- 2. Learning ObjectivesLearning Objectives Understand . . . • How to classify and select multivariate techniques. • That multiple regression predicts a metric dependent variable from a set of metric independent variables. • That discriminant analysis classifies people or objects into categorical groups using several metric predictors.
- 3. Learning ObjectivesLearning Objectives Understand . . . • How multivariate analysis of variance assesses the relationship between two or more metric dependent variables and independent classificatory variables. • How structural equation modeling explains causality among constructs that cannot be directly measured.
- 4. Learning ObjectivesLearning Objectives Understand . . . • How conjoint analysis assists researchers to discover the most importance attributes and the levels of desirable features. • How principal components analysis extracts uncorrelated factors from an initial set of variables and exploratory factor analysis reduces the number of variables to discover the underlying constructs.
- 5. Learning ObjectivesLearning Objectives Understand . . . • The use of cluster analysis techniques for grouping similar objects or people. • How perceptions of products or services are revealed numerically and geometrically by multidimensional scaling.
- 6. Wonder and CuriosityWonder and Curiosity “Wonder, connected with a principle of rational curiosity, is the source of all knowledge and discovery . . . but wonder which ends in wonder, and is satisfied with wonder, is the quality of an idiot.” Samuel Horsley English scientist and fellow Royal Society
- 7. PulsePoint:PulsePoint: Research RevelationResearch Revelation 60 The percent of workers on four continents who trust their organization’s senior leaders.
- 12. Right Questions.Right Questions. Trusted Insight.Trusted Insight. When using sophisticated techniques you want to rely on the knowledge of the researcher. Harris Interactive promises you can trust their experienced research professionals to draw the right conclusions from the collected data.
- 13. Dependency TechniquesDependency Techniques Multiple RegressionMultiple Regression Discriminant AnalysisDiscriminant Analysis MANOVAMANOVA Structural Equation Modeling (SEM)Structural Equation Modeling (SEM) Conjoint AnalysisConjoint Analysis
- 14. Uses of Multiple RegressionUses of Multiple Regression Develop self-weighting estimating equation to predict values for a DV Control for confounding Variables Test and explain causal theories
- 16. Multiple Regression ExampleMultiple Regression Example
- 18. Evaluating and Dealing withEvaluating and Dealing with MulticollinearityMulticollinearity Choose one of the variables and delete the other Create a new variable that is a composite of the others Collinearity Statistics VIF 1.000 2.289 2.289 2.748 3.025 3.067
- 19. Discriminant AnalysisDiscriminant Analysis Predicted Success Actual Group Number of Cases 0 1 Unsuccessful Successful 0 1 15 15 13 86.70% 3 20.00% 2 13.30% 12 80.00% Note: Percent of “grouped” cases correctly classified: 83.33% Unstandardized Standardized X1 X1 X1 Constant .36084 2.61192 .53028 12.89685 .65927 .57958 .97505 A. B.
- 20. MANOVAMANOVA
- 24. Multivariate Tests ofMultivariate Tests of SignificanceSignificance
- 25. Univariate Tests of SignificanceUnivariate Tests of Significance
- 26. Structural EquationStructural Equation Modeling (SEM)Modeling (SEM) Model SpecificationModel Specification EstimationEstimation Evaluation of FitEvaluation of Fit Respecification of the ModelRespecification of the Model Interpretation and CommunicationInterpretation and Communication
- 27. Structural EquationStructural Equation Modeling (SEM)Modeling (SEM)
- 28. Concept Cards for ConjointConcept Cards for Conjoint Sunglasses StudySunglasses Study
- 29. Conjoint AnalysisConjoint Analysis Brand Bolle Hobbies Oakley Ski Optiks Style* A B C A B C A A Flotation Yes No Yes Yes Yes Price $100 $72 $60 $40 $100 $72 $60 $40 $100 $72 $60 $40 $100 $72 $60 $40 * A = multiple color choices for frames, lenses, and temples. B = multiple color choices for frames, lenses, and straps (no hard temples. C = limited colors for frames, lenses, and temples.
