Multivariate analysis of variance (MANOVA) is simply an extension of the univariate Analysis of variance. In analysis of variance, we examine the one metric dependent variable with the grouping independent variable. Analysis of variance fails to compare the group when the dependent variables become more than one dependent metric variable. To account for multiple dependent variables, MANOVA bundles them together into a weighted linear combination or composite variable. These linear combinations are also called canonical variates, roots, Eigenvalues, vectors or discriminant functions. We will generally use either the canonical variates or the roots.
Once the dependent variable combines into a canonical variate, MANOVA can be performed, such as univariate ANOVA. Now, MANOVA will compare whether or not the independent variable group differs from the newly created group. In this way, MANOVA essentially tests whether or not the independent grouping variable explains a significant amount of variance in the canonical variate.
Assumptions in MANOVA:
The following are the assumptions made in MANOVA:
- Independent Random Sampling: MANOVA normally assumes that the observations are independent of one another. There is not any pattern in MANOVA for the selection of the sample. The sample is completely random.
- Level and Measurement of the Variables: MANOVA assumes that the independent variables are categorical in nature and the dependent variables are continuous variables. MANOVA also assumes that homogeneity is present between the variables that are taken for covariates.
- Linearity of dependent variable: In MANOVA, the dependent variables can be correlated to each other, or may be independent of each other. Study shows that in MANOVA, a moderately correlated dependent variable is preferred. In MANOVA, if the dependent variables are independent of each other, then we have to sacrifice the degrees of freedom and it will decrease the power of the analysis.
- Multivariate Normality: MANOVA is very sensitive with outliers and missing value. Thus, it is assumed that multivariate normality is present in the data.
- Multivariate Homogeneity of Variance: Like test analysis of variance, MANOVA also assumes that the variance between groups is equal.
Key concepts and terms
Box’s M test: In MANOVA, box’ M test is used to know the equality of covariance between the groups. Null hypothesis in MANOVA means that observed covariance matrices of the dependent variable are equal across groups.
Wilks’ lambda: In MANOVA , Wilks’ lambda test is used to know the overall significance of the model. When the overall model is significant, then we can predict the individual significance of the variable. We can use other tests as well. These other overall significance tests include Pillai’s Trace, Hotelling’s Trace, Roy’s Largest Root test, etc.
Levene’s test: In MANOVA, Levene’s test is used to know whether or not the variance between groups is equal. Insignificant value of levene’s test shows equal variance between groups.
Partial eta square: Partial eta square shows how much variance is explained by the independent variable.
Power: Power shows the probability of correctly accepting the null hypothesis.
Post hoc test: In MANOVA, when there is a significant difference between groups, then the post hoc test is performed to know the exact group means, which significantly differ from each other.
Significance: Like ANOVA, probability value is used to make statistical decisions as to whether or not the group means are equal, or if they differ from each other.
Multivariate F-statistics: F- statistics is simply derived by dividing the means sum of the square for the source variable by the source variable mean error.
Comparison between ANOVAand MANOVA:
Computation of MANOVA is more complex compared to the ANOVA. In ANOVA, we compute univariate F statistic but in MANOVA, we compute multivariate F statistics. In ANOVA, we compare grouping independent variables with one dependent variable, but in MANOVA, we compare many dependent variables with the grouping variable.
MANOVA IN SPSS: In SPSS, MANOVA can be performed using the analysis menu, selecting the “GLM” option, and then choosing the “Multivariate” option from the GLM option.
MANOVA Resources
Bray, J. H., & Maxwell, S. E. (1985). Multivariate analysis of variance. Newbury Park, CA: Sage Publications.
de Leeuw, J. (1988). Multivariate analysis with linearizable regressions. Psychometrika, 53(4), 437-454.
Gill, J. (2001). Generalized Linear Models: A Unified Approach. Thousand Oaks, CA: Sage Publications.
Hand, D. J., & Taylor, C. C. (1987). Multivariate analysis of variance and repeated measures. London: Chapman and Hall.
Huberty, C. J., & Morris, J. D. (1989). Multivariate analysis versus multiple univariate analyses Psychological Bulletin, 105(2), 302-308.
Huynh, H., & Mandeville, G. K. (1979). Validity conditions in a repeated measures design. Psychological Bulletin, 86(5), 964-973.
Meulman, J. J. (1992). The integration of multidimensional scaling and multivariate analysis with optimal transformations. Psychometrika, 57(4), 539-565.
Nelder, J. A., & Wedderburn, R. W. M. (1972). Generalized liner models. Journal of the Royal Statistical Society, 135, 370-384.
Nichols, D. P. (1993). Interpreting MANOVA parameter estimates. SPSS Keywords, 50, 8-14.
Olson, C. L. (1976). On choosing a test statistic in multivariate analyses of variance. Psychological Bulletin, 83(4), 579-586.
Powell, R. S., & Lane, D. M. (1979). CANCOR: A general least-squares program for univariate and multivariate analysis of variance and covariance. Behavior Research Methods & Instrumentation, 11(1), 87-89.
Sclove, S. L. (1987). Application of model-selection criteria to some problems in multivariate analysis. Psychometrika, 52(3), 333-343.
Smith, H. F. (1958). A multivariate analysis of covariance. Biometrics, 14, 107-127.
