A random sample of *N* experimental units is selected, and on each unit, *p* responses are taken. The responses may all be taken at the same time or may be staggered over a period of different time intervals. It is also possible to treat each of the *N* homogeneous experimental units with the same treatment at the beginning of the experiment and take the data at different time intervals, to determine the effect or absorption of the treatment over a time interval. The *j*th response on the *α*th experimental unit will be represented by *Y*_{αj} (*j* = 1, 2, …, *p*; *α* = 1, 2, …, *N*).

Let **Y**′_{α} = (*Y*_{α1}, *Y*_{α2}, …, *Y*_{αp}) be the *p*-dimensional response vector on the *α*th unit, and let *Y* = (*Y*_{αj}) be the *N* × *p* data matrix of all responses on all experimental units.

We assume **Y**_{α} ~ *IMN*_{p} (**μ**, ∑) for *α* = 1, 2, …, *N*, where **μ**′ = (*μ*_{1}, *μ*_{2}, …, *μ*_{p}) is the mean vector and ∑ is a positive definite dispersion matrix, **μ** and ∑ being unknown. The null hypothesis of interest in this setting is

The null hypothesis of (2.1.1) can be interpreted as the equality of the means of the *p* responses for the population from which the random sample of units are drawn. In the case of experimental setting where a treatment is given before the starting of the experiment, it will be interpreted as the equality of the effects over the periods for the given treatment implying ...

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