# Likelihood function for MANOVA

I'm trying to get a handle on MANOVA and Repeated Measures ANOVA. My confusion is around what the likelihood function looks like. Here's how I would formulate the likelihood function

• Notation:
• The dependent variable $Y: J$ x n matrix with j = 1,...,$J$ and i,...,n * j indexes dependent variables * i indexes observations

• Stochastic Component $$Y_i \sim N(\vec y_i|\vec \mu_i, \Sigma)$$

• where $\vec Y_i$ and $\vec \mu_i$ are $J$ x 1
• $\Sigma$ is $J$ x $J$

-Systematic Component $$\vec \mu_i =X_i\beta$$ -The likelihood is the product of multivariate normal distributions $$L(\mu,\sigma) = \prod_{i=1}^n N(\vec y_i|\vec \mu_i,\Sigma)$$ $$= \prod_{i=1}^n(2\pi )^{-\frac{J}{2}} |\Sigma|^{-\frac{1}{2}} exp \left[-\frac{1}{2}(\vec y_i-\vec \mu_i)'\Sigma^{-1}(\vec y_i-\vec \mu_i) \right]$$

• reparameterizing with $\mu_i = X\beta$

$$= \prod_{i=1}^n(2\pi )^{-\frac{J}{2}} |\Sigma|^{-\frac{1}{2}} exp \left[-\frac{1}{2}(\vec y_i-X\beta)'\Sigma^{-1}(\vec y_i-X\beta) \right]$$

Then depending on what is specified for $\Sigma$ e.g. compound symmetry vs unstructured leads to a repeated measures ANOVA or a MANOVA respectively. Is this correct?

Also, from what I've read this is usually parameterized in a different way, where a likelihood function of the conditional variance is formed. How is this different from what I wrote?