thumb|The image above depicts a visual comparison between multivariate analysis of variance (MANOVA) and univariate analysis of variance (ANOVA). In MANOVA, researchers are examining the group differences of a singular independent variable across multiple outcome variables, whereas in an ANOVA, researchers are examining the group differences of sometimes multiple independent variables on a singular outcome variable. In the provided example, the levels of the IV might include high school, college, and graduate school. The results of a MANOVA can tell us whether an individual who completed graduate school showed higher life AND job satisfaction than an individual who completed only high school or college. Results of an ANOVA can only tell us this information for life satisfaction. Analyzing group differences across multiple outcome variables often provides more accurate information as a pure relationship between only X and only Y rarely exists in nature.

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.

Model

Assume <math display="inline">n</math> <math display="inline">q</math>-dimensional observations, where the <math display="inline">i</math>’th observation <math display="inline">y_i</math> is assigned to the group <math display="inline">g(i)\in \{1,\dots,m\}</math> and is distributed around the group center <math display="inline">\mu^{(g(i))}\in \mathbb R^q</math> with multivariate Gaussian noise: <math display="block">

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,

</math> where <math display="inline">\Sigma</math> is the covariance matrix. Then we formulate our null hypothesis as

<math display="block">H_0\!:\;\mu^{(1)}=\mu^{(2)}=\dots =\mu^{(m)}.</math>

Relationship with ANOVA

MANOVA is a generalized form of univariate analysis of variance (ANOVA),

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 <math display="inline">S_{\text{model</math> and <math display="inline">S_{\text{res</math> by the same non-zero constant.

The most common statistics are summaries based on the roots (or eigenvalues) <math display="inline">\lambda_p</math> of the matrix <math display="inline">A:= S_{\text{modelS_{\text{res^{-1}</math>

  • Samuel Stanley Wilks' <math>\Lambda_\text{Wilks} = \prod_{1,\ldots,p}(1/(1 + \lambda_{p})) = \det(I + A)^{-1} = \det(S_\text{res})/\det(S_\text{res} + S_\text{model})</math> distributed as lambda (Λ)
  • the K. C. Sreedharan Pillai–M. S. Bartlett trace, <math>\Lambda_\text{Pillai} = \sum_{1,\ldots,p}(\lambda_p/(1 + \lambda_p)) = \operatorname{tr}(A(I + A)^{-1})</math>
  • the Lawley–Hotelling trace, <math>\Lambda_\text{LH} = \sum_{1,\ldots,p}(\lambda_{p}) = \operatorname{tr}(A)</math>
  • Roy's greatest root (also called Roy's largest root), <math>\Lambda_\text{Roy} = \max_p(\lambda_p) </math>

Discussion continues over the merits of each, 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.

Introducing covariates (MANCOVA)

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

See also

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

References