Chapter 6Chapter 6

Multivariate Multivariate Analysis of Analysis of VarianceVariance

Chapter 6Chapter 6

Multivariate Multivariate Analysis of Analysis of VarianceVariance

Copyright © Copyright © 20072007Prentice-Hall, Prentice-Hall, Inc.Inc.

LEARNING OBJECTIVES:LEARNING OBJECTIVES:Upon completing this chapter, you should be able to do the Upon completing this chapter, you should be able to do the

following:following:

1.1. Explain the difference between the univariate null Explain the difference between the univariate null hypothesis of ANOVA and the multivariate null hypothesis hypothesis of ANOVA and the multivariate null hypothesis of MANOVA.of MANOVA.

2.2. Discuss the advantages of a multivariate approach to Discuss the advantages of a multivariate approach to significance testing compared to the more traditional significance testing compared to the more traditional univariate approaches.univariate approaches.

3.3. State the assumptions for the use of MANOVA.State the assumptions for the use of MANOVA.

4.4. Discuss the different types of test statistics that are Discuss the different types of test statistics that are available for significance testing in MANOVA.available for significance testing in MANOVA.

5.5. Describe the purpose of post hoc tests in ANOVA and Describe the purpose of post hoc tests in ANOVA and MANOVA.MANOVA.

6.6. Interpret interaction results when more than one Interpret interaction results when more than one independent variable is used in MANOVA.independent variable is used in MANOVA.

7.7. Describe the purpose of multivariate analysis of Describe the purpose of multivariate analysis of covariance (MANCOVA).covariance (MANCOVA).

Chapter 6: Multivariate Analysis of Chapter 6: Multivariate Analysis of VarianceVariance

Chapter 6: Multivariate Analysis of Chapter 6: Multivariate Analysis of VarianceVariance

MANOVAMANOVA . . .. . . is the multivariate is the multivariate extension of the univariate techniques extension of the univariate techniques for assessing the differences between for assessing the differences between group means.group means.

MANOVA DefinedMANOVA Defined

In the univariate case, a single dependent In the univariate case, a single dependent measure is tested for equality across the groups. In measure is tested for equality across the groups. In the multivariate case, a variate is tested for equality. the multivariate case, a variate is tested for equality. In MANOVA, the researcher actually has two variates, In MANOVA, the researcher actually has two variates, one for the dependent variables and another for the one for the dependent variables and another for the independent variables. The dependent variable independent variables. The dependent variable variate is of more interest because the metric-variate is of more interest because the metric-dependent measures can be combined in a linear dependent measures can be combined in a linear combination, as we have already seen in multiple combination, as we have already seen in multiple regression and discriminant analysis. The unique regression and discriminant analysis. The unique aspect of MANOVA is that the variate optimally aspect of MANOVA is that the variate optimally combines the multiple dependent measures into a combines the multiple dependent measures into a single value that maximizes the differences across single value that maximizes the differences across groups. groups.

ANOVA versus MANOVAANOVA versus MANOVA

ANOVA vs. MANOVA ANOVA vs. MANOVA

The relationship between the univariate and The relationship between the univariate and multivariate procedures is shown below:multivariate procedures is shown below:

Number of Dependent Variables

Number of Groups inIndependent Variable

One(Univariate)

Two or More(Multivariate)

Two Groups(Specialized Case)

t-test Hotelling’s T2

Two or More Groups(Generalized Case)

Analysis of Variance(ANOVA)

Multivariate Analysis of Variance(MANOVA)

Rules of Thumb 6–1 Rules of Thumb 6–1

DECISION PROCESSES FOR MANOVADECISION PROCESSES FOR MANOVA• MANOVA is an extension of ANOVA that examines the MANOVA is an extension of ANOVA that examines the

effect of one or more nonmetric independent variables effect of one or more nonmetric independent variables on two or more metric dependent variables.on two or more metric dependent variables.

• In addition to the ability to analyze multiple dependent In addition to the ability to analyze multiple dependent variables, MANOVA also has the advantages of:variables, MANOVA also has the advantages of:

Controlling the experiment-wide error rate when Controlling the experiment-wide error rate when there is some degree of intercorrelation among there is some degree of intercorrelation among dependent variables.dependent variables.Providing more statistical power than ANOVA Providing more statistical power than ANOVA when the number of dependent variables is 5 or when the number of dependent variables is 5 or less.less.

• Nonmetric independent variables create ‘groups’ Nonmetric independent variables create ‘groups’ between which the dependent variables are compared. between which the dependent variables are compared. Many times the groups represent experimental variables Many times the groups represent experimental variables or “treatment effects.”or “treatment effects.”

