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MANOVAMultivariate Analysis of VarianceThis report aim at studying the different techniques of MANOVA and to understand its usefulness in real life situations, where there are many independent variables depending on many dependent variables.Group No. - 52/19/201091011Ashutosh Kr. Jha91016Deepinder Singh91017Divanshu Kapoor91018Gaurav Sharma91022Harsh Aggarwal91039Nishant Singh91040Pragati Saraf91049Saurabh Anand91055Soumya Saxena91057Sudhir makkar<br />Acknowledgement<br />We’d like to express our earnest gratitude towards Prof. Kaushik Paul for the stupendous guidance and support that he provided to us during the execution of this project. His role in providing a vivid insight into the dynamics of Operation Management and the present day scenario that goes beyond the realms of any text-book; have really motivated us to work that bit harder to come out with this report.<br />We would like to offer a sincere thanks to Dr. S.K.Pandey, for his extended assistance for the fulfillment of this project.<br />We’d also like to acknowledge the unending help that we received from our fellow classmates whilst the execution of this project. Their help and concern goes on to reiterate the kind of bonhomie that exists at FORE School of Management.<br />Table of Contents<br />TitlePage No.1Introduction42Assumptions for MANOVA43Decision Process for MANOVA54Box’s M Test7 Hotelling’s T28 Roy’s Greatest Characteristic Route (GCR)8 Wilks' lambda (U statistic)8 Pillai’s Criterion95Post Hoc Tests96MANOVA example97Advantages of MANOVA128Disadvantages of MANOVA139How to avoid MANOVA1310Conclusion13<br />Introduction<br />A MANOVA or multivariate analysis of variance is a way to test the hypothesis that one or more independent variables, or factors, have an effect on a set of two or more dependent variables. MANOVA answers the question ‘Does the combination of several DVs vary with respect to the IVs?’ In MANOVA a new Dependent Variable is created that attempts to maximize the differences between the treatment groups. The new DV is a linear combination of the DVs.<br /> For example, one might wish to test the hypothesis that sex and ethnicity interact to influence a set of job-related outcomes including attitudes toward co-workers, attitudes toward supervisors, feelings of belonging in the work environment, and identification with the corporate culture. As another example, you might want to test the hypothesis that three different methods of teaching writing result in significant differences in ratings of student creativity, student acquisition of grammar, and assessments of writing quality by an independent panel of judges.<br />One could use a series of univariate ANOVAs also- one for each dependent variable. MANOVA does all these univariate tests simultaneously thus used more widely in industry.<br />MANOVA Advantages over ANOVA<br />By measuring multiple DVs chances of finding a group difference increases.<br />With a single DV you “put all of your eggs in one basket”<br />Multiple measures usually do not “cost” a great deal more and one is more likely to find a difference on at least one.<br />Using multiple ANOVAs inflates type 1 error rates and MANOVA helps control for the inflation<br />Under certain (rare) conditions MANOVA may find differences that do not show up under ANOVA<br />Assumptions for MANOVA:<br />1. Sample size - The sample in each cell must be greater that the number of dependent variables<br />2. Univariate and Multivariate normality must hold for each of the dependent variable (when cell size > 30 this is less important)<br />3. Linearity - linear relationships must exist among all pairs of dependent variables and that linear combination must be distributed normally.<br />4. Homogeneity of regression - Covariates must have homogeneity of regression effect (must have equal effects on the dependent variable across the groups)<br />5. Homogeneity of variance-covariance matrix (Box's M) <br />It tests the hypothesis that the covariance matrices of the dependent variables are significantly different across levels of the independent variable. The F test from Box’s M statistics should be interpreted cautiously in that a significant result may be due to violation of the multivariate normality assumption and a non significant result may be due to small sample size and lack of power. It is fairly robust if equal sample sizes are there.<br />6. Multicollinearity and Singularity <br />When there is strong multicollinearity, one have redundant dependent measures and this decreases statistical efficiency.<br />7. Outliers - MANOVA is very sensitive to the effect of outliers because they impact on the Type I error.<br />Decision Process for MANOVA<br />BOX’s M Test<br />This test is used to test for equality of covariance matrices and provide significance levels for the test statistics which indicates the likelihood of differences within the group. For example, if a 0.01 level is considered the threshold level for indicating violations of assumption, values greater than 0.01 would be considered acceptable because they indicate no differences between groups whereas values less than 0.01 would be problematic since they indicate significant differences. Since it is sensitive to normality it should be checked for univariate normality of all dependent measures before performing test.<br />The Four most widely used measures for assessing statistical significance between groups on the independent variables are:<br /><ul><li>Roy’s greatest characteristic root
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Hotelling’s T2</li></ul>In most situations the results/conclusions are same across all four measures but in some unique instances the result will differ between measures.<br />Hotelling’s T2<br />It is a test to assess the statistical significance on the difference of the mean of two or more variables between the groups. It is a special case of MANOVA used with two groups of a treatment variable. The larger the Hotelling's trace, the more the given effect contributes to the model. <br />Roy’s Greatest Characteristic root (gcr)<br />This test measures the differences on only the first discriminant function among the dependent variables. This criterion provides advantages in power and specificity of the test but makes it less useful in certain situations where all dimensions should be considered. This test is more appropriate when the dependent variables are strongly interrelated on a single dimension, but it is also the measure most to be severely affected by violations of the assumptions.