[R-sig-ME] Assumptions of random effects for unbiased estimates

Jake Westfall jake.a.westfall at gmail.com
Wed Oct 12 03:49:45 CEST 2016

What a nice contribution from John!


On Tue, Oct 11, 2016 at 8:11 PM, Poe, John <jdpo223 at g.uky.edu> wrote:

> My reading of modern work by panel data econometricians is that they seem
> very fine with the use of mixed effects models that properly differentiate
> effects at different levels of analysis and the tools to do so have existed
> in that literature since the early 1980s. They have been borrowing heavily
> from the mixed effects literature in designing econometric models and talk
> about them in panel data textbooks. This hasn't typically filtered down to
> applied economists who tend to misunderstand what other fields do because
> other fields just tend to talk about them differently.
> The short version:
> Everyone in the mixed effects literature just uses group/grand mean
> centering and random coefficients to deal with endogeneity bias. If you are
> an economist and someone outside of econ says mixed effects models you
> should think *correlated random effects models* and not *random effects
> models*.
> The long version:
> Economists are pretty afraid error structures that are correlated with
> independent variables in general and have built up pretty elaborate
> statistical models to deal with the problem. In panel data, this manifests
> itself as wanting to avoid confounding effects at different levels of
> analysis so that within group varying effects are segregated from between
> group varying effects. It can also happen when you are omitting higher
> level random effects
> <http://methods.johndavidpoe.com/2016/09/09/independence-across-levels-in-mixed-effects-models/>
> and they are distorting the structure of the random effects that you are
> including. This is generally a good thing as you want to be able to test
> hypotheses at specific levels of analysis without confounding.
> It's a big enough theoretical concern in the discipline that they usually
> just want to remove all between group effects from the data as a *default* to
> get level one effects because it is simpler and more fool proof than
> dealing with the problem in a mixed effects setting. It's so pervasive that
> they are often socialized into not designing hypotheses for any between
> group or cross-level variation and just focus on within group (time
> varying) variability when at all possible (what economists call *within
> effects*).
> What economists refer to as fixed effects models just difference out all
> between group variation so that it cannot contaminate within group effects
> (bias level one coefficients). It's the equivalent to including group
> indicator variables in the model instead of a random effect and just
> accepting that you can't make substantive inferences about anything at the
> group level (what economists call *between effects*).
> The typical conventional wisdom in applied econometrics is to use a
> Hausman test which is a generic test comparing coefficients between a
> random effects model (with no level 2 covariates) and a model with all
> between group variability removed from the data. If there are differences
> between the two, then they prefer to go with the latter. This is bad
> practice according to econometrics textbooks but applied people don't seem
> to care (Baltagi 2013 ch 4.3). This only makes sense if you don't care
> about group invariant variables that only differ crosssectionally and/or
> you think of their effects as contamination. Panel data econometrics
> textbooks tend to argue for a wider range of options here but in practice
> not that many economists seem to use them.
> There's an alternative framework in econ for dealing with this problem
> that they call a Mundlak device (Mundlak 1978) or correlated random effects
> models (Baltagi Handbook of Panel Data 2014 ch 6.3.3 or really any panel
> data textbook) which is equivalent to a hierarchical linear model with
> group mean centering for level-one variables. This approach is used in
> econometrics by some pretty standard advanced panel data models (e.g.
> Hausman-Taylor and Arellano Bond). The other alternative that is advocated
> by panel data econometricians but doesn't seem to have filtered down to
> rank and file economists is to use random coefficients models and just
> allow the random effects to be correlated with level one variables (Hsiao
> 2014 chapter 6 and most of his other written work).
> It is important to understand that efficiency isn't the primary reason for
> use of a mixed effects model over a fixed effects model for most research.
> A common reason to use a mixed effects model is that you have hypotheses
> about variables operating at higher levels of analysis or cross-level
> interactions and those questions cannot be answered by fixed effects panel
> models that have removed all between group variability from the analysis.
> You are sacrificing the ability to test group variant hypotheses by using a
> basic fixed effects model over a mixed effects model. For nonlinear models
> like a logistic regression it can also be very difficult to use an unbiased
> fixed effects model (though there are ways in a panel setting e.g. Hahn and
> Newy 2004) and trivial to use a mixed effects model.
> Panel data econometricians almost always talk about typical practice among
> applied economists using fixed effects as flawed (see Baltagi 2013 ch.
> 4.3). Mark Nerlov's 2000 History of Panel Data Econometrics is my favorite
> example:
> The absurdity of the contention that possible correlation between some of
>> the observed explanatory variables and the individual-specific component of
>> the disturbance is a ground for using fixed effects should be clear from
>> the following example: Consider a panel of households with data on
>> consumption and income. We are trying to estimate a consumption function.
>> Income varies across households and over time. The variation across
>> households is related to ability of the main earner and other household
>> specific factors which vary little over time, that is to say, reflect
>> mainly differences in permanent income. Such permanent differences in
>> income are widely believed to be the source of most differences in
>> consumption both crosssectionally and over time, whereas, variations of
>> income over time are likely to be mostly transitory and unrelated to
>> consumption in most categories. Yet, fixed-effects regressions are
>> equivalent to using only this variation and discarding the information on
