[R-meta] Alternative view of fixed effects in meta-regression

Norman DAURELLE norm@n@d@ure||e @end|ng |rom @grop@r|@tech@|r
Thu Sep 2 09:29:10 CEST 2021 

Dear Fred, 

thank you for this example. 

Best wishes, 
Norman 


De: "Farzad Keyhan" <f.keyhaniha using gmail.com> 
À: "Lukasz Stasielowicz" <lukasz.stasielowicz using uni-osnabrueck.de> 
Cc: "r-sig-meta-analysis" <r-sig-meta-analysis using r-project.org> 
Envoyé: Samedi 28 Août 2021 21:55:31 
Objet: Re: [R-meta] Alternative view of fixed effects in meta-regression 

Dear Lukasz, 

The post you linked (had a fleeting look at it) is demonstrating my 
logic for a 2-level model i.e. random effects of the form: ~1 | ID. In 
that case, a single within and between is sufficient. But in the case 
of Tim's question, we have a 3-level model i.e. random effects of the 
form: ~ 1 | ID1/ID2 hence, additional betweens and withins. 

To build some intuition, say you have a big bag with 5 marbles in it, 
3 red ones, and 2 blue ones. 

The mean of the entire bag gives you the bag-level mean 
("X_btw_study"). The means of color-specific marbles gives you 
color-level means ("X_btw_outcome") in the big bag. 

Now, each marble --regardless of its color-- can differ from the mean 
of the big bag ("X_wthn_study"). Also, each marble --given its color-- 
can differ from its own color-level mean ("X_wthn_outcome"). 

Finally, color-level means can differ from the big bag mean 
("X_btw_outcome_study"). 

This is how my line of reasoning works. But certainly 
correction/feedback/comments are more than welcome. 

Best, 
Fred 

On Sat, Aug 28, 2021 at 2:17 PM Lukasz Stasielowicz 
<lukasz.stasielowicz using uni-osnabrueck.de> wrote: 
> 
> Dear Fred, 
> 
> isn't it sufficient to include two variables rather than four variables 
> when disentangling within-group-effects and between-group-effects? 
> Some references: 
> 
> *Bell, A., Fairbrother, M. & Jones, K. Fixed and random effects models: 
> Making an informed choice. Qual Quant 53, 1051–1074 (2019). 
> https://doi.org/10.1007/s11135-018-0802-x 
> 
> *https://strengejacke.github.io/mixed-models-snippets/random-effects-within-between-effects-model.html#the-complex-random-effect-within-between-model-rewb 
> 
> 
> According to the cited literature one could include two variables in 
> multilevel models: X_within_group and X_btw_group 
> X_btw_group refers to the group mean (e.g., mean age in the study j: x_j) 
> X_within_group refers to the difference between each observation and its 
> group mean (x_ij - x_j). 
> 
> 
> Best, 
> 
> Lukasz 
> -- 
> Lukasz Stasielowicz 
> Osnabrück University 
> Institute for Psychology 
> Research methods, psychological assessment, and evaluation 
> Seminarstraße 20 
> 49074 Osnabrück (Germany) 

