2. Wilks’ lambda

3. Pillai’s criterion

5. Turkey’s Honestly significant difference (HSD) method 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 />