The hypothesis of symmetry

As in Chapter 3 the basic problem is the comparison of a new treatment with a standard, but in contrast to Chapter 3 we don’t have two independent samples. Usually we have paired observations (Y _i , Z _i ), i = 1,..., n, where the Y-components are the measurements under the standard treatment whereas the Z -components are the measurements under the new treatment. It’s assumed that the design of the experiment allows the pairs (Y₁, Z₁),..., (Y _n , Z _n ) to be independent random variables with values in ℝ2. It’s not realistic to assume the stochastic independence of the components Y _i and Z _i , since the Y _i -measurement and the Z _i -measurement are usually taken at matched pairs or even at the same experimental unit, e.g. left-right treatments or pre-post treatments, but we assume the differences Z₁ - Y₁,..., Z _n - Y _n to be i.i.d. random variables.