 Multivariate analysis of variance

Multivariate analysis of variance compares mean vectors of several groups while accounting for correlations between their response variables. In the one-way normal model, independent observations $Y_{ri}\sim N_p({\mu }_r,\Sigma )$ share a positive-definite covariance matrix. Put $E={\sum }_{r,i}(Y_{ri}-{\bar{Y}}_r)(Y_{ri}-{\bar{Y}}_r{)}^T$ and $H={\sum }_r{n}_r({\bar{Y}}_r-\bar{Y})({\bar{Y}}_r-\bar{Y}{)}^T$. Under equal means, these matrices have independent Wishart distributions with degrees of freedom $N-g$ and $g-1$, respectively. This follows by separating the Gaussian data into orthogonal within-group and between-group projections. The Wilks lambda statistic tests equality of the means using both matrices, rather than independently testing each coordinate.

Multivariate Analysis of Variance (MANOVA) is a statistical technique used to assess whether there are any statistically significant differences between the means of multiple dependent variables across different groups or levels of one or more independent variables. It is essentially an extension of Analysis of Variance (ANOVA), which deals with a single dependent variable.