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.