**Authors: **Sami D. Gabbara & Evar L. Sadraddin & Namam J. Mahmoud

**Abstract: ** Previously a general method to analyze a nested repeated measure model was developed when the covariance matrix had a certain pattern. A case of being a number of sub-individuals of a particular individual, such as sub-field or other types of offsprings, receives several treatments. As a consequence, the observations are correlated with certain covariance matrix pattern and such a model is known as nested repeated measures model (NRMM). In this paper, a weaker assumption is used when the covariance matrix is arbitrary and has no specific pattern. Independent normally distributed individuals are taken with their own mean and common positive definite covariance matrix. It is aimed to test hypotheses about the mean. Two techniques are used for testing. The first is based on the multivariate one sample model (MOSM), when each individual receives the same treatments and hence has the same mean vector, whilst the second is based on the multivariate linear model (MLM). Different individuals receive different treatments and hence have different mean vectors. For each technique a uniformly most powerful (UMP) invariant size test is found.

**Keywords: **Multivariate Linear Model, Multivariate One Sample Model, Nested Repeated Measures Model

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doi: 10.23918/eajse.v4i2p82

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