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I am using a (Bayesian-based) mixed-effects regression model that incorporates both fixed effects and random effects.I have 3 fixed effects/covariates: 1) employment rate 2) crime rate 3) home ownership rate.

I have 2 questions:

1) Suppose we assume, I have only 2 covariates available: 1) employment rate 2) crime rate. That is, we assume covariate 3) home ownership rate was not available for whatever reason to be included in my regression. Would the use of random effect terms in my model account for the influence of unavailable covariates such as home ownership rate?

2) Would the exclusion of home ownership from the model (which we assume is relevant/significant to the dependent variable) make my interpretation of my results unreliable/misleading or would the inclusion of random effects handle this problem of excluded covariates and a potential 'incorrect'/unsound model.

(*NOTE: by 'results' I am referring to the influence/impact of regression coefficients from the 2 other variables: employment rate and crime rate on my dependent/outcome variable*)

user121
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1 Answers1

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First of all, if you are using Bayesian approach, then every parameters is considered as a random variable, so every effect is "random". The difference between "random" and "fixed" effects is more important in frequentist scenario, where we use different techniques for estimating them.

The standard linear regression model could be something like this:

$$\begin{align} y_{ij} &= \beta_0 + \beta_1 x_{ij} + \varepsilon_{ij} \\ \varepsilon_{ij} &\sim \mathcal{N}(0, \sigma^2_\varepsilon) \end{align}$$

where $\beta_0$ and $\beta_1$ are the "fixed" parameters and $\varepsilon_i$ is the noise, that is a random variable. You could expand this model for random intercept for each $j$-th group,

$$\begin{align} y_{ij} &= \beta_0 + u_j +\beta_1 x_{ij} + \varepsilon_{ij} \\ u_j &\sim \mathcal{N}(0, \sigma^2_u) \\ \varepsilon_{ij} &\sim \mathcal{N}(0, \sigma^2_\varepsilon) \end{align}$$

This means that we assume that $u_j$ is a random variable, so it has it's own "noise". Answering your question, random effect does not "account" for unobserved variables, at least not any more then the noise term in fixed-effects regression. It does model the variability better if your data is clustered (samples in $j$-th group are more similar to each other, then to other samples), but nothing more then this. So answering your second question, if you didn't observe some important variable for your model, random effects model would not do anything magical about it, you simply don't have it in your model. You didn't tell us more on what exactly the random effect would be, so it is hard to give more detailed comment on that.

Tim
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