Chapter 24 Multivariate Statistical Analysis © 2010 South-Western/Cengage Learning. All rights...

Chapter 24Chapter 24Multivariate Multivariate

Statistical AnalysisStatistical 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.

© 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 BABINCARR GRIFFIN

BUSINESS MARKET RESEARCH

EIGHTH EDITION

© 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 OUTCOMESLEARNING OUTCOMESLEARNING OUTCOMESLEARNING OUTCOMES

1.1. Understand what multivariate statistical analysis Understand what multivariate statistical analysis involves and know the two types of multivariate involves and know the two types of multivariate analysisanalysis

2.2. Interpret results from multiple regression analysisInterpret results from multiple regression analysis

3.3. Interpret results from multivariate analysis of variance Interpret results from multivariate analysis of variance (MANOVA)(MANOVA)

4.4. Interpret basic exploratory factor analysis resultsInterpret basic exploratory factor analysis results

5.5. Know what multiple discriminant analysis can be used Know what multiple discriminant analysis can be used to doto do

6.6. Understand how cluster analysis can identify market Understand how cluster analysis can identify market segmentssegments

After studying this chapter, you should be able to


What is Multivariate Data Analysis?What is Multivariate Data Analysis?

• Research that involves three or more variables, Research that involves three or more variables, or that is concerned with underlying dimensions or that is concerned with underlying dimensions among multiple variables, will involve among multiple variables, will involve multivariate statistical analysis.multivariate statistical analysis. Methods analyze multiple variables or even multiple Methods analyze multiple variables or even multiple

sets of variables simultaneously.sets of variables simultaneously. Business problems involve multivariate data analysis:Business problems involve multivariate data analysis:

most employee motivation researchmost employee motivation research customer psychographic profilescustomer psychographic profiles research that seeks to identify viable market segmentsresearch that seeks to identify viable market segments


The “Variate” in MultivariateThe “Variate” in Multivariate

• VariateVariate

A mathematical way in which a set of variables can A mathematical way in which a set of variables can be represented with one equation.be represented with one equation.

A linear combination of variables, each contributing to A linear combination of variables, each contributing to the overall meaning of the variate based upon an the overall meaning of the variate based upon an empirically derived weight.empirically derived weight.

A function of the measured variables involved in an A function of the measured variables involved in an analysis: analysis: VVkk = f (X = f (X11, X, X22, . . . , X, . . . , Xmm ) )


EXHIBIT 24.EXHIBIT 24.11 Which Multivariate Approach Is Appropriate?Which Multivariate Approach Is Appropriate?


Classifying Multivariate TechniquesClassifying Multivariate Techniques

• Dependence TechniquesDependence Techniques Explain or predict one or more dependent variables.Explain or predict one or more dependent variables. Needed when hypotheses involve distinction between Needed when hypotheses involve distinction between

independent and dependent variables.independent and dependent variables. Types:Types:

Multiple regression analysisMultiple regression analysis Multiple discriminant analysisMultiple discriminant analysis Multivariate analysis of varianceMultivariate analysis of variance Structural equations modelingStructural equations modeling


Classifying Multivariate Techniques Classifying Multivariate Techniques (cont’d)(cont’d)• Interdependence TechniquesInterdependence Techniques

Give meaning to a set of variables or seek to group Give meaning to a set of variables or seek to group things together.things together.

Used when researchers examine questions that do Used when researchers examine questions that do not distinguish between independent and dependent not distinguish between independent and dependent variables.variables.

Types:Types: Factor analysisFactor analysis Cluster analysisCluster analysis Multidimensional scalingMultidimensional scaling


Classifying Multivariate Techniques Classifying Multivariate Techniques (cont’d)(cont’d)• Influence of Measurement ScalesInfluence of Measurement Scales

The nature of the measurement scales will determine The nature of the measurement scales will determine which multivariate technique is appropriate for the which multivariate technique is appropriate for the data.data.

