[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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