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# Multivariate Analaysis of Variance (MANOVA): Sharma, Chapter 11 - Bijan Yavar

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### Multivariate Analaysis of Variance (MANOVA): Sharma, Chapter 11 - Bijan Yavar

1. 1. Multivariate Analysis of Variance (MANOVA) Lecturer: Prof.Dr. Izani Ibrahim (CRM) Presenter: Bijan Yavar (CRM) Level of Study: PHD Field of Study: Management (Crisis Management) Course: Advanced Quantitative Techniques SID: ZP01774 Researcher ID: A-3544-2010 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
2. 2. PowerPoint Template www.themegallery.com 17th December 2013 Bijan Yavar Edit your company slogan The National University of Malaysia (UKM) Graduate School of Business (GSB)
3. 3. Contents 1. Introduction (What is MANOVA?) 2. Differences Between MANOVA and ANOVA 3. Geometry of MANOVA 4. Analytic Computations for Two-Group MANOVA 5. Two-Group MANOVA 6. Multiple-Group MANOVA 7. MANOVA for Two Independent Variables of Factors 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
4. 4. 1. Introduction 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
5. 5. Introduction (What is MANOVA?) Technique for assessing group differences across multiple metric dependent variables (DVs) simultaneously, based on a set of categorical independent variables (IVs) Procedure used to test the significance of the effects of one or more IVs (categorical) on two or more DVs (continuous) The Objective of MANOVA is Very similar to some of the objectives of Discriminant Analysis. Remember, in Discriminant Analysis one of the objective was to Determine if the Groups are Significantly Different with respect to a Given set of variables 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
6. 6. 2. Differences Between MANOVA and ANOVA 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
7. 7. The Difference Between MANOVA & ANOVA 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
8. 8. The Difference Between MANOVA & ANOVA MANOVA is a Multivariate extension of ANOVA with the only Difference being that in MANOVA there are Multiple Dependent Variables ANOVA - 1 DV MANOVA - 2 or more DVs Both are used with experimental designs in which researchers manipulate or control one or more IVs to determine the effect on one DV (ANOVA) or more DVs (MANOVA) 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
9. 9. 1. Introduction 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
10. 10. Introduction (What is MANOVA?) In MANOVA the Independent Variables (IV’s) are Categorical In MANOVA the Dependent Variables (DV’s) are Continues MANOVA is an extension of ANOVA in which main effects and interactions are assessed on a combination of DVs MANOVA tests whether mean differences among groups on a combination of DVs is likely to occur by chance. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
11. 11. Introduction (What is MANOVA?) A new DV is created that is a linear combination of the individual DVs that maximizes the difference between groups. In Factorial designs a different linear combination of the DVs is created for each main effect and interaction that maximizes the group difference separately. Also when the IVs have more than one level, the DVs can be recombined to maximize paired comparisons 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
12. 12. Scenarios Scenario Number 1 A marketing manager is interested in determining if geographical region (NEWS - 1 categorical IVs at 4 levels), has an effect on consumers’ Taste preferences, Purchase Intentions, and Attitude toward the product (metric – Likert scale and has more than 2 DVs) 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
13. 13. Scenarios Scenario Number 2 A political analyst is interested in determining if party affiliation (Democratic, Republican, Independent and gender (male & female) - 2 categorical IV’s - have any effect on voters’ views on a number of issues such as a abortion, taxes, economy deficits (multiple metric DVs) 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
14. 14. 3. Geometry of MANOVA 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
15. 15. Cases – case 1: 1 IV (at 2 Levels) & 1 DV The Centroid (Or the mean) of each group (i.e Y1 and Y2) can be presented as a point in the One-Dimensional Space. If IV has an Effect on the DV, then the Means of the two groups are Different and the effect of IV is measured by the Difference Between the Two Means. Centroid For Group 1 Y1 17th December 2013 Centroid For Group 2 Figure 11.1 Y2 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
16. 16. To Explain More MANOVA is concerned with determining whether MD between group centroids is significantly greater than zero. In ANOVA case , because there are only 2 groups and 1 DV, the problem reduces to comparing the means of two group using t-test. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
17. 17. Case II: 1 IV (at 2 levels) and 2 or more DVs Since the IV is at 2 levels, there are 2 groups 2 DVs - Y and Z and for group 1 and for group 2, be the centroid for the 2 groups Fig. 11.2 - shows the centroid of each group . It can be presented a point or a vector in 2-dimensional space defined by DV (Y and Z) Figure 11.2 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
18. 18. To Explain More Mahanalobis Distance (MD) between the 2 points measures the distance between the centroids of the 2 groups. The larger the distance, the greater the difference between the 2 groups (vice versa) MANOVA reduces to computing the distance between centroids of the 2 groups and determining if the distance is statistically significant. In the case of p variables, the centroids of the 2 groups can be represented as 2 points in the p-dimensional space and the problem reduces to determining whether the distance between the 2 points is different from zero. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
19. 19. Case III - more than 1 IV and p DV Refer to example party affiliation and gender 2 DVs - tax increase (Y) and gun control (Z) 3 IVs (at 2 levels) - Democrats (M,F) , Republicans (M,F), Independents (M,F) Refer to Table 11.1 (pg. 344) There are 3 types of effects: Main effect of gender Main effect of party affiliation The interaction effect of gender and party affiliation. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
20. 20. (1) Main effect of gender In Panel I – the main effect of gender, is measured by the distance between the 2 centroids. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
21. 21. (2) Main effect of party affiliation •In Panel II - the main effect of party affiliation is measured by the distances between pairs of the 3 centroids •There will be 3 distances, each representing the distance between pairs of group 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