Selection of a multivariate technique requires Selection of a multivariate technique requires consideration of the types of measures used for both consideration of the types of measures used for both independent and dependent sets of variables.independent and dependent sets of variables.

Nominal and ordinal scales are Nominal and ordinal scales are nonmetric.nonmetric. Interval and ratio scales are Interval and ratio scales are metricmetric..


EXHIBIT 24.EXHIBIT 24.22 Which Multivariate Dependence Technique Should I Use?Which Multivariate Dependence Technique Should I Use?


EXHIBIT 24.EXHIBIT 24.33 Which Multivariate Interdependence Technique Should I Use?Which Multivariate Interdependence Technique Should I Use?


Analysis of DependenceAnalysis of Dependence

• General Linear Model (GLM)General Linear Model (GLM) A way of explaining and predicting a dependent A way of explaining and predicting a dependent

variable based on fluctuations (variation) from its variable based on fluctuations (variation) from its mean due to changes in independent variables.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.


Interpreting Multiple RegressionInterpreting Multiple Regression

• Multiple Regression AnalysisMultiple Regression Analysis An analysis of association in which the effects of two An analysis of association in which the effects of two

or more independent variables on a single, interval-or more independent variables on a single, interval-scaled dependent variable are investigated scaled dependent variable are investigated simultaneously.simultaneously.

inni eXbXbXbXbbY 3322110

• Dummy variableDummy variable The way a dichotomous (two group) independent The way a dichotomous (two group) independent

variable is represented in regression analysis by variable is represented in regression analysis by assigning a 0 to one group and a 1 to the other.assigning a 0 to one group and a 1 to the other.


Multiple Regression AnalysisMultiple Regression Analysis

• A Simple ExampleA Simple Example Assume that a toy manufacturer wishes to explain Assume that a toy manufacturer wishes to explain

store sales (dependent variable) using a sample of store sales (dependent variable) using a sample of stores from Canada and Europe. stores from Canada and Europe.

Several hypotheses are offered:Several hypotheses are offered: H1:H1: Competitor’s sales Competitor’s sales are related negatively to are related negatively to

sales.sales.

H2:H2: Sales are higher in communities with a Sales are higher in communities with a sales sales office office thanthan

when no sales office is present.when no sales office is present.

H3:H3: Grammar school enrollment Grammar school enrollment in a community is in a community is relatedrelated

positively to sales.positively to sales.


Multiple Regression Analysis (cont’d)Multiple Regression Analysis (cont’d)

• Statistical Results of the Multiple RegressionStatistical Results of the Multiple Regression Regression Equation:Regression Equation:

Coefficient of multiple determination (Coefficient of multiple determination (RR22) = 0.845) = 0.845 FF-value= 14.6, -value= 14.6, p p < 0.05< 0.05

321 7362115387018102 XXXY ....



• Regression Coefficients in Multiple RegressionRegression Coefficients in Multiple Regression Partial correlationPartial correlation

The correlation between two variables after taking into The correlation between two variables after taking into account the fact that they are correlated with other variables account the fact that they are correlated with other variables too.too.

• RR22 in Multiple Regression in Multiple Regression The coefficient of multiple determination in multiple The coefficient of multiple determination in multiple

regression indicates the percentage of variation in regression indicates the percentage of variation in YY explained by explained by allall independent variables. independent variables.



• Statistical Significance in Multiple RegressionStatistical Significance in Multiple Regression FF-test-test

Tests statistical significance by comparing the variation Tests statistical significance by comparing the variation explained by the regression equation to the residual error explained by the regression equation to the residual error variation.variation.

Allows for testing of the relative magnitudes of the sum of Allows for testing of the relative magnitudes of the sum of squares due to the regression (squares due to the regression (SSRSSR) and the error sum of ) and the error sum of squares (squares (SSESSE).).