<br />Wilks' lambda (U statistic)<br />Wilks' lambda is a test statistic used in multivariate analysis of variance (MANOVA) to test whether there are differences between the means of identified groups of subjects on a combination of dependent variables. For example, we have to test whether the mean score of two groups, graduates and diplomats, is the same across eight constructs simultaneously. Thus, there are eight dependent variables and comparing the mean of this combination for two groups. Wilks' lambda performs, in the multivariate setting, with a combination of dependent variables, the same role as the F-test performs in one-way analysis of variance. Wilks' lambda is a direct measure of the proportion of variance in the combination of dependent variables that is unaccounted for by the independent variable (the grouping variable or factor). If a large proportion of the variance is accounted for by the independent variable then it suggests that there is an effect from the grouping variable and that the groups (in this case the graduates and diplomats) have different mean values. The t-test, Hotelling's T, and the F test are special cases of Wilks' lambda. Wilks' lambda ranges from 0 to 1 and the lower the Wilks' lambda, the more the given effect contributes to the model. <br />Pillai’s Criterion<br />Multiple discriminant analysis (MDA) is the part of MANOVA where canonical roots are calculated. The Pillai-Bartlett trace is the sum of explained variances on the discriminant variates, which are the variables which are computed based on the canonical coefficients for a given root. The larger the Pillai's trace, the more the given effect contributes to the model. Pillai's trace is always smaller than Hotelling's trace. This is the most robust test of all<br />Each measure/test is preferred in differing situations:<br />Pillai’s criterion or wilk’s lambda is the preferred measure when the basic design considerations( adequate sample size, no violations of assumptions, approx. equal cell sizes) are met.<br />Pillai’s criterion is considered more robust and should be used if sample size decreases, unequal cell sizes appear or homogeneity of covariances is violated.<br />Roy’s gcr isa 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.<br />In a vast majority of situations, all of statistical measures provide similar conclusions<br />Post Hoc Tests<br />The tests of dependent variables between all possible pairs of group differences that are tested after data patterns are established. But these tests have limited power so best suited to identify large effects. Most common used post hoc procedures are:<br /><ul><li>Scheffe Method
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Turkey’s Honestly significant difference (HSD) method </li></ul>MANOVA Example<br /> A group of children with moderate learning difficulties were assessed on a number of measures-<br /> IQ, Math, Reading Accuracy, Reading Comprehension, Communication Skill. The children were divided into four groups on the basis of gender (male, female) and season of birth (summer, not summer). A MANOVA was performed using gender and season of birth as the IVs and IQ mathematics, reading accuracy, reading comprehension and communication skills as the dependent variables.<br />Box M Test is used to test for equality of covariance matrices and provide significance levels for the test statistics which indicates the likelihood of differences within the group. After looking at the below output it can be concluded that<br />Do not reject the assumption of homogeneity of variance-covariance matrices.<br />Do not reject the assumption of homogeneity of variance.<br />First we will look at the overall F test (over all three dependent variables). What we are most interested in is a statistic called Wilks’ lambda (λ), and the F value associated with that. Lambda is a measure of the percent of variance in the DVs that is *not explained* by differences in the level of the independent variable. Lambda varies between 1 and zero, and we want it to be near zero (e.g., no variance that is not explained by the IV). In the case of ours, Gender, Wilks’ lambda is .626, and has an associated F of 7.542, which is significant at p. <001.<br />Similarly we find for Season of birth Wilk’s lambda value is 0.612 and an associated F value of 7.974<br />The following table below now checks univariate analysis i.e. effect of each independent variable on each of the dependent variable i.e. IQ, Mathematical Ability, Reading Accuracy, Communication skill. <br />Advantages of MANOVA<br />It tests the effects of several independent variables and several outcome (dependent) variables within a single analysis<br />It has the power of convergence (no single operationally defined dependent variable is likely to capture perfectly the conceptual variable of interest) <br />independent variables of interest are likely to affect a number of different conceptual variables- for example: an organisation's non-smoking policy will affect satisfaction, production, absenteeism, health insurance claims, etc<br />It can provide a more powerful test of significance than available when using univariate tests<br />It reduces error rate compared with performing a series of univariate tests<br />It provides interpretive advantages over a series of univariate ANOVAs<br />Since only ‘one’ dependent variable is tested, the researcher is protected against inflating the type 1 error due to multiple comparisons. <br />Disadvantages of MANOVA<br />• Discriminant functions are not always easy to interpret - they are designed to separate groups, not to make conceptual sense. In MANOVA, each effect evaluated for significance uses different discriminant functions (Factor A may be found to influence a combination of dependent variables totally different from the combination most affected by Factor B or the interaction between Factors A and B).<br />• Like discriminant analysis, the assumptions on which it is based are numerous and difficult to assess and meet.<br />How to avoid MANOVA<br />Combine or eliminate dependent variables so that only one dependent variable need be analyzed<br />Use factor analysis to find orthogonal factors that make up the dependent variables, then use univariate ANOVAs on each factor (because the factors are orthogonal each univariate analysis should be unrelated)<br />LIMITATIONS<br />The number of people in the smallest cell should be larger than the total number of dependent variables. <br />It can be very sensitive to outliers, for small N. <br />It assumes a linear relationship (some sort of correlation) between the dependent variables. <br />MANOVA won't give you the interaction effects between the main effect and the repeated factor. <br />CONCLUSION<br />After this exercise, it is understood that MANOVA technique is really useful in real life business situations where independent variables are categorical like season of birth, gender etc. and dependent variables are more than one and are metrics in nature.<br />