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Multivariate analysis of variance

Procedure for comparing multivariate sample means

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated June 23, 2025
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In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests involving individual dependent variables separately.

Without relation to the image, the dependent variables may be k life satisfactions scores measured at sequential time points and p job satisfaction scores measured at sequential time points. In this case there are k+p dependent variables whose linear combination follows a multivariate normal distribution, multivariate variance-covariance matrix homogeneity, and linear relationship, no multicollinearity, and each without outliers.

01Model

Assume {\textstyle n {\textstyle q-dimensional observations, where the {\textstyle i’th observation {\textstyle y_{i} is assigned to the group {\textstyle g(i)\in \{1,\dots ,m\} and is distributed around the group center {\textstyle \mu ^{(g(i))}\in \mathbb {R} ^{q} with multivariate Gaussian noise: y_{i}=\mu ^{(g(i))}+\varepsilon _{i}\quad \varepsilon _{i}{\overset {\text{i.i.d.}}{\sim }}{\mathcal {N}}_{q}(0,\Sigma )\quad {\text{ for }}i=1,\dots ,n, where {\textstyle \Sigma is the covariance matrix. Then we formulate our null hypothesis as H_{0}\!:\;\mu ^{(1)}=\mu ^{(2)}=\dots =\mu ^{(m)}.

This is a graphical depiction of the required relationship amongst outcome variables in a multivariate analysis of variance. Part of the analysis involves creating a composite variable, which the group differences of the independent variable are analyzed against. The composite variables, as there can be multiple, are different combinations of the outcome variables. The analysis then determines which combination shows the greatest group differences for the independent variable. A descriptive discriminant analysis is then used as a post hoc test to determine what the makeup of that composite variable is that creates the greatest group differences.
This is a graphical depiction of the required relationship amongst outcome variables in a multivariate analysis of variance. Part of the analysis involves creating a composite variable, which the group differences of the independent variable are analyzed against. The composite variables, as there can be multiple, are different combinations of the outcome variables. The analysis then determines which combination shows the greatest group differences for the independent variable. A descriptive discriminant analysis is then used as a post hoc test to determine what the makeup of that composite variable is that creates the greatest group differences.

02Relationship with ANOVA

MANOVA is a generalized form of univariate analysis of variance (ANOVA), although, unlike univariate ANOVA, it uses the covariance between outcome variables in testing the statistical significance of the mean differences.

Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.

03Hypothesis Testing

First, define the following {\textstyle n\times q matrices:

  • {\textstyle Y: where the {\textstyle i-th row is equal to {\textstyle y_{i}
  • {\textstyle {\hat {Y}}: where the {\textstyle i-th row is the best prediction given the group membership {\textstyle g(i). That is the mean over all observation in group {\textstyle g(i): {\textstyle {\frac {1}{{\text{size of group }}g(i)}}\sum _{k:g(k)=g(i)}y_{k}.
  • {\textstyle {\bar {Y}}: where the {\textstyle i-th row is the best prediction given no information. That is the empirical mean over all {\textstyle n observations {\textstyle {\frac {1}{n}}\sum _{k=1}^{n}y_{k}

Then the matrix {\textstyle S_{\text{model}}:=({\hat {Y}}-{\bar {Y}})^{T}({\hat {Y}}-{\bar {Y}}) is a generalization of the sum of squares explained by the group, and {\textstyle S_{\text{res}}:=(Y-{\hat {Y}})^{T}(Y-{\hat {Y}}) is a generalization of the residual sum of squares. Note that alternatively one could also speak about covariances when the abovementioned matrices are scaled by 1/(n-1) since the subsequent test statistics do not change by multiplying {\textstyle S_{\text{model}} and {\textstyle S_{\text{res}} by the same non-zero constant.

The most common statistics are summaries based on the roots (or eigenvalues) {\textstyle \lambda _{p} of the matrix {\textstyle A:=S_{\text{model}}S_{\text{res}}^{-1}

Discussion continues over the merits of each, although the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that, except for the Roy's greatest root, the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases. An algorithm for the distribution of the Roy's largest root under the null hypothesis was derived in while the distribution under the alternative is studied in.

The best-known approximation for Wilks' lambda was derived by C. R. Rao.

In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square.

This is a simple visual representation of the effect of two highly correlated dependent variables within a MANOVA. If two (or more) dependent variables are highly correlated, the chances of a Type I error occurring is reduced, but the trade-off is that the power of the MANOVA test is also reduced.
This is a simple visual representation of the effect of two highly correlated dependent variables within a MANOVA. If two (or more) dependent variables are highly correlated, the chances of a Type I error occurring is reduced, but the trade-off is that the power of the MANOVA test is also reduced.

04Introducing covariates (MANCOVA)

One can also test if there is a group effect after adjusting for covariates. For this, follow the procedure above but substitute {\textstyle {\hat {Y}} with the predictions of the general linear model, containing the group and the covariates, and substitute {\textstyle {\bar {Y}} with the predictions of the general linear model containing only the covariates (and an intercept). Then {\textstyle S_{\text{model}} are the additional sum of squares explained by adding the grouping information and {\textstyle S_{\text{res}} is the residual sum of squares of the model containing the grouping and the covariates.

Note that in case of unbalanced data, the order of adding the covariates matters.

05Correlation of dependent variables

MANOVA's power is affected by the correlations of the dependent variables and by the effect sizes associated with those variables. For example, when there are two groups and two dependent variables, MANOVA's power is lowest when the correlation equals the ratio of the smaller to the larger standardized effect size.

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Sources and credits

This article is adapted from the Wikipedia article “Multivariate analysis of variance”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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