# What are the consequences of not including random effects in a linear model when they should be added?

I am dealing with repeated measures data in which there is clearly reason to incorporate random effects to account for each subject having multiple measurements.

A mixed effects model using random intercepts fits my data nicely. I also ran the same model but without the random intercept, thereby making it a standard linear regression. I realized that the population level predictions (based on the fixed effects coefficients) are virtually identical between these two models (standard vs. mixed). Interestingly, however, the Beta coefficients are rather different between these two models.

In general, considering that I am interested in making population level predictions, is the negative consequence of failing to include random intercepts when appropriate that the parameter (Beta) estimates and their associated confidence intervals will be biased?

My understanding is that failure to include random intercepts will cause issues for the assumption of independence of observations in standard multiple regression.

• Does this partially answer the question? stats.stackexchange.com/questions/48720/…
– swmo
Mar 9, 2017 at 13:39
• When you say "without random effect", you mean you are using fixed effects for each subject or that you are not using subject ids at all? Mar 9, 2017 at 14:20
• I mean that I am not using subject ids at all. e.g. y ~ blah + (1/subject) vs. y ~ blah Mar 9, 2017 at 15:14
• Well, imagine that effect of subject is HUGE and the effect of blah is tiny. By not including subject into the model you will not notice any effect of blah. Mar 9, 2017 at 15:43

There are 2 questions here:

1. ...the population level predictions (based on the fixed effects coefficients) are virtually identical between these two models (standard vs. mixed). Interestingly, however, the Beta coefficients are rather different...

I find this hard to understand. If there are multiple predictors, it might be result of multicollinearity. This can cause different models to produce different coefficients but the same predictions.

1. ...considering that I am interested in making population level predictions, is the negative consequence of failing to include random intercepts when appropriate that the parameter (Beta) estimates and their associated confidence intervals will be biased?

Failing to account for non-independence (whether through random effects or some other method) should at the very least bias estimates of uncertainty. It is also very likely to bias parameter estimates as well. Having an identical number of observations per individual might reduce or eliminate this bias.

Finally, if you are only interested in population level patterns, then you might not want to use mixed models at all. Consider Simpson's paradox: in the image below would you be interested in the positive relationship on the left, or the negative relationship on the right? Image source: Tim Copeland at https://towardsdatascience.com/what-is-simpsons-paradox-4a53cd4e9ee2

Your choice should guide the analytical approach you take. The mixed model approach would capture the positive relationships on the left. But if you're interested in the pattern on the right, I think a linear model with HAC standard errors (to adjust the estimates of uncertainty) might meet your needs better.