Easy to find references from mixed-effects models, but I could really use a nice simple paper giving the model construction for one-way MANOVA with random effects showing expected mean squares and method-of-moments approach to estimating the between and within group covariance matrices.
I'm not sure if this would be helpful, but look at section 6.2 in Neil H. Timm, Applied Multivariate Analysis, Springer. Also Sect 5.3 of John Lawson, Design and Analysis of Experiments with R, CRC press. Best wishes.
(Sorry for my English in advance, I am not a native speaker and
only in case you still look for the formulas you asked for:
If you considered your model as a 'special' case of a mixed model (without mixing of effects) you might be able to transfer the formulas you need also from the references you have.
With best wishes...
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