Chapter 7 Multivariate Analysis of Variance Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Multivariate Analysis of Variance Chapter 7 Multivariate Analysis of Variance LEARNING OBJECTIVES . . . Upon completing this chapter, you should be able to do the following: Explain the difference between the univariate null hypothesis of ANOVA and the multivariate null hypothesis of MANOVA. Discuss the advantages of a multivariate approach to significance testing compared to the more traditional univariate approaches. State the assumptions for the use of MANOVA. Discuss the different types of test statistics that are available for significance testing in MANOVA. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Multivariate Analysis of Variance Chapter 7 Multivariate Analysis of Variance LEARNING OBJECTIVES continued . . . Upon completing this chapter, you should be able to do the following: Describe the purpose of post hoc tests in ANOVA and MANOVA. Interpret interaction results when more than one independent variable is used in MANOVA. Describe the purpose of multivariate analysis of covariance (MANCOVA). Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

MANOVA Defined MANOVA . . . is the multivariate extension of the univariate techniques for assessing the differences between group means. In contrast to ANOVA, it can examine more than one dependent variable at the same time. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

ANOVA versus MANOVA In the univariate case, a single dependent measure is tested for equality across the groups. In the multivariate case, a variate is tested for equality. In MANOVA, the researcher actually has two variate – one for the dependent variables and another for the independent variables. The dependent variable variate is of more interest because the metric-dependent measures can be combined in a linear combination, as we have already seen in multiple regression and discriminant analysis. The unique aspect of MANOVA is that the variate optimally combines the multiple dependent measures into a single value that maximizes the differences across groups. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

ANOVA vs. MANOVA The relationship between the univariate and multivariate procedures is shown below: Number of Dependent Variables Number of Groups in Independent 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) Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

MANOVA Decision Process Stage 1: Objectives of MANOVA Stage 2: Research Design of MANOVA Stage 3: Assumptions in Multiple MANOVA Stage 4: Estimating the MANOVA Model and Assessing Overall Fit Stage 5: Interpreting the MANOVA Variate Stage 6: Validation of the Results Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Stage 1: Objectives of MANOVA To analyze a dependence relationship represented as the differences in a set of dependent measure across a series of groups formed by one or more categorical independent measures. To provide insights into the nature and predictive power of the independent measures as well as the interrelationships and differences in the multiple dependent measures. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

What Can We Do With MANOVA? Three types of questions suitable for MANOVA: Multiple Univariate Questions Structured Multivariate Questions Intrinsically Multivariate Questions Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

DECISION PROCESSES FOR MANOVA Rules of Thumb 7–1 DECISION PROCESSES FOR MANOVA MANOVA is an extension of ANOVA that examines the effect of one or more nonmetric independent variables on two or more metric dependent variables. In addition to the ability to analyze multiple dependent variables, MANOVA also has the advantages of: Controlling the experiment-wide error rate when there is some degree of intercorrelation among dependent variables. Providing more statistical power than ANOVA when the number of dependent variables is 5 or less. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Rules of Thumb 7–1 continued . . . DECISION PROCESSES FOR MANOVA Nonmetric independent variables create ‘groups’ between which the dependent variables are compared. Many times the groups represent experimental variables or “treatment effects.” Researchers should include only dependent variables that have strong theoretical support. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Stage 2: Issues in the Research Design of MANOVA Sample Size Requirements – Overall and by Group. Factorial Designs – Two or More Treatments. Selecting Treatments – types and number; interaction effects Using covariates – ANCOVA AND MANCOVA Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Sample Size Requirements by Group Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

RESEARCH DESIGN OF MANOVA Rules of Thumb 7–2 RESEARCH DESIGN OF MANOVA Cells (groups) are formed by the combination of independent variables. For example, a three-category nonmetric variable (e.g., low, medium, high) combined with a two-category nonmetric variable (e.g., gender of male versus female) will result in a 3 x 2 design with six cells (groups). Sample size per group is a critical design issue: Minimum sample size per group must be greater than the number of dependent variables. The recommended minimum cell size is 20 observations per cell (group). Researchers should try to have approximately equal sample sizes per cell (group). Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

RESEARCH DESIGN OF MANOVA Rules of Thumb 7–2 continued . . . RESEARCH DESIGN OF MANOVA Covariates and blocking variables are effective ways of controlling for external influences on the dependent variables that are not directly represented in the independent variables: An effective covariate is one that is highly correlated with the dependent variable(s) but not correlated with the independent variables. The maximum number of covariates in a model should be (.10 x Sample Size) – (Number of Groups – 1). Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Objectives of Covariance Analysis The objective of the covariate is to eliminate any effects that . . . affect only a portion of the respondents, or vary among the respondents. Similar to the use of a blocking factor, covariates can achieve two specific purposes . . . eliminate some systematic error outside the control of the researcher that can bias the results, and account for differences in the responses due to unique characteristics of the respondents. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Stage 3: Assumptions of ANOVA and MANOVA Independence of observations Equality of variance – covariance matrices for all groups Normality Linearity and multicollinearity among the dependent variables Sensitivity to outliers Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