22. 22. (3) Interaction Effect of Gender and Party Affiliation Panel III - if the effect of gender is independent of party affiliation, the vectors joining the respective centroids should be parallel. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
23. 23. (4) Interaction Effect of Gender and Party Affiliation Panel IV - if the effect of gender is not independent of party affiliation, the vectors joining the respective points will not be parallel. The magnitude of the interaction effect between 2 variables is indicated by the extent to which the vector are nonparallel. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
24. 24. 4. Analytic Computations for Two-Group MANOVA 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
25. 25. Analytic Computations for Two-Group MANOVA MULTIVARIATE SIGNIFICANT TEST SIGNIFICANT TEST MD2, T2,F Eigenvalue Wilks’ Lambda Hotelling’s trace Pillai’s criterion Roy’s Largest Root F-ratio UNIVARIATE SIGNIFICANT TEST MD2, T2,F Analytic computation EFFECT SIZE 17th December 2013 Partial eta square Partial eta square Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
26. 26. Analytic Computations for Two-Group MANOVA MULTIVARIATE SIGNIFICANT TEST UNIVARIATE SIGNIFICANT TEST SIGNIFICA NT TEST MD2, T2,F Eigenvalue Wilks’ Lambda Hotelling’s trace Pillai’s criterion Roy’s Largest Root F-ratio MD2, T2,F EFFECT SIZE Partial eta square Partial eta square Analytic computation 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
27. 27. 5, 6 and 7. Rest of the Topics 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
28. 28. Multivariate Significance Test The first step- determine if the 2 groups are significantly different with respect to the variables Are the centroids of the 2 groups significantly different? The null and alternative hyphothesis for multivariates statistical significance testing in MANOVA are ; H0 = µv1 = µv2 = µv3 ……. = µvk Null hypothesis formally states that the difference between the centroid is zero 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
29. 29. Similarities Between MANOVA and Discriminant Analysis Objective of discriminant analysis To identify a linear combination of (discriminant function) of the variables that would give the maximum separation between 2 groups. Statistical test is performed to determine if the groups are significantly different with respect to the linear combination (discriminant scores) In MANOVA, we test whether the centroids of the 2 groups are significantly different. Although a linear combination of , which provides the maximum separation between the 2 groups, is not computed in MANOVA, the multivariate significance tests implicitly test whether the mean score of the 2 groups obtained from such a linear combination are significantly different. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
30. 30. Briefly In the case of ONE IV, there is NO difference between MANOVA and discriminant analysis In the case of more than one IV, MANOVA provide additional insights into the effects of IVs on DVs that are not provided by discriminant analysis. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
31. 31. MANOVA vs Discriminant Analysis MANOVA is applied to experimental situations where all, or at least some, IVs are manipulated and subjects are randomly assigned to group, usually with equal cell sized Discriminant function analysis developed in the context of non-experimental research where groups are formed naturally and are not usually the same size 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
32. 32. MANOVA vs Discriminant Analysis (cont’d) MANOVA asks if mean differences among groups on the combined DV are larger than expected by chance. Discriminant function analysis asks if there is some combination of variables that reliably separates groups. Should be noted that there is NO mathematical distinction between MANOVA and discriminant function analysis. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
33. 33. MANOVA Advantages Over ANOVA By measuring multiple DVs you increase your chances for finding a group difference With a single DV you “put all of your eggs in one basket” Multiple measures usually do not “cost” a great deal more and you are more likely to find a difference on at least one. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
34. 34. MANOVA Advantages Over ANOVA Using multiple ANOVAs inflates type 1 error rates and MANOVA helps control for the inflation Under certain (rare) conditions MANOVA may find differences that do not show up under ANOVA 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
35. 35. Assumptions of MANOVA Normal Distribution: - The dependent variable should be normally distributed within groups. Overall, the F test is robust to non-normality, if the non-normality is caused by skewness rather than by outliers. Tests for outliers should be run before performing a MANOVA, and outliers should be transformed or removed. Linearity - MANOVA assumes that there are linear relationships among all pairs of dependent variables, all pairs of covariates, and all dependent variable-covariate pairs in each cell. Therefore, when the relationship deviates from linearity, the power of the analysis will be compromised. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
36. 36. Assumptions of MANOVA (Cont’d) Homogeneity of Variances: - Homogeneity of variances assumes that the dependent variables exhibit equal levels of variance across the range of predictor variables. Remember that the error variance is computed (SS error) by adding up the sums of squares within each group. If the variances in the two groups are different from each other, then adding the two together is not appropriate, and will not yield an estimate of the common within-group variance. Homoscedasticity can be examined graphically or by means of a number of statistical tests. Homogeneity of Variances and Covariances: - In multivariate designs, with multiple dependent measures, the homogeneity of variances assumption described earlier also applies. However, since there are multiple dependent variables, it is also required that their intercorrelations (covariances) are homogeneous across the cells of the design. There are various specific tests of this assumption. 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
37. 37. T2 and F 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
38. 38. Pillai Trace and Hotelling Trace 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
39. 39. F - Test 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
40. 40. Questions 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
41. 41. Reference Sharma, Subhash (1996). Applied Multivariate Techniques, U.S.A., John Wiley & Sons Inc. (Chapter 11: Multivariate Analysis of Variance, PP. 342 – 373). 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
42. 42. For Listening 17th December 2013 Bijan Yavar The National University of Malaysia (UKM) Graduate School of Business (GSB)
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