MSE

MSR

knSSe

kSSrF

1/

/



• Degrees of Freedom (Degrees of Freedom (d.f.d.f.)) kk = number of independent variables = number of independent variables nn = number of observations or respondents = number of observations or respondents

• Calculating Degrees of Freedom (Calculating Degrees of Freedom (d.f.d.f.)) d.f.d.f. for the numerator = for the numerator = kk d.f.d.f. for the denominator = for the denominator = n - kn - k - 1 - 1


FF-test-test

MSE

MSR

knSSe

kSSrF

1/

/


EXHIBIT 24.EXHIBIT 24.44

Interpreting Interpreting Multiple Regression Multiple Regression ResultsResults


ANOVA (n-way) and MANOVAANOVA (n-way) and MANOVA

• Multivariate Analysis of Variance (MANOVA)Multivariate Analysis of Variance (MANOVA) A multivariate technique that predicts multiple A multivariate technique that predicts multiple

continuous dependent variables with multiple continuous dependent variables with multiple categorical independent variables.categorical independent variables.


ANOVA (n-way) and MANOVA (cont’d)ANOVA (n-way) and MANOVA (cont’d)

Interpreting N-way (Univariate) ANOVAInterpreting N-way (Univariate) ANOVA1.1. Examine overall model Examine overall model FF-test result. If significant, -test result. If significant,

proceed.proceed.

2.2. Examine individual Examine individual FF-tests for individual variables.-tests for individual variables.

3.3. For each significant categorical independent For each significant categorical independent variable, interpret the effect by examining the group variable, interpret the effect by examining the group means.means.

4.4. For each significant, continuous covariate, interpret For each significant, continuous covariate, interpret the parameter estimate (the parameter estimate (bb).).

5.5. For each significant interaction, interpret the means For each significant interaction, interpret the means for each combination.for each combination.


Discriminant AnalysisDiscriminant Analysis

• A statistical technique for predicting the probability A statistical technique for predicting the probability that an object will belong in one of two or more that an object will belong in one of two or more mutually exclusive categories (dependent mutually exclusive categories (dependent variable), based on several independent variables.variable), based on several independent variables. To calculate discriminant scores, the linear function To calculate discriminant scores, the linear function

used is:used is:

niniii XbXbXbZ 2211


Discriminant Analysis ExampleDiscriminant Analysis Example

332211 XbXbXbZ

321 0007001300690 XXX ...


EXHIBIT 24.EXHIBIT 24.55 Multivariate Dependence Techniques SummaryMultivariate Dependence Techniques Summary


Factor AnalysisFactor Analysis

• Statistically identifies a reduced number of Statistically identifies a reduced number of factors from a larger number of measured factors from a larger number of measured variables.variables.

• Types:Types: Exploratory factor analysis (EFA)Exploratory factor analysis (EFA)—performed —performed

when the researcher is uncertain about how many when the researcher is uncertain about how many factors may exist among a set of variables.factors may exist among a set of variables.

Confirmatory factor analysis (CFA)Confirmatory factor analysis (CFA)—performed —performed when the researcher has strong theoretical when the researcher has strong theoretical expectations about the factor structure before expectations about the factor structure before performing the analysis.performing the analysis.


EXHIBIT 24.EXHIBIT 24.66 A Simple Illustration of Factor AnalysisA Simple Illustration of Factor Analysis


Factor Analysis (cont’d)Factor Analysis (cont’d)

• How Many FactorsHow Many Factors Eigenvalues are a measure of how much variance is Eigenvalues are a measure of how much variance is

explained by each factor.explained by each factor. Common rule:Common rule:

Base the number of factors on the number of eigenvalues Base the number of factors on the number of eigenvalues greater than 1.0.greater than 1.0.

• Factor LoadingFactor Loading Indicates how strongly a measured variable is Indicates how strongly a measured variable is

correlated with a factor.correlated with a factor.



