[R-sig-ME] Cluster-robust SEs & random effects -- seeking some clarification

Ben Bolker bbo|ker @end|ng |rom gm@||@com
Sun Jul 31 01:48:35 CEST 2022

I haven't been following the whole thread that carefully, but I want to 
emphasize that

   posXsymp~treatment + pairID + (1 | union)

is *not*, by any definition I'm familiar with, a "random-slopes model"; 
that is, it only estimates a single population-level treatment 
effect/doesn't allow the effect of treatment to vary across groups 
defined by 'union'.  You would need a random-effect term of the form 
(treatment | union).

   Reasons why you might *not* want to do this:

  * if treatment only varies across and not within levels of union 
("union is nested within treatment" according to some terminology), then 
this variation is unidentifiable
  * maybe you have decided that you don't have enough data/want a more 
parsimonious model.

   Schielzeth and Forstmeier, among many others (this is the example I 
know of), have cautioned about the consequences of leaving out 
random-slopes terms.

Schielzeth, Holger, and Wolfgang Forstmeier. “Conclusions beyond 
Support: Overconfident Estimates in Mixed Models.” Behavioral Ecology 
20, no. 2 (March 1, 2009): 416–20. https://doi.org/10.1093/beheco/arn145.

On 2022-07-30 7:43 p.m., J.D. Haltigan wrote:
> Addendum:
> It just occurred to me on my walk that I think I am getting a bit lost in
> some of the differences in nomenclature across scientific silos. In the
> original model that they specified, which treated the 'pairID' variable as
> a control variable for which they controlled for 'fixed effects' of
> control/treatment villages (in their own language in the paper) using
> cluster-robust SEs, I think this is indeed a 'random-intercepts only' model
> in the language of Hamaker et al. They implement the 'absorb' command in
> STATA which I believe aggregates across the pairIDs to generate an
> 'omnibus' F-test of sorts for the pairID variable (in the ANOVA
> nomenclature). I say this as when I specify the pairID variable in the lmer
> model I shared (or in a fixest model I conducted to replicate the original
> Abalauck results in R), I get the estimates for all the pairs (i.e., there
> is no way to aggregate across them--though I think formally the models are
> the same if we are unconcerned about any one pairID [treatment/control
> village pair].
> So, in the lmer model I shared where I specify a specific random effects
> term for the 'cluster' variable, I think this indeed is allowing for random
> slopes across the clusters which implies the treatment effect may vary
> across the clusters (and we might anticipate it will for various reasons I
> can elaborate on). More generally: we are generalizing to *any* universe of
> villages (say in the entire world) where the treatment intervention (masks)
> may vary across villages. This is the crux of invoking the random effects
> model (i.e., random slopes model).
> I realize this is a mouthful, but I think the way these terms (e.g.,
> random/fixed effects models etc.) are used across disciplines makes things
> a bit confusing.
> On Sat, Jul 30, 2022 at 5:25 PM J.D. Haltigan <jhaltiga using gmail.com> wrote:
>> This is a very helpful walkthrough, James. My responses are italicized
>> under yours to maintain thread readability. The key is Generalizability
>> here and (as I also note in my last reply) the idea is to Generalize to a
>> universe of "any villages or clusters." That is, the target population we
>> are generalizing to is *any* random population.
>> On Sat, Jul 30, 2022 at 3:01 PM James Pustejovsky <jepusto using gmail.com>
>> wrote:
>>> Hi J.D.,
>>> A few comments/reactions inline below.
>>> James
>>> On Wed, Jul 27, 2022 at 5:37 PM J.D. Haltigan <jhaltiga using gmail.com> wrote:
>>>> ...
>>> In the original investigation, the authors did not invoke a random
>>>> effects model (but did use the pairIDs to control for fixed effects as
>>>> noted and with robust SEs). Thus, in the original investigation there was
>>>> *no* specification of a random effects model for the 'cluster' variable. We
>>>> know from some other work there were some biases in village mapping and
>>>> other possible sources of between-cluster variation that might be
>>>> anticipated to have influence--at the random intercepts level--so we are
>>>> looking into how specifying 'cluster' as a random effect might change the
>>>> fixed effects estimates for the treatment intervention effect. In the
>>>> Hamaker et al. language, it is indeed a 'random intercepts' only model.
>>> I don't follow how using a random intercepts model improves the
>>> generalizability warrant here. The random intercepts model is essentially
>>> just a re-weighted average of the pair-specific effects in the original
>>> analysis, where the weights are optimally efficient if the model is
>>> correctly specified. That last clause carries a lot of weight here--correct
>>> specification means 1) treatment assignment is unrelated to the random
>>> effects, 2) the treatment effect is constant across clusters, 3)
>>> distributional assumptions are valid (i.e., homoskedasticity at each level
>>> of the model).
>>> If the effects are heterogeneous, then I would think that including
>>> random slopes on the treatment indicator would provide a better basis for
>>> generalization. But even then, the warrant is still pretty vague---what is
>>> the hypothetical population of villages from which the observed villages
>>> are sampled?
>> *In the most basic model (without baseline controls) the model takes the
>> form: myModel = lmer(posXsymp~treatment + pairID + (1 | union), data =
>> myData). I believe--correct me if I am wrong--that this reflects a
>> random-intercepts only model, but I may be mistaken. If I am, and this is
>> allowing for random slopes on the treatment indicator, then I will need to
>> rethink my statements.  *
>>>> Given this, however, does it also make sense to include the cluster
>>>> robust SEs for the fixed effects which would account for possible
>>>> heterogeneity of treatment effects (i.e., slopes) across clusters?s
>>>> If you're committed to the random intercepts model, then yes I think so
>>> because using cluster robust SEs at least acknowledges the possibility of
>>> heterogeneous treatment effects.
>> *If the above model does allow for both random intercepts and slopes, then
>> perhaps the use of cluster robust SEs is redundant in some sense since the
>> random slopes would be modeling the heterogeneity in treatment effects?*
>>>> Bottom line: in their original analyses, clusters are seen as
>>>> interchangeable from a conceptual perspective (rather than drawn from a
>>>> random universe of observations). When one scales up evidence to a universe
>>>> of observations that are random (as they would be in the intended universe
>>>> of inference in the real-world), then we are better positioned, I think, to
>>>> adjudicate whether the mask intervention effect is 'practically
>>>> significant' (in addition to whether the focal effect remains marginally
>>>> significant from a frequentist perspective).
>>> As noted above, this argument is a bit vague to me. If there's concern
>>> about generalizability, then my first question would be: what is the target
>>> population to which you are trying to generalize?
>> *Essentially, the target population we are trying to generalize to is a
>> random selection of villages. Any random selection of villages. In other
>> words, villages should not be seen as interchangeable. We are interested in
>> whether the effects generalize to any randomly selected village. *
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Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Director, School of Computational Science and Engineering
(Acting) Graduate chair, Mathematics & Statistics

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