- 30. Conjoint ResultsConjoint Results Participant 8 in SunglassesParticipant 8 in Sunglasses StudyStudy
- 31. Conjoint Results forConjoint Results for Sunglasses StudySunglasses Study
- 32. Interdependency TechniquesInterdependency Techniques Factor AnalysisFactor Analysis Cluster AnalysisCluster Analysis Multidimensional ScalingMultidimensional Scaling
- 34. Factor MatricesFactor Matrices A Unrotated Factors B Rotated Factors Variable I II h2 I II A B C D E F Eigenvalue Percent of variance Cumulative percent 0.70 0.60 0.60 0.50 0.60 0.60 2.18 36.3 36.3 -.40 -.50 -.35 0.50 0.50 0.60 1.39 23.2 59.5 0.65 0.61 0.48 0.50 0.61 0.72 0.79 0.75 0.68 0.06 0.13 0.07 0.15 0.03 0.10 0.70 0.77 0.85
- 35. Orthogonal Factor RotationsOrthogonal Factor Rotations
- 36. Correlation Coefficients, MetroCorrelation Coefficients, Metro U MBA StudyU MBA Study Variable Course V1 V2 V3 V10 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 Financial Accounting Managerial Accounting Finance Marketing Human Behavior Organization Design Production Probability Statistical Inference Quantitative Analysis 1.00 0.56 0.17 -.14 -.19 -.21 -.44 0.30 -.05 -.01 0.56 1.00 -.22 0.05 -.26 -.00 -.11 0.06 0.06 0.06 .017 -.22 1.00 -.48 -.05 -.56 -.04 0.07 -.32 0.42 -.01 0.06 0.42 -.10 -.23 -.05 -.08 -.10 0.06 1.00
- 37. Factor Matrix, Metro UFactor Matrix, Metro U MBA StudyMBA Study Variable Course Factor 1 Factor 2 Factor 3 Communality V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 Eigenvalue Percent of variance Cumulative percent Financial Accounting Managerial Accounting Finance Marketing Human Behavior Organization Design Production Probability Statistical Inference Quantitative Analysis 0.41 0.01 0.89 -.60 0.02 -.43 -.11 0.25 -.43 0.25 1.83 18.30 18.30 0.71 0.53 -.17 0.21 -.24 -.09 -.58 0.25 0.43 0.04 1.52 15.20 33.50 0.23 -.16 0.37 0.30 -.22 -.36 -.03 -.31 0.50 0.35 0.95 9.50 43.00 0.73 0.31 0.95 0.49 0.11 0.32 0.35 0.22 0.62 0.19
- 38. Varimax Rotated Factor MatrixVarimax Rotated Factor Matrix Variable Course Factor 1 Factor 2 Factor 3 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 Financial Accounting Managerial Accounting Finance Marketing Human Behavior Organization Design Production Probability Statistical Inference Quantitative Analysis 0.84 0.53 -.01 -.11 -.13 -.08 -.54 0.41 0.07 -.02 0.16 -.10 0.90 -.24 -.14 -.56 -.11 -.02 0.02 0.42 -.06 0.14 -.37 0.65 -.27 -.02 -.22 -.24 0.79 0.09
- 39. Cluster AnalysisCluster Analysis Select sample to clusterSelect sample to cluster Define variablesDefine variables Compute similaritiesCompute similarities Select mutually exclusive clustersSelect mutually exclusive clusters Compare and validate clusterCompare and validate cluster
- 41. Cluster MembershipCluster Membership ________Number of Clusters ________ Film Country Genre Case 5 4 3 2 Cyrano de Bergerac Il y a des Jours Nikita Les Noces de Papier Leningrad Cowboys . . . Storia de Ragazzi . . . Conte de Printemps Tatie Danielle Crimes and Misdem . . . Driving Miss Daisy La Voce della Luna Che Hora E Attache-Moi White Hunter Black . . . Music Box Dead Poets Society La Fille aux All . . . Alexandrie, Encore . . . Dreams France France France Canada Finland Italy France France USA USA Italy Italy Spain USA USA USA Finland Egypt Japan DramaCom DramaCom DramaCom DramaCom Comedy Comedy Comedy Comedy DramaCom DramaCom DramaCom DramaCom DramaCom PsyDrama PsyDrama PsyDrama PsyDrama DramaCom DramaCom 1 4 5 6 19 13 2 3 7 9 12 14 15 10 8 11 18 16 17 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 1 1 1 1 2 2 2 2 3 3 3 3 3 4 4 4 4 3 3 1 1 1 1 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
- 43. Similarities Matrix of 16Similarities Matrix of 16 RestaurantsRestaurants
- 44. Positioning of SelectedPositioning of Selected RestaurantsRestaurants
- 45. Key TermsKey Terms • Average linkage method • Backward elimination • Beta weights • Centroid • Cluster analysis • Collinearity • Communality • Confirmatory factor analysis • Conjoint analysis • Dependency techniques • Discriminant analysis • Dummy variable • Eigenvalue • Factor analysis
- 46. Key Terms (cont.)Key Terms (cont.) • Factors • Forward selection • Holdout sample • Interdependency techniques • Loadings • Metric measures • Multicollinearity • Multidimensional scaling (MDS) • Multiple regression • Multivariate analysis • Multivaria analysis of variance (MANOVA) • Nonmetric measures • Path analysis
- 47. Key Terms (cont.)Key Terms (cont.) • Path diagram • Principal components analysis • Rotation • Specification error • Standardized coefficients • Stepwise selection • Stress index • Structural equation modeling • Utility score
Editor's Notes
- Multivariate techniques are those statistical techniques which focus upon and bring out bold relief the structure of simultaneous relationships among three or more phenomena.
