Can random effects apply only to categorical variables?

Question

This questions might sound stupid, but... is it correct that random effects could apply only to categorical variables (like individual id, population id, ...), e.g. say $x_i$ is categorical variable:

$y_i$ ~ $\beta_{x_i}$

$\beta_{x_i}$ ~ $Norm(\mu, \delta^2)$

but from the principle the random effect cannot apply to continuous variable (like height, mass...), say $z_i$:

$y_i$ ~ $\alpha + \beta \cdot z_{i}$

because then there is only one coefficient $\beta$ which cannot be constrained? Sounds logical but I wonder why it is never mentioned in statistical literature! Thanks!

EDIT: But what if I constrain $z_i$ like $z_i$ ~ $Norm(\mu, \delta^2)$? Is it then random effect? But this is different from the constrain I put on $\beta_{x_i}$ - here I constrain the variable whereas in the previous example I constrained the coefficient! It starts to look as a big mess to me... Anyway, it doesn't make much sense to put this constraint, because $z_i$ are known values, so maybe this idea is completely odd :-)

Answer 1

This is a good and a very basic question.

The interpretation of random effects is very domain-specific and is dependent on the modeling choice (the statistical model or being a Bayesian or frequentist). For a very good discussion, see page 245, Gelman and Hill (2007). For a Bayesian everything is random (even though parameters may have a true fixed value, they are modeled as random), and a frequentist can also choose a parameter value to be a fixed effect that would have been otherwise modeled as random (see Casella, 2008, discussion about blocks to be fixed or random in example 3.2).

Edit (after comment)

Data are fixed after you observe them. If they are continuous, they should be modeled as continuous. You can model categorical variables as categorical and sometimes as continuous (like in an ordinal variable setting). The parameters are unknown and they may be modeled as fixed or random. The parameters essentially relate response to predictors. If you want individual predictor's slope (or its coefficient in a linear model) to vary for each response, model it as random, otherwise model it as fixed. Similarly, if you want the intercept to vary regarding groups, then they should be modeled as random; otherwise they should be fixed.