[R] Conservative "ANOVA tables" in lmer

Peter Dalgaard p.dalgaard at biostat.ku.dk
Thu Sep 7 17:20:29 CEST 2006

Martin Maechler <maechler at stat.math.ethz.ch> writes:

> >>>>> "DB" == Douglas Bates <bates at stat.wisc.edu>
> >>>>>     on Thu, 7 Sep 2006 07:59:58 -0500 writes:
>     DB> Thanks for your summary, Hank.
>     DB> On 9/7/06, Martin Henry H. Stevens <hstevens at muohio.edu> wrote:
>     >> Dear lmer-ers,
>     >> My thanks for all of you who are sharing your trials and tribulations
>     >> publicly.
>     >> I was hoping to elicit some feedback on my thoughts on denominator
>     >> degrees of freedom for F ratios in mixed models. These thoughts and
>     >> practices result from my reading of previous postings by Doug Bates
>     >> and others.
>     >> - I start by assuming that the appropriate denominator degrees lies
>     >> between n - p and and n - q, where n=number of observations, p=number
>     >> of fixed effects (rank of model matrix X), and q=rank of Z:X.
>     DB> I agree with this but the opinion is by no means universal.  Initially
>     DB> I misread the statement because I usually write the number of columns
>     DB> of Z as q.
>     DB> It is not easy to assess rank of Z:X numerically.  In many cases one
>     DB> can reason what it should be from the form of the model but a general
>     DB> procedure to assess the rank of a matrix, especially a sparse matrix,
>     DB> is difficult.
>     DB> An alternative which can be easily calculated is n - t where t is the
>     DB> trace of the 'hat matrix'.  The function 'hatTrace' applied to a
>     DB> fitted lmer model evaluates this trace (conditional on the estimates
>     DB> of the relative variances of the random effects).
>     >> - I then conclude that good estimates of P values on the F ratios lie
>     >>   between 1 - pf(F.ratio, numDF, n-p) and 1 - pf(F.ratio, numDF, n-q).
>     >>   -- I further surmise that the latter of these (1 - pf(F.ratio, numDF,
>     >>   n-q)) is the more conservative estimate.
> This assumes that the true distribution (under H0) of that "F ratio"
> *is*  F_{n1,n2}  for some (possibly non-integer)  n1 and n2.
> But AFAIU, this is only approximately true at best, and AFAIU,
> the quality of this approximation has only been investigated
> empirically for some situations. 
> Hence, even your conservative estimate of the P value could be
> wrong (I mean "wrong on the wrong side" instead of just
> "conservatively wrong").  Consequently, such a P-value is only
> ``approximately conservative'' ...
> I agree howevert that in some situations, it might be a very
> useful "descriptive statistic" about the fitted model.

I'm very wary of ANY attempt at guesswork in these matters. 

I may be understanding the post wrongly, but consider this case: Y_ij
= mu + z_i + eps_ij, i = 1..3, j=1..100

I get rank(X)=1, rank(X:Z)=3,  n=300

It is well known that the test for mu=0 in this case is obtained by
reducing data to group means, xbar_i, and then do a one-sample t test,
the square of which is F(1, 2), but it seems to be suggested that
F(1, 297) is a conservative test???!

   O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)                  FAX: (+45) 35327907

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