- See the text Instructors Manual (downloadable from the text website) for ideas for using this research-generated statistic.
- If criterion and predictor variables exist in the research question, we will have an assumption of dependence. Multiple regression, multivariate analysis of variance, and discriminant analysis are techniques where criterion or dependent variables and predictor or independent variables are present. If the variables are interrelated without designating some as dependent and others independent, then interdependence of the variables is assumed. Factor analysis, cluster analysis, and multidimensional scaling are example of interdependency techniques.
- Exhibit 19-1 In the flowchart shown in Exhibit 19-1, this section displays the appropriate techniques for dependency of both metric and nonmetric variables. Metric measures are those using interval or ratio measures while nonmetric use ordinal or nominal measures. From the chart, we can see that if the research question includes dependent and independent variables, and the dependent variables are nonmetric, we should use conjoint analysis. However, if the dependent variables are metric, we can use MANOVA, canonical analysis, or structural equations modeling depending on whether the independent variables are metric or nonmetric.
- Exhibit 19-1 (cont.)
- Exhibit 19-1 (cont.) Multidimensional scaling develops a perceptual map of the locations of some objects relative to others. This map specifies how the objects differ. Cluster analysis identifies homogeneous subgroups or cluster. Factor analysis looks for patterns among the variables to discover if an underlying combination of the original variables can summarize the original set.
- This is a list of the dependency techniques discussed on the following slides.
- Multiple regression is a statistical tool used to develop a self-weighting estimating equation that predicts values for a dependent variable from the values of independent variables. It is used as a descriptive tool in three types of situations. These are listed in the slide. First, it is often used to develop a self-weighting estimating equation that predicts values for a DV from the values for several IVs. For instance, one could try to predict company sales on the basis of new housing starts, new marriage rates, annual disposable income, and a time factor. Second, a descriptive application of multiple regression calls for controlling for confounding variables to better evaluate the contribution of other variables. For instance, one could control the brand of a product and the store in which it is bought to study the effects of price as an indicator of product quality. A third use is test and explain causal theories. In this approach, often referred to as path analysis, regression is used to describe an entire structure of linkages that have been advanced from a causal theory.
- Multiple regression is an extension of the bivariate linear regression presented in Chapter 19. Although dummy variables (nominal variables coded 0,1) may be used, all other variables must be interval or ratio. The generalized equation is shown in the slide. The regression coefficients are stated either in raw score units (the actual X values) or as standardized coefficients (X values stated in terms of their standard scores). In either case, the value of the regression coefficient states the amount that Y varies with each unit change of the associated X variable when the effects of all other X variables are being held constant. When the regression coefficients are standardized, they are called beta weights and their values indicate the relative importance of the associated X values. An example is provided on the next slide.
