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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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  • $\begingroup$ 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? $\endgroup$
    – Peter Ellis
    Commented Dec 18, 2012 at 18:57
  • $\begingroup$ @Peter Ellis: thanks for the comment! I've edited. $\endgroup$
    – abc
    Commented Dec 18, 2012 at 19:15

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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.

On the other hand, formula B would estimate residuals from a simpler model - one with no treatment effect in it - if you are interested in the random component of such a model. As a lot of inference is done with this simpler model as the null hypothesis, there is an argument for paying considerable attention to the distribution of the residuals of this simpler model eg checking for non-breach of assumptions of normality, constant variance, etc (if your model has such assumptions - they are usual but you have not made them explicit here).

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  • $\begingroup$ 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 $\hat\epsilon_i = Y_i - \bar Y$? $\endgroup$
    – abc
    Commented Dec 22, 2012 at 6:22
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    $\begingroup$ Formula B only gives the residuals if the model has no reference to treatment at all. Obviously this would be no good to fit as the only model we test (or, as you say, why bother with the experiment). As I said in the answer, sometimes this is our null hypothesis (treatment has no effect) so it should be taken seriously, but it isn't the model we think that best fits the data unless the treatment really has no impact. $\endgroup$
    – Peter Ellis
    Commented Dec 22, 2012 at 6:25

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