# Multivariate analysis of variance (MANOVA)

The Multivariate analysis of variance (MANOVA) procedure provides regression analysis and analysis of variance for multiple dependent variables by one or more factor variables or covariates. The factor variables divide the population into groups. Using this general linear model procedure, you can test null hypotheses about the effects of factor variables on the means of various groupings of a joint distribution of dependent variables. You can investigate interactions between factors as well as the effects of individual factors. In addition, the effects of covariates and covariate interactions with factors can be included. For regression analysis, the independent (predictor) variables are specified as covariates.

Both balanced and unbalanced models can be tested. A design is balanced if each cell in the model
contains the same number of cases. In a multivariate model, the sums of squares due to the effects
in the model and error sums of squares are in matrix form rather than the scalar form found in
univariate analysis. These matrices are called SSCP (sums-of-squares and cross-products) matrices.
If more than one dependent variable is specified, the multivariate analysis of variance using
Pillai's trace, Wilks' lambda, Hotelling's trace, and Roy's largest root criterion with approximate
*F* statistic are provided as well as the univariate analysis of variance for each dependent
variable. In addition to testing hypotheses, Multivariate analysis of variance (MANOVA) produces
estimates of parameters.

Commonly used *a priori* contrasts are available to perform hypothesis testing.
Additionally, after an overall *F* test has shown significance, you can use post hoc tests to
evaluate differences among specific means. Estimated marginal means give estimates of predicted mean
values for the cells in the model, and profile plots (interaction plots) of these means allow you to
visualize some of the relationships easily. The post hoc multiple comparison tests are performed for
each dependent variable separately.

Residuals, predicted values, Cook's distance, and leverage values can be saved as new variables in your data file for checking assumptions. Also available are a residual SSCP matrix, which is a square matrix of sums of squares and cross-products of residuals, a residual covariance matrix, which is the residual SSCP matrix divided by the degrees of freedom of the residuals, and the residual correlation matrix, which is the standardized form of the residual covariance matrix.

WLS Weight allows you to specify a variable used to give observations different weights for a weighted least-squares (WLS) analysis, perhaps to compensate for different precision of measurement.

- Example
- A manufacturer of plastics measures three properties of plastic film: tear resistance, gloss, and opacity. Two rates of extrusion and two different amounts of additive are tried, and the three properties are measured under each combination of extrusion rate and additive amount. The manufacturer finds that the extrusion rate and the amount of additive individually produce significant results but that the interaction of the two factors is not significant.
- Methods
- Type I, Type II, Type III, and Type IV sums of squares can be used to evaluate different hypotheses. Type III is the default.
- Statistics
- Post hoc range tests and multiple comparisons: least significant difference,
Bonferroni, Sidak, Scheffé, Ryan-Einot-Gabriel-Welsch multiple
*F*, Ryan-Einot-Gabriel-Welsch multiple range, Student-Newman-Keuls, Tukey's honestly significant difference, Tukey's*b*, Duncan, Hochberg's GT2, Gabriel, Waller Duncan*t*test, Dunnett (one-sided and two-sided), Tamhane's T2, Dunnett's T3, Games-Howell, and Dunnett's*C*. Descriptive statistics: observed means, standard deviations, and counts for all of the dependent variables in all cells; the Levene test for homogeneity of variance; Box's*M*test of the homogeneity of the covariance matrices of the dependent variables; and Bartlett's test of sphericity. - Plots
- Spread-versus-level, residual, and profile (interaction).

## Data considerations

- Data
- The dependent variables should be quantitative. Factors are categorical and can have numeric values or string values. Covariates are quantitative variables that are related to the dependent variable.
- Assumptions
- For dependent variables, the data are a random sample of vectors from a
multivariate normal population; in the population, the variance-covariance matrices for all cells
are the same. Analysis of variance is robust to departures from normality, although the data should
be symmetric. To check assumptions, you can use homogeneity of variances tests (including Box's
*M*) and spread-versus-level plots. You can also examine residuals and residual plots. - Related procedures
- Use the Explore procedure to examine the data before doing an analysis of variance. For a single dependent variable, use Analysis of covariance (ANCOVA). If you measured the same dependent variables on several occasions for each subject, use Repeated measures ANOVA.

## Obtaining Multivariate analysis of variance (MANOVA) tables

This feature requires Custom Tables and Advanced Statistics.

- From the menus choose:
- Click Select variables under the Dependent variables section and select at least two dependent variables. Click OK after selecting the variables.
- Click Select variables under the Fixed factor variables section and select one or more independent categorical variables that represent potential causes for variation in the dependent variable.
- Optionally, you can click Covariate variables to
select continuous variables that may have an influence on the dependent variables.
Click Select variables under the Covariate variables section and select covariate variables. Click OK after selecting the variables.

- Optionally, click Select variable under the WLS weight section to specify a single weight variable for weighted least-squares analysis. If the value of the weighting variable is zero, negative, or missing, the case is excluded from the analysis. A variable already used in the model cannot be used as a weighting variable. Click OK after selecting the variable.
- Optionally, you can select the following options from the
Additional settings menu:
- Click Model to analyze a full factorial model (all interactions and main effects), a custom model (subset of interactions and main effects), or to specify particular terms (for example, for nested designs).
- Click Contrasts to test for differences among the factor variables.
- Click Post hoc tests to use post hoc range tests that produce multiple comparisons between factor means.
- Click Statistics to select which statistics to include in the procedure.
- Click EM means to select the factors and interactions for which you want estimates of the population marginal means in the cells. The means are adjusted for covariates (when covariates are specified).
- Click Plots to enable the display of charts in the output and to select the charting settings.
- Click Options to specify null hypothesis settings and control of the treatment of missing data.
- Click Save to dataset to add values predicted by the model, residuals, and related measures to the dataset as new variables.
- Click Bootstrap for deriving robust estimates of standard errors and confidence intervals for estimates such as the mean, median, proportion, odds ratio, correlation coefficient or regression coefficient.

- Click Run analysis.

This procedure pastes ANOVA-M command syntax.