- Exhibit 19-2 In this example, multiple regression is used to evaluate the key drivers of customer usage (DV) for hybrid mail for an e-business called SuperLetter. Super Letter provides a service that enables users to create a document on any PC and send it in a secure, encrypted mode over the Internet to a distant international terminal near the addressee, where it will be printed, processed, and delivered via a local postal service. The independent variables are customer perceptions of 1) cost/speed valuation, 2) security, and 3) reliability. The output from SPSS is provided in the slide. In model 1, cost/speed is entered. In model 2, security is added. In model 3, reliability is added. Looking at the R2 column, you can see that the cost/speed variable explains 77% of customer usage. The adjusted R2 for model 3 is .871. R2 is adjusted to reflect the model’s goodness of fit for the population. The standard error of model 3 is .4937. This is the standard deviation of actual values of Y about the estimated Y values. Analysis of variance measures whether or not the equation represents a set of regression coefficients that are statistically significant from zero. Unstandardized regression coefficients for all three models are shown in the lower table in the column headed B. The equation can be constructed as Y= -.093 + .448X1 + .315X2 + 254X3. Standardized regression coefficients are shown in the column labeled Beta. When these are used, the regression Y intercept is zero. Standard error is a measure of the sampling variability of each regression coefficient.
- Most statistical packages provide various methods for selecting variables for the equation. The equation can be built with all variables or specific combinations, or you can select a method that sequentially adds or removes variables. Forward selection starts with the constant and adds variables that result in the largest R2 increase. Backward elimination begins with a model containing all independent variables and removes the variable that changes R2 the least. Stepwise selection is the most popular method and it combines forward and backward sequential approaches. The independent variable that contributes the most to explaining the dependent variable is added first. Subsequent variables are included based on their incremental contribution over the first and on whether they meet the criterion for entering the equation (significance level). Variables may be removed at each step if they meet the removal criterion. Pictured: Probability Plot from Exhibit 18-4
- Collinearity exists when two independent variables are highly correlated. Multicollinearity exists when more than two independent variables are highly correlated. If these conditions exist, the effects on multiple regression are damaging. The estimated regression coefficients may fluctuate widely. Correlations of .80 or greater should be dealt with using the techniques listed in the slide. Exhibit 19-2 contains a column called Collinearity Statistics that shows a variable inflation factor (VIF) index. This column is shown in the slide. The VIF is a measure of the effect of the other independent variables on a regression coefficient. Large values, usually 10.0 or more, suggest collinearity or multicollinearity. With the hybrid-mail example, multicollinearity is not a problem.
- Exhibit 19-3 Discriminant analysis joins a nominally scaled criterion or dependent variable with one or more independent variables that are interval- or ratio-scaled. Once the discriminant equation is found, it can be used to predict the classification of a new observation. The is done by calculating a linear function of the form Use formula from page 578 Where Di is the score on discriminant function i. The dis are weighting coefficients; do is a constant. The Xs are the values of the discriminating variables used in the analysis. A single discriminant equation is required if the categorization calls for two groups. If three groups are involved, two discriminant equations are required. The most common use is to classify persons or objects into various groups or to analyze known groups to determine the relative importance of specific factors. Exhibit 19-3 provides an example based on the selection process for employees at KDL. The independent variables are years of prior work experience, GPA, and employment test scores. An algorithm determines how well these variables classify employees who have been judged as successful and unsuccessful. Exhibit 19-3 shows the standardized and unstandardized discriminant function coefficients which indicate that employment test scores are the most power in discriminating group membership. Using the discriminant equation, D =.659X1 + .580X2 = .975X3, you can now predict whether future candidates are likely to be successful at KDL.
- Exhibit 19-4 Multivariate analysis of variance, MANOVA, is a commonly used multivariate technique. MANOVA assesses the relationship between two or more dependent variables and classificatory variables or factors. It is similar to ANOVA but with the added ability to handle several dependent variables simultaneously. It uses special matrices to test for differences among groups. The variance between groups is determined by partitioning the total SSCP matrix and testing for significance. The F ratio, generalized to a ratio of the within-group variance and total-group variance matrices, tests for equality among treatment groups. MANOVA examines similarities and differences among the multivariate mean scores of several populations. The null hypothesis is that all of the centroids are equal. Exhibit 19-4 shows graphically three populations whose centroids are unequal, allowing the research to reject the null. A centroid refers to the multivariate mean scores in MANOVA.
- Exhibit 19-5 In this example, the manager for CalAudio, a manufacturer of MP3 players, is concerned about brand loyalty. He fears that the quality of players is affecting repurchase decisions. Engineering takes a sample of 15 MP3 players made with the old manufacturing method and 15 made with a new method. The players are measured for adherence to product specifications and are stress-tested to determine time before failure. Exhibit 19-5 shows the mean and standard deviation of the dependent variables for the old method (1) and the new method (2). The new method extended the time before failure to 181 weeks, and improved adherence to specifications from 90 to 95. However, it slowed manufacturing speed by almost 30 minutes. Exhibit 19-6 continues the analysis and is shown on the next slide.
