Multivariate analysis of covariance
Multivariate analysis of covariance (MANCOVA) is an extension of analysis of covariance (ANCOVA) methods to cover cases where there is more than one dependent variable and where the control of concomitant continuous independent variables – covariates – is required. The most prominent benefit of the MANCOVA design over the simple MANOVA is the 'factoring out' of noise or error that has been introduced by the covariant. A commonly used multivariate version of the ANOVA F-statistic is Wilks' Lambda (Λ), which represents the ratio between the error variance (or covariance) and the effect variance (or covariance). Goals. Similarly  to all tests in the ANOVA family, the primary aim of the MANCOVA is to test for significant differences between group means. The process of characterising a covariate in a data source allows the reduction of the magnitude of the error term, represented in the MANCOVA design as "MS". Subsequently, the overall Wilks' Lambda will become larger and more likely to be characterised as significant. This grants the researcher more statistical power to detect differences within the data. The multivariate aspect of the MANCOVA allows the characterisation of differences in group means in regards to a linear combination of multiple  dependent variables, while simultaneously controlling for covariates. "Example situation where MANCOVA is appropriate:" Suppose a scientist is interested in testing two new drugs for their effects on depression and anxiety scores. Also suppose that the scientist has information pertaining to the overall responsivity to drugs for each patient; accounting for this covariate will grant the test higher sensitivity in determining the effects of each drug on both dependent variables. Assumptions. Certain assumptions must be met for the MANCOVA to be used appropriately: Logic of MANOVA. Analogous to ANOVA, MANOVA is based on the product of model variance matrix, formula_1 and  inverse of the error variance matrix, formula_2, or formula_3. The hypothesis that formula_4 implies that the product formula_5. Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis. The most common statistics are summaries based on the roots (or eigenvalues) formula_6 of the formula_7 matrix: Covariates. In statistics, a covariate represents a source of variation that has not been controlled in the experiment and is believed to affect the dependent variable. The aim  of such techniques as ANCOVA is to remove the effects of such uncontrolled variation, in order to increase statistical power and to ensure an accurate measurement of the true relationship between independent and dependent variables. An example is provided by the analysis of trend in sea-level by Woodworth (1987). Here the dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available. The primary independent variable was "time". Use was made of a "covariate" consisting of yearly values of annual mean atmospheric pressure at  sea level. The results showed that inclusion of the covariate allowed improved estimates of the trend against time to be obtained, compared to analyses which omitted the covariate.