• Researchers should include only dependent variables Researchers should include only dependent variables that have strong theoretical support.that have strong theoretical support.

MANOVA Decision ProcessMANOVA Decision Process

Stage 1: Objectives of MANOVAStage 1: Objectives of MANOVA

Stage 2: Research Design of MANOVAStage 2: Research Design of MANOVA

Stage 3: Assumptions in Multiple Stage 3: Assumptions in Multiple MANOVAMANOVA

Stage 4: Estimating the MANOVA Model Stage 4: Estimating the MANOVA Model and Assessing Overall Fitand Assessing Overall Fit

Stage 5: Interpreting the Stage 5: Interpreting the MANOVAVariateMANOVAVariate

Stage 6: Validation of the ResultsStage 6: Validation of the Results

Stage 1: Objectives of MANOVAStage 1: Objectives of MANOVA

1.1. To analyze a dependence relationship To analyze a dependence relationship represented as the differences in a set of represented as the differences in a set of dependent measure across a series of groups dependent measure across a series of groups formed by one or more categorical independent formed by one or more categorical independent measures.measures.

2.2. To provide insights into the nature and To provide insights into the nature and predictive power of the independent measures predictive power of the independent measures as well as the interrelationships and differences as well as the interrelationships and differences in the multiple dependent measures.in the multiple dependent measures.

What Can We Do With MANOVA?What Can We Do With MANOVA?

Three types of questions suitable for Three types of questions suitable for MANOVA:MANOVA:

• Multiple Univariate Questions.Multiple Univariate Questions.

• Structured Multivariate Questions.Structured Multivariate Questions.

• Intrinsically Multivariate Intrinsically Multivariate Questions.Questions.

Stage 2: Issues in the Research Design Stage 2: Issues in the Research Design of MANOVAof MANOVA

• Sample Size Requirements – Sample Size Requirements – Overall and by Group.Overall and by Group.

• Factorial Designs – Two or More Factorial Designs – Two or More Treatments.Treatments.

• Selecting Treatments.Selecting Treatments.

Rules of Thumb 6–2 Rules of Thumb 6–2

RESEARCH DESIGN OF MANOVARESEARCH DESIGN OF MANOVA• Cells (groups) are formed by the combination of Cells (groups) are formed by the combination of

independent variables. For example, a three-independent variables. For example, a three-category nonmetric variable (e.g., low, medium, category nonmetric variable (e.g., low, medium, high) combined with a two-category nonmetric high) combined with a two-category nonmetric variable (e.g., gender of male versus female) will variable (e.g., gender of male versus female) will result in a 3 x 2 design with six cells (groups).result in a 3 x 2 design with six cells (groups).

• Sample size per group is a critical design issue:Sample size per group is a critical design issue: Minimum sample size per group must be Minimum sample size per group must be

greater than the number of dependent greater than the number of dependent variables.variables.

The recommended minimum cell size is 20 The recommended minimum cell size is 20 observations per cell (group).observations per cell (group).

Researchers should try to have Researchers should try to have approximately equal sample sizes per cell approximately equal sample sizes per cell (group).(group).

Rules of Thumb 6–2 continued . . . Rules of Thumb 6–2 continued . . .

RESEARCH DESIGN OF MANOVARESEARCH DESIGN OF MANOVA• Covariates and blocking variables are effective Covariates and blocking variables are effective

ways of controlling for external influences on the ways of controlling for external influences on the dependent variables that are not directly dependent variables that are not directly represented in the independent variables:represented in the independent variables:

An effective covariate is one that is highly An effective covariate is one that is highly correlated with the dependent variable(s) correlated with the dependent variable(s) but not correlated with the independent but not correlated with the independent variables.variables.

The maximum number of covariates in a The maximum number of covariates in a model should be (.10 x Sample Size) – model should be (.10 x Sample Size) – (Number of Groups – 1).(Number of Groups – 1).

Objectives of Covariance AnalysisObjectives of Covariance Analysis

The objective of the covariate is to eliminate The objective of the covariate is to eliminate any effects that:any effects that:

1.1. affect only a portion of the respondents, affect only a portion of the respondents, or or

2.2. vary among the respondents. vary among the respondents.

Similar to the use of a blocking factor, Similar to the use of a blocking factor, covariates can achieve two specific covariates can achieve two specific purposes:purposes:

1.1. eliminate some systematic error outside eliminate some systematic error outside the control of the researcher that can bias the control of the researcher that can bias the results, andthe results, and

2.2. account for differences in the responses account for differences in the responses due to unique characteristics of the due to unique characteristics of the respondents. respondents.

