[R-sig-ME] Adding Level for non-repeated measurements

[Viechtbauer, Wolfgang (SP)]

Dear Joe,

Meta-analysis is different. In a meta-analysis, the sampling variances (one per estimate) are pre-specified and this allows us to add a random effect corresponding to each estimate to the model. In a multilevel model with a normally distributed response variable, you cannot do this. Well, you can do this, but this random effect is the same as the error term and hence completely confounded.

Best,
Wolfgang

>-----Original Message-----
>From: Tip But [mailto:fswfswt using gmail.com]
>Sent: Friday, 19 March, 2021 18:06
>To: David Duffy
>Cc: r-sig-mixed-models; Viechtbauer, Wolfgang (SP)
>Subject: Re: [R-sig-ME] Adding Level for non-repeated measurements
>
>Dear David,
>
>Thank you for your response. As my toy example showed, we do have a normally
>distributed response variable.
>
>As to 1), I have seen (e.g., see variable `id` in:
>https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2018-July/000896.html) that
>what you refer to as "individual-specific" random-effects are used in, for
>example, multi-level meta-regression models with a normally distributed response
>variable.
>
>In the context of multi-level meta-regression models with a normally distributed
>response variable, the addition of "effectSize-specific" (="individual-specific")
>random-effects often account for the variation at the level of individual
>estimates of effect size. That is: "effectSize ~ 1 + (1 | studyID / effectSizeID)"
>where the data looks like:
>
>studyID      effectSizeID       effectSize
>1                   1                        .2
>1                   2                        .1
>2                   3                        .4
>3                   4                        .3
>3                   5                        .6
>.                    .                          .
>.                    .                          .
>.                    .                          .
>
>So, I reasoned if  "(1 | studyID / effectSizeID)" is possible in the context of
>multi-level meta-regression models with a normally distributed response variable,
>then,  "(1 | sch_id / stud_id)" is possible in the context of multi-level models
>with a normally distributed response variable where the data looks like:
>
>sch_id       stud_id             score
>1                   1                        9
>1                   2                        6
>2                   3                        8
>3                   4                        5
>3                   5                        3
>.                    .                          .
>.                    .                          .
>.                    .                          .
>### Is my reasoning flawed here?
>
>As to 2), I can certainly allow the variances in each "sch_id" to be different.
>But does this address the correlations among students in each school, correct?
>
>Many thanks,
>Joe
>
>On Fri, Mar 19, 2021 at 2:57 AM David Duffy <David.Duffy using qimrberghofer.edu.au>
>wrote:
>Joe wrote:
>
>> I have a cross-sectional (i.e., non-repeated measurements) dataset from
>> students ("stud_id") nested within many schools ("sch_id").
>> 1- Given above, should we possibly add an additional random-effect for
>> "stud_id"? If yes, why?
>> 2- Given above, should we also allow residuals in each school (e_ij) to
>> correlate? If yes, why? (I have a bit of a conceptual problem understanding
>> this part given the cross-sectional nature of our study.)
>
>I think this is more a slightly-harder-than-elementary stats question rather than
>a "technical" query. If this was some types of
>GLMM, then the answer to 1 would be yes eg poisson GLMM then an individual-
>specific random effect adds in one type of
>extra-poisson variation. This is not the case for the gaussian (hopefully you see
>why). As to 2, consider how the *variance* of your
>measurement could be different within each school.