# [R-sig-ME] meaning of lmer formula

Ben Bolker bbolker at gmail.com
Thu Sep 7 04:07:53 CEST 2017

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On 17-09-06 05:17 PM, Don Cohen wrote:
> Ben Bolker writes:
>
>  > > Does (input1 + input2 | group) mean the same as
>  > > (input1 | group) + (input2 | group) or does it also include a
>  > > correlation between input1 and input2 ?
>  >   The latter.
>
> It occurred to me after I wrote the questions that
> (input1 | group) + (input2 | group) might also include that
> correlation between input1 and input2 - I gather now it does not.

It does not.  Separate terms in lme4 formulas are always independent.

(input1|group) + (input2|group) is problematic because both terms
include an intercept.  (0+input2|group) can be helpful, but doesn't do
what you think when the variable on the LHS (e.g. input2) is a factor.
>
>  > > Back to my question about additive shifts, is that extra correlation
>  > > required in order to maintain invariance over additive shifts?
>  >   Yes.
>
> One more question (so far):
> If there are two different groupings, would
>  (input1 + input2 | group1) + (input1 + input2 | group2)
> involve correlations between group1 and group2 ?

No.

>
> If I think of (1+input1 | group) as a 2x2 matrix, and
> (1+input1+input2 | group) as 3x3
> then is (1+input1+input2 | group1)+(1+input1+input2 | group2)
> two independent 3x3's or a full 6x6 (or something else) ?
>

If input1 and input2 are both numeric (or 2-level factors) then they're
two independent 3x3s.

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