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Suppose that we have a response variable that is measured on each subject. There are 2 categorical variables (factors): one between-subject factor Group with two levels (control and treatment) and one within-subject factor Category with three levels (high, moderate, and low). Such a 2 $\times$ 3 mixed ANOVA can be analyzed through a general linear model (GLM) by dummy coding the factors.

To demonstrate the GLM formulation, assume there are six subjects, three in each group, and we have the following equation with effect coding:

$$ \begin{bmatrix} y_{11} \\ y_{12} \\ y_{13} \\ y_{21} \\ y_{22} \\ y_{23} \\ y_{31} \\ y_{32} \\ y_{33} \\ y_{41} \\ y_{42} \\ y_{43} \\ y_{51} \\ y_{52} \\ y_{53} \\ y_{61} \\ y_{62} \\ y_{63} \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0\\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0\\ 1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & -1 & -1 & -1 & -1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & -1 & -1 & 0 & 0\\ 1 & 1 & 0 & 1 & 0 & 1 & -1 & -1 & 0 & 0\\ 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 0 & 0 \\ 1 & -1 & 1 & 0 & -1 & 0 & 0 & 0 & 1 & 0 \\ 1 & -1 & 0 & 1 & 0 & -1 & 0 & 0 & 1 & 0\\ 1 & -1 & -1 & -1 & 1 & 1 & 0 & 0& 1 & 0 \\ 1 & -1 & 1 & 0 & -1 & 0 & 0 & 0& 0 & 1 \\ 1 & -1 & 0 & 1 & 0 & -1 & 0 & 0& 0 & 1 \\ 1 & -1 & -1 & -1 & 1 & 1 & 0 & 0& 0 & 1 \\ 1 & -1 & 1 & 0 & -1 & 0 & 0 & 0& -1 & -1 \\ 1 & -1 & 0 & 1 & 0 & -1 & 0 & 0& -1 & -1 \\ 1 & -1 & -1 & -1 & 1 & 1 & 0 & 0& -1 & -1 \end{bmatrix}\\ \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \\ \beta_5 \\ \beta_6 \\ \beta_7 \\ \beta_8 \\ \beta_9 \end{bmatrix}+\begin{bmatrix} \epsilon_{11} \\ \epsilon_{12} \\ \epsilon_{13} \\ \epsilon_{21} \\ \epsilon_{22} \\ \epsilon_{23} \\ \epsilon_{31} \\ \epsilon_{32} \\ \epsilon_{33} \\ \epsilon_{41} \\ \epsilon_{42} \\ \epsilon_{43} \\ \epsilon_{51} \\ \epsilon_{52} \\ \epsilon_{53} \\ \epsilon_{61} \\ \epsilon_{62} \\ \epsilon_{63} \end{bmatrix} $$

The $F$-statistic for each effect (e.g., main effects and interactions) is formulated with terms each of which is associated with a partial model with some columns removed from the model matrix.

Suppose I would like to also consider each subject's body weight as a covariate. I barely see any discussions of modeling such a covariate under the GLM framework. This is not really an ideal approach because it would presume that all the three categories have the same body weight effect. So what is exactly the difficulty here when a covariate is incorporated in a GLM? Broken orthogonality or something else?

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This is a repeated measures design, not a vanilla option GLM. You can add a continuous covariate, which does what covariates do. It is difficult to picture what exactly the covariate is doing in terms of the columns of your matrix. Imagine, however, that you are modelling the residuals of your response with respect to the covariate, on the one hand, against the residuals of the dummy variables (on the other) against the covariate (on the other hand).

Covariates usually do break orthogonality, since they are often measurements that the experimenter observers, rather than levels that are preset. They raise all the usual theoretical issues about "correlation not being causality", confounders, and everything else. This does not prevent people from including covariates in repeated measures models.

I am sure that any decent textbook on repeated measures would discuss these issues.

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  • $\begingroup$ Thanks for the help! I've found only one source where it discusses the issue of modeling a covariate in a mixed ANOVA. It argues that a covariate would not have any impact on the repeated-measures factor, and the author suggests the following: Do 2 separate analyses, one with the between-subjects factor plus the covariate, and the other with the repeated-measures factor but no covariate. Why such a kludge approach? Such suggestion made me think that there might be some more serious difficulty in modeling a covariate under GLM. Any books disucssing the issues? $\endgroup$ – bluepole Jun 5 '14 at 22:59

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