Stage 3: MANOVA Stage 3: MANOVA AssumptionsAssumptions

• IndependenceIndependence

• Equality of variance – covariance Equality of variance – covariance matricesmatrices

• NormalityNormality

• Linearity and multicollinearity Linearity and multicollinearity among the dependent variablesamong the dependent variables

• Sensitivity to outliersSensitivity to outliers

Rules of Thumb 6–3 Rules of Thumb 6–3

MANOVA/ANOVA ASSUMPTIONSMANOVA/ANOVA ASSUMPTIONS• For the multivariate test procedures used with MANOVA For the multivariate test procedures used with MANOVA

to be valid:to be valid: Observations must be independent.Observations must be independent. Variance-covariance matrices must be equal (or Variance-covariance matrices must be equal (or

comparable) for all treatment groups.comparable) for all treatment groups. The dependent variables must have a multivariate The dependent variables must have a multivariate

normal distribution.normal distribution.• Multivariate normality is assumed, but many times hard Multivariate normality is assumed, but many times hard

to assess. Univariate normality does not guarantee to assess. Univariate normality does not guarantee multivariate normality, but if all variables meet the multivariate normality, but if all variables meet the univariate normality requirement then departures from univariate normality requirement then departures from multivariate normality are inconsequential.multivariate normality are inconsequential.

• ANOVA F-tests are generally robust if violations of these ANOVA F-tests are generally robust if violations of these assumptions are modest.assumptions are modest.

Why Use MANOVA?Why Use MANOVA?

• Control experimental error Control experimental error rate.rate.

• Test for difference between Test for difference between multiple dependent variables.multiple dependent variables.

Stage 4: Estimation of the MANOVA Stage 4: Estimation of the MANOVA Model and Assessing Overall FitModel and Assessing Overall Fit

• Selecting criteria for significance Selecting criteria for significance tests.tests.

• Assessing statistical power.Assessing statistical power.

Rules of Thumb 6–4 Rules of Thumb 6–4

MANOVA ESTIMATIONMANOVA ESTIMATION• The four most widely used measures for assessing statistical The four most widely used measures for assessing statistical

significance between groups on the independent variables are:significance between groups on the independent variables are: Roy’s Greatest Characteristic RootRoy’s Greatest Characteristic Root Wilk’s LambdaWilk’s Lambda Pillai’s Criterion Pillai’s Criterion Hotelling’s Trace Hotelling’s Trace

• In most situations the results/conclusions will be the same In most situations the results/conclusions will be the same across all four measures, but in some unique instances results across all four measures, but in some unique instances results will differ between the measures.will differ between the measures.

• Maintaining adequate statistical power is critical:Maintaining adequate statistical power is critical: Power in the .80 range for the selected alpha level is Power in the .80 range for the selected alpha level is

acceptable.acceptable. When the effect size is small, the researcher should use larger When the effect size is small, the researcher should use larger

sample sizes per group to maintain acceptable levels of sample sizes per group to maintain acceptable levels of statistical power.statistical power.

• The General Linear Model (GLM) is widely used in testing The General Linear Model (GLM) is widely used in testing ANOVA or MANOVA modelsANOVA or MANOVA models. GLM is available on most statistical . GLM is available on most statistical packages like SPSS and SAS.packages like SPSS and SAS.

Stage 5: Interpretation of the Stage 5: Interpretation of the MANOVA ResultsMANOVA Results

• Evaluating CovariatesEvaluating Covariates

• Assessing the Dependent Assessing the Dependent VariateVariate

• Identifying Differences Identifying Differences between Individual Groupsbetween Individual Groups

Identifying Differences Between Individual Identifying Differences Between Individual GroupsGroups

• Post Hoc MethodsPost Hoc Methods

• A Priori or Planned A Priori or Planned ComparisonsComparisons

Post Hoc MethodsPost Hoc Methods

• Scheffe Scheffe

• Tukey’s honestly significant Tukey’s honestly significant difference (HSD)difference (HSD)

• Tukey’s extension of the Fisher Tukey’s extension of the Fisher least significant difference (LSD)least significant difference (LSD)

• Duncan’s multiple-range testDuncan’s multiple-range test

• Newman-Kuels testNewman-Kuels test

Rules of Thumb 6–5Rules of Thumb 6–5

INTERPRETING AND VALIDATING MANOVA RESULTSINTERPRETING AND VALIDATING MANOVA RESULTS• When covariates are involved in a GLM model:When covariates are involved in a GLM model:

Analyze the model both with and without the covariates. Analyze the model both with and without the covariates. If the covariates do not improve the statistical power or If the covariates do not improve the statistical power or

have no effect on the significance of the treatment have no effect on the significance of the treatment effects, then they can be dropped from the final analysis.effects, then they can be dropped from the final analysis.