MANOVA/ANOVA ASSUMPTIONS Rules of Thumb 7–3 MANOVA/ANOVA ASSUMPTIONS For the multivariate test procedures used with MANOVA to be valid: Observations must be independent. Variance-covariance matrices must be equal (or comparable) for all treatment groups. The dependent variables must have a multivariate normal distribution. Multivariate normality is assumed, but many times hard to assess. Univariate normality does not guarantee multivariate normality, but if all variables meet the univariate normality requirement then departures from multivariate normality are inconsequential. ANOVA F-tests are generally robust if violations of these assumptions are modest. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Why Use MANOVA? Control experimental error rate. Test for difference between multiple dependent variables. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Stage 4: Estimation of the MANOVA Model and Assessing Overall Fit Selecting criteria for significance tests. Assessing statistical power. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Criteria for Statistical Tests Statistical Measures Roy’s greatest characteristic root Wilks’ Lambda Pillai’s criterion and Hotelling’s trace Statistical Power Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Selecting a Statistical Measure Rules of Thumb 7–4 Selecting a Statistical Measure Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Selecting a Statistical Measure Rules of Thumb 7–4 continued . . . Selecting a Statistical Measure Roy’s gcr is a more powerful test statistic if the researcher is confident that all assumptions are strictly met and the dependent measures are representative of a single dimension of effects. In a vast majority of situations, all of the statistical measures provide similar conclusions. When faced with conflicting conditions, however, statistical measures can be selected that meet the situation faced by the researcher. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Rules of Thumb 7–5 MANOVA ESTIMATION The four most widely used measures for assessing statistical significance between groups on the independent variables are: Roy’s Greatest Characteristic Root Wilk’s Lambda Pillai’s Criterion Hotelling’s Trace In most situations the results/conclusions will be the same across all four measures, but in some unique instances results will differ between the measures. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Rules of Thumb 7–5 continued . . . MANOVA ESTIMATION Maintaining adequate statistical power is critical: Power in the .80 range for the selected alpha level is acceptable. When the effect size is small, the researcher should use larger sample sizes per group to maintain acceptable levels of statistical power. The General Linear Model (GLM) is widely used in testing ANOVA or MANOVA models. GLM is available on most statistical packages like SPSS and SAS. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Stage 5: Interpretation of the MANOVA Results Interpret the effects of covariates, if used. Assess which dependent variables exhibited differences across the groups of each treatment. Identify whether the groups differ on a single dependent variable or the entire dependent variate. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Assessing Effects on the Dependent Variate Main effects of the treatments Impacts of interaction terms Statistical significance Types of significant interactions Ordinal interactions Disordinal interactions Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Interpreting Covariates and Interaction Effects Rules of Thumb 7–6 Interpreting Covariates and Interaction Effects When covariates are involved in a GLM model: Analyze the model both with and without the covariates. If the covariates do not improve the statistical power or have no effect on the significance of the treatment effects, then they can be dropped from the final analysis. Any time two or more independent variables (treatments) are included in the analysis, interactions must be examined before drawing conclusions about main effects for any independent variable: If the interactions are not statistically significant, then main effects can be interpreted directly since the difference between treatments is considered constant across combinations of levels. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Rules of Thumb 7–6 Continued . . . If the interaction is statistically significant and the differences are not constant across combinations of levels, then the interaction must be determined to be ordinal or disordinal: Ordinal interactions mean that the direction of differences do not vary by level (e.g., males always less than females) even though the difference between males/females varies by level on the other treatment. In this case, the size of the main effect (e.g., males versus females) should only be described separately for each level of the other treatment. Significant disordinal interactions occur when the direction of an observed main effect changes with the level of another treatment (e.g., males greater than females for one level and less than females for another level). Disordinal interactions interfere with the interpretation of main effects. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Identifying Differences Between Individual Groups Post Hoc Methods A Priori or Planned Comparisons Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Post Hoc Methods Scheffe Tukey’s honestly significant difference (HSD) Tukey’s extension of the Fisher least significant difference (LSD) Duncan’s multiple-range test Newman-Kuels test Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Stage 6: Validation of the Results Replication. Use of covariates? Assessing causation? Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Interpreting Differences between Individual Groups Rules of Thumb 7–7 Interpreting Differences between Individual Groups When the independent variable has more than two groups, two types of procedures can be used to isolate the source of differences: Post-hoc tests examine potential statistical differences among all possible combinations of group means. Post-hoc tests have limited power and thus are best suited to identify large effects. Planned comparisons are appropriate when a priori theoretical reasons suggest that certain groups will differ from another group or other groups. Type I error is inflated as the number of planned comparisons increases. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.

Description of HBAT Primary Database Variables Variable Description Variable Type Data Warehouse Classification Variables X1 Customer Type nonmetric X2 Industry Type nonmetric X3 Firm Size nonmetric X4 Region nonmetric X5 Distribution System nonmetric Performance Perceptions Variables X6 Product Quality metric X7 E-Commerce Activities/Website metric X8 Technical Support metric X9 Complaint Resolution metric X10 Advertising metric X11 Product Line metric X12 Salesforce Image metric X13 Competitive Pricing metric X14 Warranty & Claims metric X15 New Products metric X16 Ordering & Billing metric X17 Price Flexibility metric X18 Delivery Speed metric Outcome/Relationship Measures X19 Satisfaction metric X20 Likelihood of Recommendation metric X21 Likelihood of Future Purchase metric X22 Current Purchase/Usage Level metric X23 Consider Strategic Alliance/Partnership in Future nonmetric Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall.