Should the difference between control and treatment be modelled explicitly or implicitly? Given the following experimental setup:
Multiple samples are taken from a subject and each sample is treated multiple ways (including a control treatment). What is mainly interesting is the difference between the control and each treatment.
I can think of two simple models for this data. With sample $i$, treatment $j$, treatment 0 being the control, let $Y_{ij}$ be the data, $\gamma_i$ be the baseline for sample $i$, $\delta_j$ be the difference for treatment $j$. The first model looks at both the control and difference:
$$
Y_{ij}=\gamma_i+\delta_j+\epsilon_{ij}
$$
$$
\delta_0=0
$$
Whilst the second model only looks at the difference. If we precalculate $d_{ij}$ beforehand
$$
d_{ij}=Y_{ij}-Y_{i0}
$$
then
$$
d_{ij}=\delta_j+\varepsilon_{ij}
$$
My question is what are the fundamental differences between these two setups? In particular, if the levels are meaningless in themselves and only the difference matters, is the first model doing too much and is perhaps underpowered?
 A: The $\epsilon_{ij}$ are likely to be correlated in the second model but not the first.
In the first, these terms represent measurement error and deviations from the additive model.  With reasonable care--such as by randomizing the sequence of measurements--those errors can be made independent when the model is accurate.  Whence
$$d_{ij} = Y_{ij} - Y_{i0} = \gamma_i + \delta_j + \epsilon_{ij} - (\gamma_i + \delta_0 + \epsilon_{i0}) = \delta_j + (\epsilon_{ij} - \epsilon_{i0}).$$
(Note that this contradicts the last equation in the question, because it is wrong to assume $\epsilon_{i0}=0$.  Doing so would force us to concede that the $\gamma_i$ are random variables rather than parameters, at least once we acknowledge the possibility of measurement error for the control.  This would lead to the same conclusions below.)
For $j, k \ne 0$, $j \ne k$ this implies
$$Cov(d_{ij}, d_{ik}) = Cov(\epsilon_{ij} - \epsilon_{i0}, \epsilon_{ik} - \epsilon_{i0}) = Var(\epsilon_{i0}) \ne 0.$$
The correlation can be substantial.  For iid errors, a similar calculation shows it equals 0.5.  Unless you are using procedures that explicitly and correctly handle this correlation, favor the first model over the second.