• Factor RotationFactor Rotation A mathematical way of simplifying factor analysis A mathematical way of simplifying factor analysis

results to better identify which variables “load on” results to better identify which variables “load on” which factors.which factors.

Most common procedure is varimax rotation.Most common procedure is varimax rotation.

• Data Reduction TechniqueData Reduction Technique Approaches that summarize the information from Approaches that summarize the information from

many variables into a reduced set of variates formed many variables into a reduced set of variates formed as linear combinations of measured variables.as linear combinations of measured variables.

The rule of parsimony:The rule of parsimony: an explanation involving an explanation involving fewer components is better than one involving many fewer components is better than one involving many more.more.



• Creating Composite Scales with Factor ResultsCreating Composite Scales with Factor Results When a clear pattern of loadings exists, the When a clear pattern of loadings exists, the

researcher may take a simpler approach by summing researcher may take a simpler approach by summing the variables with high loadings and creating a the variables with high loadings and creating a summated scale.summated scale. Very low loadings suggest a variable does not contribute Very low loadings suggest a variable does not contribute

much to the factor.much to the factor. The reliability of each summated scale is tested by computing The reliability of each summated scale is tested by computing

a coefficient alpha estimate.a coefficient alpha estimate.



• CommunalityCommunality A measure of the percentage of a variable’s variation A measure of the percentage of a variable’s variation

that is explained by the factors.that is explained by the factors. A relatively high communality indicates that a variable A relatively high communality indicates that a variable

has much in common with the other variables taken has much in common with the other variables taken as a group.as a group.

Communality for any variable is equal to the sum of Communality for any variable is equal to the sum of the squared loadings for that variable.the squared loadings for that variable.



• Total Variance ExplainedTotal Variance Explained Squaring and totaling each loading factor; dividing the Squaring and totaling each loading factor; dividing the

total by the number of factors provides an estimate of total by the number of factors provides an estimate of variance in a set of variables explained by a factor.variance in a set of variables explained by a factor. This explanation of variance is much the same as This explanation of variance is much the same as RR22 in in

multiple regression.multiple regression.


Cluster AnalysisCluster Analysis

• Cluster analysisCluster analysis A multivariate approach for grouping observations A multivariate approach for grouping observations

based on similarity among measured variables.based on similarity among measured variables. Cluster analysis is an important tool for identifying market Cluster analysis is an important tool for identifying market

segments. segments.

Cluster analysis classifies individuals or objects into a small Cluster analysis classifies individuals or objects into a small number of mutually exclusive and exhaustive groups.number of mutually exclusive and exhaustive groups.

Objects or individuals are assigned to groups so that there is Objects or individuals are assigned to groups so that there is great similarity within groups and much less similarity great similarity within groups and much less similarity between groups.between groups.

The cluster should have high internal (within-cluster) The cluster should have high internal (within-cluster) homogeneity and external (between-cluster) heterogeneity.homogeneity and external (between-cluster) heterogeneity.


EXHIBIT 24.EXHIBIT 24.77 Clusters of Individuals on Two DimensionsClusters of Individuals on Two Dimensions


EXHIBIT 24.EXHIBIT 24.88 Cluster Analysis of Test-Market CitiesCluster Analysis of Test-Market Cities


Multidimensional ScalingMultidimensional Scaling

• Multidimensional ScalingMultidimensional Scaling Measures objects in multidimensional space on the Measures objects in multidimensional space on the

basis of respondents’ judgments of the similarity of basis of respondents’ judgments of the similarity of objects.objects.


EXHIBIT 24.EXHIBIT 24.99 Perceptual Map of Six Graduate Business Schools: Simple SpacePerceptual Map of Six Graduate Business Schools: Simple Space


EXHIBIT 24.EXHIBIT 24.1010 Summary of Multivariate Techniques for Analysis of Summary of Multivariate Techniques for Analysis of InterdependenceInterdependence

Chapter 24 Multivariate Statistical Analysis © 2010 South-Western/Cengage Learning. All rights...

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