> > Date: Fri, 27 Aug 2021 20:31:55 -0500 
> > From: Farzad Keyhan <f.keyhaniha using gmail.com> 
> > To: Timothy MacKenzie <fswfswt using gmail.com> 
> > Cc: R meta <r-sig-meta-analysis using r-project.org> 
> > Subject: Re: [R-meta] Alternative view of fixed effects in 
> > meta-regression 
> > Message-ID: 
> > <CAEvy2r2yqKiR4UkL=9PNDC2n4c78sawxL+PXODzELCEoWnyfqQ using mail.gmail.com> 
> > Content-Type: text/plain; charset="utf-8" 
> > 
> > Dear Tim, 
> > 
> > Unconditional 3-level models (i.e., models with no moderator) fit by 
> > --rma.mv()-- assume: (A) normality of individual effects within 
> > studies, (B) normality of level-specific effects, and that (C) the 
> > relationship among the effects at each level is univariate linear. 
> > 
> > (If your model is a multivariate one, then those relationships are 
> > assumed to be multivariate linear). 
> > 
> > Applying these assumptions to the model that you referred to (j cases 
> > nested in k studies) would mean that the potential linear relationship 
> > between case-specific effects can be estimated by adding a moderator 
> > (e.g., --Age_jk--) that can vary at the case level. 
> > 
> > Now, if you add a moderator that varies among the cases, then, your 
> > fixed-effect coefficient for --Age_jk-- would detonate the amount of 
> > change in case-specific true effects (which are averages of individual 
> > effect sizes for each case) relative to 1 year increase in --Age_jk--. 
> > 
> > Or equivalently: “the difference in average effect sizes between cases 
> > that differ in age by one year”. 
> > 
> > So, you can add moderators at any level, and interpret the fixed 
> > effects for those moderators as: the amount of change in 
> > level-specific true effects relative to 1 unit increase in those 
> > moderators. 
> > 
> > To (partially) answer your final question, for moderators that can 
> > vary between more than one level, a single regression coefficient is a 
> > mix of the moderators’ effects on more than one levels’ true effects. 
> > Thus, it is a good idea to disentangle these effects. In the context 
> > of multilevel meta-regression, I’m not sure if there is a suggested 
> > procedure to do so. But *conceptually* something like what follows 
> > *might* make sense: 
> > 
> > 1- Create a variable called “X_btw_study”: Average X in each study. 
> > 2- Create a variable called “X_btw_outcome”: Average X in each 
> > outcome in each study. 
> > 3- Create a variable called “X_btw_outcome_study”: Subtract (1) from (2). 
> > 4- Create a variable called “X_wthn_study”: Subtract (1) from each 
> > X value in each study. 
> > 5- Create a variable called “X_wthn_outcome”: Subtract (2) from X 
> > value of that outcome in each study. 
> > 6- Fit the following model: >> rma.mv(yi ~ X_btw_study + 
> > X_btw_outcome + X_btw_outcome_study + X_wthn_study + X_wthn_outcome, 
> > random=~1 | study/outcome) << 
> > 
> > In my conceptual description above, I divided X into five parts 
> > between two levels. But I leave it to other meta-regression experts to 
> > comment on whether I've missed something or if they know of a 
> > practical way of to deal with moderators that can vary across more 
> > than one level 
> > 
> > Best, 
> > Fred 
> > 
> > On Sat, Aug 21, 2021 at 9:09 AM Timothy MacKenzie <fswfswt using gmail.com> wrote: 
> >> 
> >> Dear Colleagues, 
> >> 
> >> I have some clarification questions. 
> >> 
> >> In multilevel models, what do the fixed-effect coefficients exactly 
> >> predict? (change in the 'observed' effect yi for 1 unit of increase in 
> >> moderator X OR change in some form of 'true effect' [depending on the 
> >> random-part specification] for 1 unit of increase in moderator X) 
> >> 
> >> The reason I ask this is the bottom of p.26 of this paper ( 
> >> https://osf.io/4fe6u/). In this paper, Dr. Pustejovsky describes a 3-level 
> >> model (j cases in k studies): 
> >> 
> >> Rjk = Y0 + Y1(Age)jk + Vk + Ujk + ejk 
> >> 
> >> Then, he interprets Age's fixed effect coefficient as: *"the difference in 
> >> average effect sizes between cases [level 2] that differ in age by one 
> >> year"*. 
> >> 
> >> I wonder how this interpretation is possible and can be extended to other 
> >> models (see below)? 
> >> 
> >> Say X is a continuous moderator that can vary between 'studies' and 
> >> 'outcomes'. How can we apply Dr. Pustojuvsky's logic to the interpretation 
> >> of 'X' fixed coefficient separately in: 
> >> 
> >> (A) 'rma.mv(yi ~ X, random=~1 | study)' 
> >> vs. 
> >> (B) 'rma.mv(yi ~ X, random=~1 | study/outcome)' differ? 
> >> 
> >> Thank you very much, 
> >> Tim 
> >> 
> >> [[alternative HTML version deleted]] 
> >> 
> >> _______________________________________________ 
> >> R-sig-meta-analysis mailing list 
> >> R-sig-meta-analysis using r-project.org 
> >> https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis 

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