Instatistics,multivariate analysis of variance (MANOVA) is a procedure for comparingmultivariate sample means. As a multivariate procedure, it is used when there are two or moredependent variables,^[1] and is often followed by significance tests involving individual dependent variables separately.^[2]

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

Assume$n$$q$-dimensional observations, where the$i$’th observation$y_i$ is assigned to the group$g(i)\in \{1,\dots ,m\}$ and is distributed around the group center${\mu }^{(g(i))}\in {\mathbb{R}}^q$ withmultivariate Gaussian noise:$${\displaystyle y_i={\mu }^{(g(i))}+{\epsilon }_i{\epsilon }_i\stackrel{\text{i.i.d.}}{\sim }{\mathcal{N}}_q(0,\mathrm{\Sigma })\text{ for }i=1,\dots ,n,}$$ where$\mathrm{\Sigma }$ is thecovariance matrix. Then we formulate ournull hypothesis as$${\displaystyle H_0:{\mu }^{(1)}={\mu }^{(2)}=\cdots ={\mu }^{(m)}.}$$

MANOVA is a generalized form of univariateanalysis of variance (ANOVA),^[1] although, unlikeunivariate ANOVA, it uses thecovariance between outcome variables in testing the statistical significance of the mean differences.

Wheresums of squares appear in univariate analysis of variance, in multivariate analysis of variance certainpositive-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 abouterror distributions, the counterpart of the sum of squares due to error has aWishart distribution.

First, define the following$n\times q$ matrices:

• $Y$: where the$i$-th row is equal to$y_i$

• $\hat{Y}$: where the$i$-th row is the best prediction given the group membership$g(i)$. That is the mean over all observation in group$g(i)$:$\frac{1}{\text{size of group }g(i)}\sum _{k:g(k)=g(i)}{y}_k$.

• $\overline{Y}$: where the$i$-th row is the best prediction given no information. That is theempirical mean over all$n$ observations$\frac{1}{n}\sum _{k=1}^n{y}_k$

Then the matrix$S_{\text{model}}:=(\hat{Y}-\overline{Y}{)}^T(\hat{Y}-\overline{Y})$ is a generalization of the sum of squares explained by the group, and$S_{\text{res}}:=(Y-\hat{Y}{)}^T(Y-\hat{Y})$ is a generalization of theresidual sum of squares.^[3][4]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$S_{\text{model}}$ and$S_{\text{res}}$ by the same non-zero constant.

The most common^[3][5] statistics are summaries based on the roots (or eigenvalues)${\lambda }_p$ of the matrix$A:=S_{\text{model}}{S}_{\text{res}}^{-1}$

• Samuel Stanley Wilks'${\displaystyle {\mathrm{\Lambda }}_{\text{Wilks}}=\prod _{1,\dots ,p}(1/(1+{\lambda }_p))=det(I+A{)}^{-1}=det(S_{\text{res}})/det(S_{\text{res}}+S_{\text{model}})}$ distributed aslambda (Λ)
• theK. C. Sreedharan Pillai–M. S. Bartletttrace,${\displaystyle {\mathrm{\Lambda }}_{\text{Pillai}}=\sum _{1,\dots ,p}({\lambda }_p/(1+{\lambda }_p))=\mathrm{tr}(A(I+A{)}^{-1})}$^[6]
• theLawley–Hotelling trace,${\displaystyle {\mathrm{\Lambda }}_{\text{LH}}=\sum _{1,\dots ,p}({\lambda }_p)=\mathrm{tr}(A)}$
• Roy's greatest root (also calledRoy's largest root),${\displaystyle {\mathrm{\Lambda }}_{\text{Roy}}=\underset{p}{max}({\lambda }_p)}$

Discussion continues over the merits of each,^[1] 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 thenull 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 thenull hypothesis was derived in^[7] while the distribution under the alternative is studied in.^[8]

The best-knownapproximation for Wilks' lambda was derived byC. R. Rao.

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

One can also test if there is a group effect after adjusting for covariates. For this, follow the procedure above but substitute$\hat{Y}$ with the predictions of thegeneral linear model, containing the group and the covariates, and substitute$\overline{Y}$ with the predictions of the general linear model containing only the covariates (and an intercept). Then$S_{\text{model}}$ are the additional sum of squares explained by adding the grouping information and$S_{\text{res}}$ is the residual sum of squares of the model containing the grouping and the covariates.^[4]

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

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.^[9]

• Permutational analysis of variance for a non-parametric alternative
• Discriminant function analysis
• Canonical correlation analysis
• Multivariate analysis of variance (Wikiversity)
• Repeated measures design

Wikiversity has learning resources aboutMultivariate analysis of variance

• Analysis of variance
• Design of experiments

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