Assumptions of random effects for unbiased estimates
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>
_______________________________________________ R-sig-mixed-models at r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models