>> the consumption-income relationship contained the cross-section variation
>> among the household means.
> See the last couple of pages of this lecture
> <http://www.johndavidpoe.com/wp-content/uploads/2012/09/Blalock-Lecture.pdf> for
> the citations in the econometrics and multilevel literature that I
> referenced.
> On Tue, Oct 11, 2016 at 3:32 PM, Jake Westfall <jake.a.westfall at gmail.com>
> wrote:
>> Hi Laura and Ben,
>> I like this paper on this topic:
>> http://psych.colorado.edu/~westfaja/FixedvsRandom.pdf
>> What it comes down to essentially is that if the cluster effects are
>> correlated with the "time-varying" (i.e., within-cluster varying) X
>> predictor -- so that, for example, some clusters have high means on X and
>> others have low means on X -- then there is the possibility that the
>> average within-cluster effect (which is what the fixed effect model
>> estimates) differs from the overall effect of X, not conditional on the
>> clusters. An extreme example of this is Simpson's paradox. Now since the
>> estimate from the random-effects model can be seen as a weighted average
>> of
>> these two effects, it will generally be pulled to some extent away from
>> the
>> fixed-effect estimate toward the unconditional estimate, which is the bias
>> that econometricians fret about. However, if the cluster effects are not
>> correlated with X, so that each cluster has the same mean on X, then this
>> situation is not possible, so the random-effect model will give the same
>> unbiased estimate as the fixed-effect model.
>> A simple solution to this problem is to retain the random-effect model,
>> but
>> to split the predictor X into two components, one representing the
>> within-cluster variation of X and the other representing the
>> between-cluster variation of X, and estimate separate slopes for these two
>> effects. One can even test whether these two slopes differ from each
>> other,
>> which is conceptually similar to what the Hausman test does. As described
>> in the paper linked above, the estimate of the within-cluster component of
>> the X effect equals the estimate one would obtain from a fixed-effect
>> model.
>> As for the original question, I can't speak for common practice in
>> ecology,
>> but I suspect it may be like it is in my home field of psychology, where
>> we
>> do worry about this issue (to some extent), but we discuss it using
>> completely different language. That is, we discuss it in terms of whether
>> there are different effects of the predictor at the within-cluster and
>> between-cluster levels, and how our model might account for that.
>> Jake
>> On Tue, Oct 11, 2016 at 1:50 PM, Ben Bolker <bbolker at gmail.com> wrote:
>> >
>> >   I didn't respond to this offline, as it took me a while even to start
>> > to come up to speed on the question.  Random effects are indeed defined
>> > from *very* different points of view in the two communities
>> > ([bio]statistical vs. econometric); I'm sure there are points of
>> > contact, but I've been having a hard time getting my head around it all.
>> >
>> > Econometric definition:
>> >
>> > The wikipedia page <https://en.wikipedia.org/wiki/Random_effects_model>
>> > and CrossValidated question
>> > <http://stats.stackexchange.com/questions/66161/why-do-
>> > random-effect-models-require-the-effects-to-be-uncorrelated-
>> with-the-inpu>
>> > were both helpful for me.
>> >
>> >  In the (bio)statistical world fixed and random effects are usually
>> > justified practically in terms of shrinkage estimators, or
>> > philosophically in terms of random draws from an exchangeable set of
>> > levels: e.g. see
>> > <http://stats.stackexchange.com/questions/4700/what-is-
>> > the-difference-between-fixed-effect-random-effect-and-mixed-
>> effect-mode/>
>> > for links.
>> >
>> >   I don't think I can really write an answer yet.  I'm still trying to
>> > understand at an intuitive or heuristic level what it means for
>> > Cov(x_it,c_i)=0, where x_it is a set of explanatory variables over time
>> > for an individual subject and c_i is the conditional mode (=BLUP in
>> > linear mixed-model-land) for the deviation of the individual i from the
>> > population mean ... or more particularly what it means for that
>> > condition to be violated, which is the point at which fixed effects
>> > would become preferred.
>> >
>> >   As a side note, some statisticians (Andrew Gelman is the one who
>> > springs to mind) have commented on the possible overemphasis on bias.
>> > (All else being equal unbiased estimators are preferred to biased
>> > estimators but all else is not always equal). Two examples: (1)
>> > penalized estimators such as lasso/ridge regression (closely related to
>> > mixed models) give biased parameter estimates with lower mean squared
>> > error. (2) When estimating variability, one has to choose a particular
>> > scale (variance, standard error, log(standard error), etc.) on which one
>> > would prefer to get an unbiased answer.
>> >
>> > On 16-10-11 12:02 PM, Laura Dee wrote:
>> > > Dear all,
>> > > Random effects are more efficient estimators – however they come at
>> the
>> > > cost of the assumption that the random effect is not correlated with
>> the
>> > > included explanatory variables. Otherwise, using random effects leads
>> to
>> > > biased estimates (e.g., as laid out in Woolridge
>> > > <https://faculty.fuqua.duke.edu/~moorman/Wooldridge,%20FE%20
>> and%20RE.pdf
>> > >'s
>> > > Econometrics text). This assumption is a strong one for many
>> > > observational datasets, and most analyses in economics do not use
>> random
>> > > effects for this reason. *Is there a reason why observational
>> ecological
>> > > datasets would be fundamentally different that I am missing? Why is
>> this
>> > > important assumption (to have unbiased estimates from random effects)
>> > > not emphasized in ecology? *
>> > >
>> > > Thanks!
>> > >
>> > > Laura
>> > >
>> > > --
>> > > Laura Dee
>> > > Post-doctoral Associate
>> > > University of Minnesota
>> > > ledee at umn.edu <mailto:ledee at umn.edu>
>> > > lauraedee.com <http://lauraedee.com>
>> >
>> > _______________________________________________
>> > R-sig-mixed-models at r-project.org mailing list
>> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>> >
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> --
> Thanks,
> John
> John Poe
> Doctoral Candidate
> Department of Political Science
> Research Methodologist
> UK Center for Public Health Services & Systems Research
> University of Kentucky
> 111 Washington Avenue, Room 203a
> Lexington, KY 40536
> www.johndavidpoe.com

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