<Dd> ε ∼ N (0, σ 2). (\ displaystyle \ varepsilon \ thicksim N (0, \ sigma ^ (2)). \,) </Dd> <P> In a randomized controlled experiment, the treatments are randomly assigned to experimental units, following the experimental protocol . This randomization is objective and declared before the experiment is carried out . The objective random - assignment is used to test the significance of the null hypothesis, following the ideas of C.S. Peirce and Ronald Fisher . This design - based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University . Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox . </P> <P> In its simplest form, the assumption of unit - treatment additivity states that the observed response y i, j (\ displaystyle y_ (i, j)) from experimental unit i (\ displaystyle i) when receiving treatment j (\ displaystyle j) can be written as the sum of the unit's response y i (\ displaystyle y_ (i)) and the treatment - effect t j (\ displaystyle t_ (j)), that is </P> <Dl> <Dd> y i, j = y i + t j . (\ displaystyle y_ (i, j) = y_ (i) + t_ (j).) </Dd> </Dl>

When is the analysis of variance procedure used