# What is an appropriate formula for residuals calculation in a model describing a single-factor repeated measures design?

For the repeated measures design D.C.Montgomery in his "Design and Analysis of Experiments" book provides the mathematical / statistical (linear) model (with changes): $y_{ij} = \mu + \beta_i + \tau_j + \epsilon_{ij}$, where $\beta$ is a parameter associated with the $i$th subject, $\tau$ is the effect of the $j$th treatment, and $\epsilon$ is the independent error.
In Weixing Song's document it is stated that to check the model's assumptions we should operate with residuals calculated as (A): $$\hat\epsilon_{ij} = Y_{ij} - \bar Y_{i.} - \bar Y_{.j} + \bar Y.$$

The formula (B) on Wikipedia is $$\hat\epsilon_i = Y_i - \bar Y.$$

My question is:
what formula is used to correctly calculate residuals in the model describing a single-factor repeated measures design to check its assumptions (multivariate normality, homoscedasticity, sphericity, linearity, additivity)?

Thank you.

P.S.
E.Vonesh and V.M.Chinchilli in their "Linear and Nonlinear Models for the Analysis of Repeated Measurements" monograph state that the model for single-sample RM design may be written as $y_{i} = \mu + \epsilon_{i}$.

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Looks like you changed your question half way through the last sentence. Are you asking a yes no question (starts with 'does') or a 'why' question? – Peter Ellis Dec 18 '12 at 18:57
@Peter Ellis: thanks for the comment! I've edited. – stan Dec 18 '12 at 19:15
I've removed the CW status for this question (if you don't want this to happen again try to group your multiple edits). – chl Dec 26 '12 at 9:00

If I understand the notation correctly here, $\bar Y$ is the overall mean. So $Y_{i} - \bar Y$ will give you the difference between an individual's score and the overall mean but does not relate at all to the treatment effect. So no, you can't use formula B to estimate residuals from the model you have specified.
Dear @Peter Ellis, I think you consider $y_{j} = \mu + \beta_j + \epsilon_{j}$ under "a simpler model... with no treatment effect". I'm not sure but I guess that RM designs were developed to reveal changes in the very treatment factor rather than among subjects (i.e. "random component"). So perhaps the simpler model should look like $y_{i} = \mu + \tau_i + \epsilon_{i}$? And as far as I understand eventually I have to recheck assumptions calculated as $\hat\epsilon_i = Y_i - \bar Y$? – stan Dec 22 '12 at 6:22