Why are the covariate terms suppressed in mixed effects model and ANOVA?

Question

I am used to seeing the linear mixed-effects model in the form: $$Y_{ij} = \mu + \alpha_i + \gamma_j + (\alpha\gamma)_{ij} + \epsilon_{ij}$$ assuming $ \gamma_j i.i.d \sim N(0,\sigma^2_{\gamma})$ and $ (\alpha\gamma)_{ij} i.i.d \sim N(0,\sigma^2_{\alpha\gamma})$

where $\alpha_i$ is the fixed effect of machine $i$ and $\gamma_j$ is the random effect of worker $j$ and $(\alpha\gamma)_{ij}$ is the corresponding random interaction. And assuming the data looks like below:

## Classes 'nffGroupedData', 'nfGroupedData', 'groupedData' and 'data.frame':   54 obs. of  3 variables:
##  $ Worker: Factor w/ 6 levels "1","2","3","4",..: 1 1 1 2 2 2 3 3 3 4 ...
##  $ Machine: Factor w/ 3 levels "A","B","C": 1 1 1 1 1 1 1 1 1 1 ...
##  $ y  : num  52 52.8 53.1 51.8 52.8 53.1 60 60.2 58.4 51.1 ...

and also I see ANOVA in the following form:

$$Y_{ij} = \mu + \alpha_i + \epsilon_{ij}$$

where $\alpha_i$ is known as the treatment effect and $\epsilon_{ij}\;iid\;\sim N(0,\sigma^2)$

I start to ask myself why is it that there are no covariates $x_{ij}$ anywhere in the both model, only the coefficients are there. Clearly although Worker and Machine are factors, they are still covariates and hence should appear as $x_{ij}$ somewhere in the model. I am sensing that I might have misunderstood something here. Why are the covariate terms suppressed in the model at the top ? How do I reconcile the notations in this case with linear regression model $Y_i = \beta_0 + \beta_1 X_1 + \epsilon_i$?

Please guide me here. Thanks.

Answer 1

I can understand your confusion. It really all depends on how you define your notation, and what background you have in these topics.

Models like this stem from 4 different strands of methodological research: hierarchical linear models (eg Bryk & Raudenbush, 1992), multilevel models (eg Snijders & Bosker, 2012) and mixed effects models (eg Pinheiro & Bates, 2000), and the analysis of planned experiments (eg ANOVA, random effects ANOVA, factorial experimens etc). The first two are very similar, but the first evolved mostly in the USA, whereas the 2nd evolved mostly in europe; the third is fairly ubiquitous, while the 4th can be thought of as a class of more general models of which the first thre are specific examples.

The notation of your first model is a textbook example of BOTH a two-factor factorial experiment (with two random factors) AND a two-factor mixed effects model (with one random factor)

$$Y_{ij} = \mu + \alpha_i + \gamma_j + (\alpha\gamma)_{ij} + \epsilon_{ij}$$

This model says we have 2 factors, machine and worker in your case, where 3, and 6 machines are workers, respecively, have been sampled from a wider population of such machines and workers. This would then be a variance components model. Of course the usual objections to the number of levels of both factors would apply. Alternatively (and this is the way you using it) we can say that machine is a fixed effect, and then we have a mixed effects model.

So we can see that using this notation, we are using the subsripts to denote the levels of each factor:

$$ \left\{\begin{array}{c} \begin{align} i&=1,2,3 \\ j&=1,2, \dots,6 \\ \end{align} \end{array}\right. $$

That is the usual type of notation in experimental design analysis, hence the similar notation in your ANOVA model: $$ Y_{ij} = \mu + \alpha_i + \epsilon_{ij} $$

However, in the last model you wrote:

$$ Y_i = \beta_0 + \beta_1 X_1 + \epsilon_i $$

the assumption is that $X_1$ is the name of continuous variable, not a level 1 of a factor called $X$ . If it were a factor then we could use the notation above, or we could create seperate dummy variables.