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I understand that to run a multilevel model like

$ y_{ij} = \gamma_{00} + \gamma_{10}x_1{_{ij}} + u_{0j} + e_{ij} $

in lme4 I can use

lmer("y ~ 1 + x + (1 | group)", data = data)

or get the same result with 

lmer("y ~ x + (1 | group)", data = data)

That is, if I don't state what should be done, lme4 assumes I want the overall intercept. I noticed that I can obtain

$ y_{ij} = \gamma_{10}x_1{_{ij}} + u_{0j} + e_{ij} $

by running

lmer("y ~ 0 + x + (1 | group)", data = data)

At first I thought that if $\gamma_{00}$ was going to be close to 0 anyway it wouldn't make much difference whether I included it or not, but by analogy with this answer it seems like it would potentially bias the other other parameters even if the intercept is statistically non-significant.

I'm aware from this question that it can sometimes be appropriate to remove the intercept in regular regression, and so assume it can be good idea to also remove the intercept in multilevel modelling if we have some good reason to believe the data-generating process is forced to go through the origin.

But are there any reasons for removing the overall intercept that are specific to multilevel modelling?

user1205901 - Reinstate Monica
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  • Isn’t the answer same as for regular linear regression https://stats.stackexchange.com/q/7948/35989 ? – Tim May 22 '20 at 04:39
  • Thanks - I'd missed that post. I've edited the OP to take it into account. – user1205901 - Reinstate Monica May 22 '20 at 13:08
  • The first answer to the linked post says "The **shortest** answer: **never**", so it doesn't exactly say that "it can sometimes be appropriate". Any reason why you think the above shouldn't apply to multilevel models? – Tim May 22 '20 at 13:22
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    While the first answer only alludes in a general sense to instances in which it can be appropriate, the second most upvoted answer is quite different in tone, and gives 3 examples. I don't have any huge reason to think that things might be different in multilevel modelling, beyond the fact that many things seem different in MLM. – user1205901 - Reinstate Monica May 22 '20 at 22:54

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