- Exhibit 19-7 When MANOVA is applied properly, the dependent variables are correlated If the dependent variables are unrelated, there would be no necessity for a multivariate test and we could simply use separate F tests for the dependent variables. Bartlett’s test of sphericity helps us decide if we should continue with MANOVA or return to separate univariate tests. In Exhibit 19-7, shown in the slide, we look for a determinant value that is close to 0. This implies that one or more dependent variables is a linear function of another. The determinant has a chi-square transformation that simplifies testing for statistical significance. Since the observed significance is below that set for the model, we are able to reject the null hypothesis and conclude that there are dependencies among the dependent variables. Next, we test the equality of means that considers the dependent variables for the two manufacturing methods in our example. This test is analogous to a t-test or an F test for multivariate data. This is continued on the next slide.
- Exhibit 19-6 contains homogeneity-of-variance tests for separate dependent variables and a multivariate test. The former are known as univariate tests. The multivariate test is a comparable version that tests the variables simultaneously to determine whether MANOVA should proceed. The significance levels of Cochran’s C and Bartlett-Box F do not allow us to reject any of the tests for dependent variables considered separately. This means the two methods have equal variances in each dependent variable. This fulfills the univariate assumptions for homogeneity of variance. We then consider the variances and covariances simultaneously with Box’s M. Again, we are unable to reject the homegeneity-of-variance assumption regarding the matrices. This satisfies the multivariate assumptions.
- Exhibit 19-8 The sums-of-squares and cross-products matrices are used to test equality of means. Exhibit 19-8, shown in the slide, shows three tests. All the tests provided are compared to the F distribution for interpretation. Since the observed significance level is less than .05 for the Hotelling T2 test, we reject the null hypothesis that said methods 1 and 2 provide equal results with respect to the dependent variables. Similar results are shown from the Pillai trace and Wilk’s statistic. The final component of MANOVA is discussed on the next slide.
- Exhibit 19-9 To detect where the differences lie, we can examine the results of univariate F tests in Exhibit 19-9. Since there are only two methods examined in this example, the F is equivalent to a t2 for a two-sample t-test. The significance levels for these tests do not reflect that several comparisons are being made. There are statistically significant differences in all three dependent variables resulting from the new manufacturing method.
- Structural equation modeling (SEM) implies a structure for the covariances between observed variables, and accordingly it is sometimes called covariance structure modeling. More commonly, researchers refer to SEM as LISREL models, the name of the first and most widely cited SEM computer program. SEM has advantages. Multiple and interrelated relationships can be estimated simultaneously and it can represent observed concepts, latent variables, in these relationships and account for measurement error in the estimation process. This section provides a broad conceptual introduction to SEM. Researchers using SEM must follow 5 basic steps, which are shown in the slide. Step one is specification, or formal statement of the model’s parameters. These parameters are specified as fixed or free. Fixed parameters have values set by the researcher and are not estimated from the data. When there is a hypothesized, but unknown relation between the variables, the parameters are set free to be estimated from the data. Researchers must be careful to avoid specification error, a bias that overestimates the importance of the variables included in the model. After the model has been specified, the researcher must obtain estimates of the free parameters from the observed data. This is often accomplished using an iterative method such as maximum likelihood estimation (MLE). Following convergence, the researcher must evaluate goodness-of-fit criteria which indicate whether the model should be rejected. If the model is not rejected, the researcher will continue the analysis and interpret the path coefficients in the model. Model respecification follows when there is poor fit. Finally, the hypotheses and results are presented in the form of a path diagram.
- Exhibit 19-10 A research consultant, hired by MindWriter, is investigating the relationship between customer satisfaction and service quality and the degree to which customer satisfaction and service quality predict customer purchase intention. The research proposed three possible relationship (models) among the variables. Following the basic steps for SEM, the researcher finds that none of the competing models has a satisfactory fit. Model 2 is most promising in that it has the lowest chi-square value and the highest value for the adjusted-goodness-of-fit index. After examining the second model’s residual matrices and modification index, the researcher finds that the model could achieve a better fit if relation between service quality and purchase intention were not fixed. The researcher then respecifies the model to free the parameters. This results in an acceptable goodness-of-fit. The results suggest that good service quality leads to customer satisfaction and that both variables have a direct effect on purchase intention. In the exhibit, the direction of the single-pointed arrows from service quality and satisfaction to purchase intention denotes that purchase intention is a dependent variable in its relation to both service quality and satisfaction. However, while satisfaction is independent to purchase intention, it is dependent in its relation to service quality.