• Any time two or more independent variables (treatments) are Any time two or more independent variables (treatments) are included in the analysis, interactions must be examined included in the analysis, interactions must be examined before drawing conclusions about main effects for any before drawing conclusions about main effects for any independent variable:independent variable: If the interactions are not statistically significant, then If the interactions are not statistically significant, then

main effects can be interpreted directly since the main effects can be interpreted directly since the difference between treatments is considered constant difference between treatments is considered constant across combinations of levels.across combinations of levels.

If the interaction is statistically significant and the If the interaction is statistically significant and the differences are not constant across combinations of differences are not constant across combinations of levels, then the interaction must be determined to be levels, then the interaction must be determined to be ordinal or disordinal:ordinal or disordinal:

Rules of Thumb 6–5 Rules of Thumb 6–5 Continued . . .Continued . . .

o Ordinal interactions mean that the direction of differences do Ordinal interactions mean that the direction of differences do not vary by level (e.g., males always less than females) even not vary by level (e.g., males always less than females) even though the difference between males/females varies by level on though the difference between males/females varies by level on the other treatment. In this case, the size of the main effect the other treatment. In this case, the size of the main effect (e.g., males versus females) should only be described (e.g., males versus females) should only be described separately for each level of the other treatment.separately for each level of the other treatment.

o Significant disordinal interactions occur when the direction of Significant disordinal interactions occur when the direction of an observed main effect changes with the level of another an observed main effect changes with the level of another treatment (e.g., males greater than females for one level and treatment (e.g., males greater than females for one level and less than females for another level). Disordinal interactions less than females for another level). Disordinal interactions interfere with the interpretation of main effects.interfere with the interpretation of main effects.

• When the independent variable has more than two groups, two When the independent variable has more than two groups, two types of procedures can be used to isolate the source of types of procedures can be used to isolate the source of differences:differences:

Post-hoc tests examine potential statistical differences among all Post-hoc tests examine potential statistical differences among all possible combinations of group means. Post-hoc tests have limited possible combinations of group means. Post-hoc tests have limited power and thus are best suited to identify large effectspower and thus are best suited to identify large effects..

Planned comparisons are appropriate when a priori theoretical Planned comparisons are appropriate when a priori theoretical reasons suggest that certain groups will differ from another group reasons suggest that certain groups will differ from another group or other groups. Type I error is inflated as the number of planned or other groups. Type I error is inflated as the number of planned comparisons increases.comparisons increases.

Stage 6: Validation of the Stage 6: Validation of the ResultsResults

• Replication.Replication.

• Use of covariates?Use of covariates?

• Assessing causation?Assessing causation?

Variable DescriptionVariable Description Variable TypeVariable TypeData Warehouse Classification VariablesData Warehouse Classification VariablesX1X1 Customer TypeCustomer Type nonmetric nonmetric X2X2 Industry TypeIndustry Type nonmetric nonmetric X3X3 Firm SizeFirm Size nonmetric nonmetric X4X4 RegionRegion nonmetricnonmetricX5X5 Distribution SystemDistribution System nonmetricnonmetricPerformance Perceptions VariablesPerformance Perceptions VariablesX6X6 Product QualityProduct Quality metricmetricX7X7 E-Commerce Activities/WebsiteE-Commerce Activities/Website metricmetricX8X8 Technical SupportTechnical Support metricmetricX9X9 Complaint ResolutionComplaint Resolution metricmetricX10X10 Advertising Advertising metricmetricX11X11 Product LineProduct Line metricmetricX12X12 Salesforce ImageSalesforce Image metricmetricX13X13 Competitive PricingCompetitive Pricing metricmetricX14X14 Warranty & ClaimsWarranty & Claims metricmetricX15X15 New ProductsNew Products metricmetricX16X16 Ordering & BillingOrdering & Billing metricmetricX17X17 Price FlexibilityPrice Flexibility metricmetricX18X18 Delivery SpeedDelivery Speed metricmetricOutcome/Relationship MeasuresOutcome/Relationship MeasuresX19X19 SatisfactionSatisfaction metric metric X20X20 Likelihood of RecommendationLikelihood of Recommendation metric metric X21X21 Likelihood of Future PurchaseLikelihood of Future Purchase metric metric X22X22 Current Purchase/Usage LevelCurrent Purchase/Usage Level metric metric X23X23 Consider Strategic Alliance/Partnership in FutureConsider Strategic Alliance/Partnership in Future nonmetricnonmetric

Description of HBAT Primary Database VariablesDescription of HBAT Primary Database Variables