From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>

Date: Thu 07 Sep 2006 - 15:20:29 GMT

Date: Thu 07 Sep 2006 - 15:20:29 GMT

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

> >>>>> "DB" == Douglas Bates <bates@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@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 ุster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk) FAX: (+45) 35327907 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.Received on Fri Sep 08 01:26:32 2006

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