Hi,
I have the following conceptual / interpretative question regarding
random effects:
A mixed effects model was fit on biological data, with observations
coming from different species. There is a clear overall effect of
certain predictors (entering the model as fixed effect), but as
different species react slightly differently, the predictor also enters
the model as random effect and with species as grouping variable. The
resulting model is very fine.
Now comes the tricky part however: I can inspect not only the variance
parameter estimate for the random effect, but also the 'coefficients'
for each species. If I do this, suppose I find out that they make
biologically sense, and maybe actually more sense then they should:
For each species vast biological knowledge is available, regarding
traits etc. So I can link the random effect coefficients to that
knowledge, see the deviation from the generic predictor impact (the
fixed effect) and relate it to the traits of my species.
However I see the following problem with that approach: If I have no
knowledge of the species traits, or the species names are anonymous to
me, it makes sense to treat the species-specific deviations as
realizations of a random variable (principle of exchangeability). Once I
know however the species used in the study and have the biological
knowledge at hand, it does not make so much sense any more; I can
predict whether for that particular species the generic predictor impact
will be amplified, or not. That is, I can predict if more likely the
draw from the assumed normal distribution of the random effects will be
> 0, or < 0 - which is of course complete contradictory and nonsense if
I assume I have a random draw from a N(0, sigma) distribution.
Integrating the biological knowledge as fixed effect however might be
tremendously difficult, as species traits can sometimes not readily be
quantified in a numeric way.
I could defer issue to the species traits and say, once the species
evolved their traits were drawn randomly from a population. This however
causes problems with ideas of evolution and phylogenetic relationships
among the species.
Maybe my question can be rephrased the following way:
Does it ever make sense to _interpret_ the coefficients of the random
effects for each group and link it to properties of the grouping
variable? The assumption of a realization of a random variable seems to
render that quite problematic. However, this means that the more
ignorant I am , and the less knowledge I have, the more the random
realization seems to become realistic - which is at odds with scientific
investigations.
Suppose the mixed model is one of the famous social sciences studies
analysing pupil results on tests at different schools, with schools
acting as grouping variable for a random effect intercept. If I have no
knowledge about the schools, the random effect assumption makes sense.
If I however investigate the schools in detail (either a priori or a
posterior), say teaching quality of the teachers, socio-economic status
of the school area etc, it will probably make sense to predict which
ones will have pupils performing above average, and which below average.
However then probably these factors leading me to the predictions should
enter the model as fixed effects, and maybe I don't need and school
random effect any more at all. But this means actually the school
deviation from the global mean is not the realization of a random
variable, but instead the result of something quite deterministic, but
which is usually just unknown, or can only be measured with extreme,
impractical efforts. So the process might not be random, just because
so little is known about the process, the results appear as if they
would be randomly drawn (from a larger population distribution). Again,
is ignorance / lack of deeper knowledge the key to using random effects
- and the more knowledge I have, the less ?
many thanks,
Thomas
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