I'm not clear on why we use random effects. I see two cases: Either we do it because we want to model the heterogeneity of cluster or group-level effects within a population sample. Or, I see it chosen out convenience: Because we don't care about individual cluster or group-level effects, and thus using random effects and only modeling a single coefficient for the random effect is more parsimonious and therefore leads to a better model in principle.

For example, in Introduction to Regression, Montgomery says:

The units used for a specific random effect represent a random sample from a much larger population of possible units. For example, patients in a biomedical study often are random effects. The analyst selects the patients for the study from a large population of possible people. The focus of all statistical inference is not on the specific patients selected; rather, the focus is on the population of all possible patients. The key point underlying all random effects is this focus on the population and not on the specific units selected for the study.

However, I also see random effects simply described in terms of principles of simplicity:

If interest lies in mean intercept and slope (α, β) and sex difference δs but not individual subjects then wasteful to include subject specific fixed effects ai and bi

So my question is: Is it appropriate to consider an effect "random" if we have included all group-level factors (all groups)? In this case, we would just be modeling the heterogeneity of the groups as defined by the subsamples and their sampling error, rather than the sampling error of the groups themselves, if that makes sense. Is this still a "random" effect?

I'm still getting my head around this so apologies if that doesn't make sense.

I'm not clear on why we use random effects. I see two cases: Either we do it because we want to model the heterogeneity of cluster or group-level effects within a population sample. Or, I see it chosen out convenience: Because we don't care about individual cluster or group-level effects, and thus using random effects and only modeling a single coefficient for the random effect is more parsimonious and therefore leads to a better model in principle.

For example, in Introduction to Regression, Montgomery says:

The units used for a specific random effect represent a random sample from a much larger population of possible units. For example, patients in a biomedical study often are random effects. The analyst selects the patients for the study from a large population of possible people. The focus of all statistical inference is not on the specific patients selected; rather, the focus is on the population of all possible patients. The key point underlying all random effects is this focus on the population and not on the specific units selected for the study.

However, I also see random effects simply described in terms of principles of simplicity:

If interest lies in mean intercept and slope $(\alpha, \beta)$ and sex difference $δ$s but not individual subjects then wasteful to include subject specific fixed effects $a_i$ and $b_i$.

So my question is: Is it appropriate to consider an effect "random" if we have included all group-level factors (all groups)? In this case, we would just be modeling the heterogeneity of the groups as defined by the subsamples and their sampling error, rather than the sampling error of the groups themselves, if that makes sense. Is this still a "random" effect?