- Exhibit 19-11 The cards shown in Exhibit 19-11 were administered to a small sample of 10 subjects who were asked to order their cards from most to least desirable. The data produced the results presented in Exhibits 19-12 and 19-13 shown on the next slide.
- Conjoint analysis typically uses input from nonparametric independent variables. Its objective is to secure utility scores (also called partworths) that represent the importance of each aspect of a product or service in the subjects’ overall preference ratings. Utility scores are computed from the subjects’ rankings or ratings of a set of cards. Each card in the deck describes one possible configuration of combined product attributes. The first step is to select the attributes most pertinent to the purchase decision. This may require an exploratory study or can be done through expert interviews. The attributes selected are the independent variables, called factors. The possible values for an attribute are called factor levels. After selecting the factors and their levels, a computer program determines the number of product descriptions necessary to estimate the utilities. SPSS procedures build a file structure for all possible combinations, generate the subset required for testing, produce the card descriptions, and analyze the results. In the example started in the slide, four factors relevant to the choice of sun protection products for watersports enthusiasts are considered. This is a 4 (brand) * 3 (style) * 2 (flotation) * 4 (price) design resulting in a 96-option study. The algorithm selected 16 cards to estimate the utilities for the full concept. Four holdout cards were administered but evaluated separately. The cards are shown in Exhibit 19-11.
- Exhibit 19-12 Exhibit 19-12 contains the results of the eighth participant’s preferences. The individual is an avid windsurfer and flotation was the most important attribute for her, followed by style, price, and brand. From her preferences, we can compute her maximum utility score. The prediction that responders would prefer less expensive prices did not hold for the eighth respondent as revealed by the asterisk next to the price factor in the exhibit. She reversed herself on price to get flotation. The results for the sample are presented in Exhibit 19-13 on the next slide.
- Exhibit 19-13 Exhibit 19-13 shows that the sample placed price first in importance, followed by flotation, style, and brand. Group utilities may be calculated just as for the individual. Pearson’s r and Kendall’s tau measure the relationship between observed and estimated preferences. Here we can see that this relationship is significant. Conjoint analysis is an effective tool used by researchers to match preferences to known characteristics of market segments and design or target a product accordingly.
- This is a list of the interdependency techniques discussed on the following slides.
- Exhibit 19-14 Factor analysis is a general term for several specific computational techniques. All have the objective of reducing to a manageable number many variables that belong together and have overlapping measurement characteristics. It begins with the construction of a new set of variables bad on the relationships in the correlation matrix. The most frequently used approach is principal components analysis. This method transforms a set of variables into a new set of composite variables that are not correlated with each other. These linear combinations of variables, called factors, account for the variance in the data as a whole. The best combination makes up the first principal component and so on. This process continues until all the variance is accounted for. The output might look like the exhibit shown above.
- Exhibit 19-15 The values in Exhibit 19-13 are correlation coefficients between the factor and the variables (.70 is the correlation coefficient between variable A and factor I). These coefficients are called loadings. Eigenvalues are the sum of the variances of the factor values. When divided by the number of variables, an eigenvalue yields an estimate of the amount of total variance explained by the factor. For example, factor I accounts for 36% of the total variance. If a factor has a low eigenvalue, it adds little to the explanation of variance and is discarded. The column headed “h2” gives the communalities, or estimates of the variance in each variable that is explained by the two factors. For variable A, this column indicates that 65% of the variance in Variable A are explained with the two factors. In this case, the unrotated factor loadings are not informative because the variables are not loading clearly on one factor or another. We will attempt to get a more clear loading using rotation. You can see from the column of rotated factor loadings that each variable has now clearly loaded on either factor I or factor II. Rotation is discussed further in the following slide.
