# Multivariate Analysis of Variance (MANOVA)

Use this model to carry out a MANOVA (Multivariate ANalysis Of VAriance) of two or more balanced or unbalanced factors.

## Description of the MANOVA in XLSTAT

### General description

The MANOVA uses the same conceptual framework as the ANOVA. The main difference comes from the nature of the dependent variables: instead of a single one, we can study many of them. With the MANOVA, explanatory variables are often called factors. Effects of factors are estimated on a combination of several response variables.

The advantage of the MANOVA as opposed to several simultaneous ANOVAs lies in the fact that it takes into account correlations between response variables, which results in a richer use of the information contained in the data.

The MANOVA tests the presence of significant differences among combinations of levels of factors on several response variables. MANOVA also enables the simultaneous tests of all hypotheses tested by an ANOVA and is more likely to detect differences between levels of factors.

Furthermore, the computation of several ANOVAs instead of one MANOVA increases the Type I error, which is the probability that the null hypothesis will be wrongly rejected.

The potential covariation between response variables is not taken into account with several ANOVAs. Instead, the MANOVA is sensitive to both the difference of averages between levels of factors and the covariation between explanatory variables. And a potential correlation between response variables is more likely to be detected when these variables are studied together, as is the case with a MANOVA.

### Balanced and unbalanced MANOVA

We speak about balanced MANOVA when the numbers of categories are equal for all combinations of factors. When the numbers of all categories for one of the combinations of factors are not equal, then the MANOVA is said to be unbalanced. XLSTAT can handle both cases.

## Results of the MANOVA in XLSTAT

**Summary statistics**: The tables of descriptive statistics show the simple statistics for all the variables selected. The number of observations, missing values, the number of non-missing values, the mean and the standard deviation (unbiased) are displayed for the dependent variables (in blue) and the quantitative explanatory variables. For qualitative explanatory variables, the names of the various categories are displayed together with their respective frequencies.

**Means of level factor**: This table provides for each level factor and quantitative variable the mean value.

**SSCP matrices**: These tables are displayed to give a general view of the effects of the factors and interactions between factors.

**Wilks’ test (Rao’s approximation)**: This table provides the results of Wilks' Lambda test which tests the hypothesis of equality of the mean vectors for the different levels.

**Hotelling-Lawley test**: This table provides the results of Hotelling-Lawley trace test, which tests the hypothesis of equality of the mean vectors for the different levels.

**Pillai’s test**: This table provides the results of Pillai’s trace test, which tests the hypothesis of equality of the mean vectors for the the different levels.

**Roy’s test**: This table provides the results of Roy’s greatest root test which tests the hypothesis of equality of the mean vectors for the different levels.

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