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

Wed Oct 12 03:11:19 CEST 2016

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%
> 20and%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>
> >
> > _______________________________________________
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> > 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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