# MANOVA

Multivariate analysis of variance (MANOVA) is an extension of the univariate analysis of variance.  In Analysis of Variance, we examine the one scale dependent variable with the grouping independent variable.  Analysis of variance is used to compare the group with only one dependent variable.  To account for multiple dependent variables, MANOVA bundles them together into a weighted linear combination or composite variable.  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 simultaneously explains a significant amount of variance in the dependent variable.

Assumptions:

1. Independent Random Sampling: MANOVA assumes that the observations are independent of one another, there is not any pattern for the selection of the sample, the sample is completely random.
2. Level and Measurement of the Variables: MANOVA assumes that the independent variables are categorical and the dependent variables are continuous or scale variables.
3. Linearity of dependent variable: The dependent variables can be correlated to each other, or may be independent of each other. Study shows that a moderately correlated dependent variable is preferred, 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.
4. Multivariate Normality: Multivariate normality is present in the data.
5. Multivariate Homogeneity of Variance: Variance between groups is equal.

Key concepts and terms:

• Box’s M test: Used to know the equality of covariance between the groups.  Null hypothesis means that observed covariance matrices of the dependent variable are equal across groups.  This assumption is often notated in practice.
• Wilks’ lambda: Used to know the overall significance of the model. When the overall model is significant, then we can state that simultaneous differences exist on the dependent variable by the independent variable(s).  Other overall significance tests include: Pillai’s Trace, Hotelling’s Trace, and Roy’s Largest Root test.
• Levene’s test: Used to examine whether or not the variance between independent variable groups are equal.  Non-significant values of Levene’s test show 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: When there is a significant difference between groups, then post hoc tests are performed to know the exact group means, (i.e., which specific independent variable level significantly differ from each other).
• Significance: Like ANOVA, the 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: The F- statistic is derived by essentially dividing the means sum of the square for the source variable by the source variable mean error.

Comparison between ANOVA and 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.

SPSS: Can be performed using the analysis menu, selecting the “GLM” option, and then choosing the “Multivariate” option from the GLM option.

Resources

Bray, J. H., & Maxwell, S. E. (1985). Multivariate analysis of variance. Newbury Park, CA: Sage Publications. View

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. View

Hand, D. J., & Taylor, C. C. (1987). Multivariate analysis of variance and repeated measures. London: Chapman and Hall. View

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.

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