MANCOVA

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).

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_error. 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.

Certain assumptions must be met for the MANCOVA to be used appropriately:

Analogous to ANOVA, MANOVA is based on the product of model variance matrix, $\Sigma _{\text{model}}$ $\Sigma _{\text{model}}$ and inverse of the error variance matrix, $\Sigma _{\text{res}}^{-1}$ $\Sigma _{\text{res}}^{-1}$ , or $A=\Sigma _{\text{model}}\times \Sigma _{\text{res}}^{-1}$ $A=\Sigma _{\text{model}}\times \Sigma _{\text{res}}^{-1}$ . The hypothesis that $\Sigma _{\text{model}}=\Sigma _{\text{residual}}$ $\Sigma _{\text{model}}=\Sigma _{\text{residual}}$ implies that the product $A\sim I$ $A \sim I$ . 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.

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