# GLM Multivariate Analysis

The GLM Multivariate 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, GLM Multivariate 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).

GLM Multivariate 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 GLM Univariate. If you measured the same dependent variables
on several occasions for each subject, use GLM Repeated Measures.

Obtaining GLM Multivariate Tables

This feature requires SPSS® Statistics Standard Edition or the Advanced Statistics Option.

- From the menus choose:
- Select at least two dependent variables.

Optionally, you can specify Fixed Factor(s), Covariate(s), and WLS Weight.

This procedure pastes GLM: Multivariate command syntax.