© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Multivariate Methods I: Factor, Cluster, and Discriminant Analyses CHAPTER 10 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 1. What are the benefits of using Multivariate methods in analyzing data?

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Benefits of Multivariate Methods  Marketing problems are rarely, if ever, completely described by one or two variables  Quick and convenient  Improved understanding of statistical concepts by researchers and managers  Researchers take the time to educate themselves fully about the ins and outs of standard data analysis methodologies.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 2. Differentiate between interdependence and dependence methods of multivariate analyses.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Uses of Multivariate Methods  To understand marketing outcomes, we need to consider multiple variables.  dependence methods – one or more variables are designated as being predicted by (dependent on) a set of independent variables  interdependence methods – the interrelationship among all the variables taken together are studied  Factor analysis tells us which VARIABLES are similar to one another and how they should be grouped.  Cluster analysis tells us which CASES (PEOPLE or OBJECTS) are similar and how they should be grouped.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 3. Differentiate between Factor analysis and Cluster analysis.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Factor vs. Cluster Analysis  Factor analysis reduces the number of variables  Cluster analysis reduces the number of cases.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 4. Define Factor analysis.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Factor Analysis  extracts a small set of factors from a larger number of intercorrelated variables  aids in data “reduction”  helps identify underlying structure of data  finds optimal weights (loadings) for scaling data  transforms the independent variables so the resulting factors are uncorrelated  provides quantities better suited to be inputs to a regression  is used to:  develop personality scales  determine how to combine geodemographic and psychographic variables to best explain behavior patterns  identify critical product attributes and similarities

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 6. What are the possible applications of factor analysis?

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Factor Analysis  aids in data “reduction”  helps identify underlying structure of data  finds optimal weights (loadings) for scaling data  transforms the independent variables so the resulting factors are uncorrelated  provides quantities better suited to be inputs to a regression  develop personality scales  determine how to combine geodemographic and psychographic variables to best explain behavior patterns  identify critical product attributes and similarities

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 7. What are the steps in factor analysis?

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1. establish correlations between variables of interest 2. extract principal components, linear combinations of the variables in the correlation matrix that explain as much variance as possible z = b 1 X 1 + b 2 X 2 + …+ b m X m  eigenvalue: the size of a factor relative to the average factor. Factors with larger eigenvalues are kept and those with lower eigenvalues are discarded. The sum-of-squared factor loadings on all variables is the eigenvalue for that factor.  communality: the proportion of the variation in each variable that is explained by a particular group of factors. The sum-of-squared loadings across the set of factors in a solution is the communality for that variable. Steps in Factor Analysis

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Steps in Factor Analysis (continued)  scree plot: graph of factors vs. variance explained that helps identify how many factors are needed  factor loadings: correlations between original variables and factors derived from them. A large value (near -1 or 1) means the factor “loads high” on that variable. 3. rotate the original solution to yield new factors that correlate well with some variables and poorly with others. This aids interpretation.  orthogonal rotation preserves the factors as uncorrelated varimax rotation reorients the factors so that their loadings are as near -1, 0, or 1 as possible  oblique rotation allows the factors to intercorrelate

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Steps in Factor Analysis (continued) FIGURE 10.5 Scree Plot of Eigenvalues for Factors

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 8. Define Cluster analysis.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Cluster Analysis and Latent Class Analysis  groups cases (objects, items, people) into clusters  latent clusters must be discerned through analysis  seeks intermediate level of aggregation between placing all objects in a single group and placing each object in its own group  can be used to:  develop consumer segments based upon profiles  identify suitable test market locations  determine similar markets in various countries  aid media selection by finding similar groups of subscribers

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Running the Cluster Analysis  clustering routine standardizes the data and uses the chosen distance metric to calculate distances (dissimilarities) between each pair of items  variables must be made scale-independent (may be re- weighted afterward to assign relative importance)  transform the data by creating z-scores (for each variable, subtracting its sample mean and dividing by its sample standard deviation)  linearly transform each variable to have range 0 to 1, or -1 to 1 by dividing each value by the maximum, the mean, or the standard deviation

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Running the Cluster Analysis (continued)  choose distance metric for calculating distance between each pair of items  the Euclidean Distance  Squared Euclidean Distance – sum-of-squared deviations  Pearson correlation – correlation between two items' coordinates  Cosine – cosine of the angle between two items' coordinates  Chebyshev distance – maximum absolute difference between two items’ coordinates  City Block – sum of the absolute differences between two items’ coordinates

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Hierarchical vs. Nonhierarchical Clustering  Hierarchical clustering maintains objects in the same cluster as new, larger clusters are formed  important when structural (tree-like) breakdown of a set of objects is needed  useful for comparing different clustering solutions  Non-hierarchical (k-means) clustering allows basis of clustering to shift and objects to switch clusters  works well for fixed or hypothesized number of clusters  better at handling very large numbers of cases  Choosing the best number of clusters for a given situation involves balancing the project’s need for accuracy of classification against simplicity.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interpreting a Hierarchical Cluster Analysis  agglomeration schedule: analysis output that shows the order in which the clusters join with one another  normalized centroid distance (NCD) : the radius of a circle/sphere put around each point. It gradually increases from having each case in its own cluster to having one cluster of all cases  dendrogram: a graph of a hierarchical clustering solutions that aids visualization of the clustering structure in order to decide how many clusters to actually use

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interpreting a Hierarchical Cluster Analysis (continued) FIGURE 10.8 Hierarchical Cluster Analysis of Beer Brands

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interpreting a Hierarchical Cluster Analysis (continued) Linear discriminant functions (dashed lines) discriminate the clusters using simple linear functions of the variables. FIGURE Calories (Y) vs, Price(X) for Four Clusters

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Q. 9. Define Discriminant Analyses (DA).

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Discriminant Analysis Discriminant analysis (DA) is appropriate for analyzing nominal dependent variable and interval independent variables. It is used:  to estimate discriminant functions that maximally distinguish categories of nominal dependent variables  to determine whether and how groups differ  to predict which categories a new customer might fit based on observed data (e.g., demographics)  to test profiles as predictors of target behavior  to identify the most useful predictors in distinguishing groups

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Discriminant Analysis (continued)  discriminant function (DF): a linear combination of variables: DF = v 1 X 1 + v 2 X 2 + … + v m X m  coefficients derived to maximize in order to make the mean scores across categories of the dependent variable as different as possible  output includes “confusion matrix” that categorizes correct and incorrect predictions by cross-tabulating the dependent variable category that the discriminant function predicts a subject will be in with the category that the subject is actually in.

© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Analysis of Variance (ANOVA) Analysis of Variance (ANOVA) is a collection of methods that test whether a set of means are the same.  specialized form of regression applicable when analyzing one intervally scaled dependent variable and one or more nominally scaled independent variable  online appendix with detailed description of common and useful forms of ANOVA is available at ModernMarketingResearch.com