# Does “random effect” really exist in real data when we use random/mixed effect model? [closed]

If I understand correctly, here is a standard case when we need the mixed effect model:

We are interested in studying the how drugs influence human health conditions, so we collected information about many patients' drug usage, health condition, and other factors such as demographical information to study. We build a linear mixed model:

$$y = X\beta + Zu + \epsilon$$

where $X$ stands for the covariates about drugs, $Z$ stands for other demographical information, $\beta$ stands for fixed effects, $u$ stands for random effects and $u\sim N(0, \sigma_u^2)$, and $\epsilon$ stands for the residue errors.

My question is about the fact that demographical information does not influence health conditions randomly. In reality, it should be a fixed effect. Since there are many different definitions of fixed and random effects, as given by John Salvatier's answer, let me clarify a little bit: In reality, I believe the data should be: $$y=X\beta + Z\alpha + \epsilon$$ where both $\alpha$ and $\beta$ are fixed, so the var$(y)$ should be as simple as $\sigma_\epsilon^2$.

I do not object to the fact of applying mixed models to these data, but my question is that in many statistic papers I read so far, when researchers talk about the consistency or bias/variance of fitting mixed models, it looks like they mostly assume the underlying data distribution to be a mixed effect model, then since random effects may not even exist in the real world, I don't understand the practical value of these analyses.

1. Of course, all these analyses have research values as they spread new findings/ideas/methods, but does it sound like they are analyzing a problem that does not exist?

2. Are there papers that discuss the statistical properties of the mixed model when they believe the underlying true data distributions are all fixed effects?

• Your quote does not look like "a standard case" for using mixed models. Mixed aka hierarchical models are used when there are some clusters/hierarchies in the data. I don't see anything like that in your description. The whole Q is unclear to me. – amoeba Jul 13 '18 at 21:15
• How about let's say $Z$ stands for gender, then male/female forms two clusters? – Haohan Wang Jul 13 '18 at 21:26
• This would be fixed effect. Your whole question seems to be based on your confusion about what a random effect is. – amoeba Jul 13 '18 at 21:34
• Hmm, my main confusion is about what happens if we model fixed effect as random effect. I'm pretty sure there are some works doing so, but I'm not sure if they can be called "standard case". – Haohan Wang Jul 13 '18 at 21:45
• I wouldn't think most demographic information would be used as a random effect. Maybe something like county of residence... – Sal Mangiafico Jul 13 '18 at 22:12