[R-sig-ME] Fwd: lmer stand dev of coefficients

[Andrew Robinson]

Hi all,

This article might help:

The BLUPs are not "best" when it comes to bootstrapping

Jeffrey S. Morris 

Statistics & Probability Letters 56 (2002) 425-430

In the setting of mixed models, some researchers may construct a
semiparametric bootstrap by sampling from the best linear unbiased
predictor residuals.  This paper demonstrates both mathematically and
by simulation that such a bootstrap will consistently underestimate
the variation in the data in finite samples.

Cheers,

Andrew


On Sun, Dec 21, 2008 at 10:59:01AM -0600, Douglas Bates wrote:
> On Sun, Dec 21, 2008 at 9:40 AM, Daniel Ezra Johnson
> <danielezrajohnson at gmail.com> wrote:
> > ---------- Forwarded message ----------
> > From: Daniel Ezra Johnson <danielezrajohnson at gmail.com>
> > Date: Sun, Dec 21, 2008 at 3:39 PM
> > Subject: Re: [R-sig-ME] lmer stand dev of coefficients
> > To: Douglas Bates <bates at stat.wisc.edu>
> 
> > Can you explain briefly what circumstances would lead these quantities
> > to be quite different?
> 
> First, I misspoke. (Note to self: Don't try to answer questions on
> theory before the second cup of coffee.)  The standard deviation of
> the BLUPs (or, as I prefer to call them, the conditional modes) of the
> random effects are not an estimate of the conditional standard
> deviation of the random effects given the data.  I can only make sense
> of the conditional standard deviation of a particular random effect
> and that would be much smaller than the observed standard deviation of
> the conditional modes.
> 
> What I should have said is somewhat more subtle.  We know that the
> conditional modes of the random effects have less variability than the
> corresponding individual estimates of a parameter.  I enclose a script
> and its output for a particularly simple example - a random-effects
> model fit to the Dyestuff data from the lme4 package.  The design is a
> balanced, one-way classification so the estimate of the mean yield is
> simply the mean of the Yield variable.
> 
> We see that the conditional modes are always smaller in magnitude than
> the deviations of the individual means from the overall mean.  The
> fact that the ratio is constant is a consequence of the balanced
> design.   We say that the conditional modes are shrunk towards zero
> because the random effects have a finite variance.
> 
> The conditional modes are also shrunk relative to what would be
> expected from the unconditional variance of the random effects, but I
> find it more difficult to explain why.  It makes sense to me that the
> mle of the unconditional standard deviation would be larger than the
> standard deviation of the conditional modes but of the way the way the
> likelihood criterion is formulated.
> 
> Perhaps someone else can explain why.
> 
> > Suppose the random effect grouping factor is Subject.
> >
> > On what basis would the software estimate the unconditional SD of (the
> > population of) Subjects to be something quite different (and as you
> > say, usually larger) than that of the particular group of Subjects in
> > the data?
> >
> > Dan
> >
> > On Sun, Dec 21, 2008 at 3:32 PM, Douglas Bates <bates at stat.wisc.edu> wrote:
> >> On Sun, Dec 21, 2008 at 3:55 AM, Iasonas Lamprianou
> >> <lamprianou at yahoo.com> wrote:
> >>> Dear friends
> >>> when I use sd(coef(mymodel)$myvariable) I get 0.21
> >>> However, the summary of the model gives
> >>> Error terms:
> >>>  Groups      Name        Std.Dev.
> >>>  myvariable (Intercept) 0.33
> >>>  Residual               0.76
> >>>
> >>> Why dont I get the same value (0.21 instead of 0.33)?
> >>
> >> Because they are estimates of different quantities:
> >> sd(coef(mymodel)$myvariable) is an estimate (although it is not
> >> entirely clear what the properties of such an estimate would be) of
> >> the conditional standard deviation of the random effects given the
> >> data, whereas 0.33 is the maximum likelihood estimate or REML estimate
> >> of the unconditional standard deviation of the random effects.  We
> >> would expect the conditional standard deviation to be smaller than the
> >> unconditional standard deviation.
> >>
> >> P.S. If you are starting a new topic on the mailing list you don't
> >> need to quote a previous message to the list and especially not an
> >> entire digest message.
> >>
> >> _______________________________________________
> >> R-sig-mixed-models at r-project.org mailing list
> >> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >>
> >
> > _______________________________________________
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> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >

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-- 
Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-6410
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599
http://www.ms.unimelb.edu.au/~andrewpr
http://blogs.mbs.edu/fishing-in-the-bay/