- Exhibit 19-16 When factors are intentionally rotated to result in no correlation between the factors in the final solution, the software is called orthogonal. When the factors are not manipulated to be zero correlation but may reveal the degree of correlation that exists naturally, it is called oblique. The variables in Exhibit 19-16 can be plotted in two-dimensional space. Two axes divide the space and the points are positioned relative to these axes. The location of the axes is arbitrary and represent only one of an infinite number of reference frames that could be used to reproduce the matrix. As long as one does not change the intersection points and keeps the axes at right angles, one can use the orthogonal method to rotate the axes and find a better position for the reference axes. “Better” means a matrix that makes the factors as pure as possible in terms of the factor loadings. An example is shown on the next slide.
- Exhibit 19-17 The chair of Metro U’s MBA program is interested in determining if distinct types of people are involved in the study. Using a sample of 21 grade reports, he calculates a correlation matrix between the grades of all courses, factor analyzes the matrix, and rotates the matrix. Exhibit 19-17 shows part of the correlation matrix. The extraction of components is shown in Exhibit 19-18 on the next slide.
- Exhibit 19-18 While there are 10 courses in the analysis, only three factors appear. Recall that communalities indicate the amount of variance in each variable that is explained by the factors. Thus, these three factors account for about 73% of the variance in grades in the financial accounting course. The eigenvalue row is a measure of the explanatory power of each factor. The three factors account for about 43% of the total variance in course grades. To further clarify the factors, a varimax (orthogonal) rotation is used to secure the matrix shown on the next slide
- Exhibit 19-19 The rotation does clarify the relationship among course grades. We could interpret the results as showing three kinds of students, the accounting, finance, and marketing types.
- Cluster analysis is a set of techniques for grouping similar objects or people. It is useful for segmentation research. It starts with a undifferentiated group of people, events, or objects and attempts to reorganize them into homogeneous subgroups. The five steps basic to most cluster studies are listed in the slide. An example is shown on the following slide.
- Exhibit 19-20 The example shown above shows a cluster analysis of individuals based on three dimensions: age, income, and family size. A full example is illustrated on the following two slides.
- Exhibit 19-21 The object of this cluster is to classify 19 films into homogeneous subgroups. The production companies were American, Canadian, French, Italian, Spanish, Finnish, Egyptian, and Japanese. Three genres were represented: comedy, dramatic comedy, and psychological drama. The exhibit includes the film name, country of origin, and genre. It also lists the clusters for each using the average linking method. Exhibit 19-22 shows the study’s dendogram on the next slide.
- Exhibit 19-22 This exhibit shows that the clustering procedure begins with 19 films and continues until all the films are in an undifferentiated group. The solid vertical line shows the point at which the clustering solution best represents the data. This determination was guided by coefficients provided by SPSS for each stage of the procedure. Five clusters explain this data set. The first cluster shown in the exhibit has three French-language films and one Canadian film, all of which are dramatic comedies. Cluster 2 consists of comedy films. Cluster 3 is composed of dramatic comedies from American and Italian film companies. Later, a Spanish film is added to this cluster. Cluster 4 includes three American psychological dramas and a Finnish film. Cluster 5 includes two very different dramatic comedies. This cluster analysis classified productions based on reviewers’ ratings.
- Exhibit 19-23 Multidimensional scaling (MDS) creates a special description of a respondent’s perception about a product, service, or other object of interest on a perceptual map. Items perceived to be similar will fall close together on the perceptual map. There are three types of attribute space, each representing a multidimensional map. There is objective space in which an object can be positioned in terms of its measurable attributes. There is subjective space were perceptions of the object may be positioned. With a third map, we can describe respondents’ preferences using the object’s attributes. This represents their ideal. MDS is illustrated with a study of 16 restaurants. We created a metric algorithm measuring the similarities among the 16 restaurants by asking patrons questions on a 5-point metric scale about different dimensions of service quality and price. The matrix of similarities is shown in the slide. Higher numbers reflect the items that are more dissimilar. A computer program is used to analyze the data matrix and generate a perceptual map. This is shown in Exhibit 19-24 on the next slide.
- Exhibit 19-24 The objective is to find a multidimensional spatial pattern that best reproduces the original order of the data. The computer program presents these relationships as a geometric configuration so that all distances between pairs of points closely correspond to the original matrix. Determining how many dimensions to use is complex. The more dimensions in space, the more likely the result will closely match the input data. The goal, though, is to secure the structure that provides a good fit with the fewest dimensions. MDS is best understood using two or at most three dimensions. In the restaurant example, we conclude that two dimensions represent an acceptable geometric configuration.