# Re: [R] Conservative "ANOVA tables" in lmer

From: Douglas Bates <bates_at_stat.wisc.edu>
Date: Thu 07 Sep 2006 - 15:11:47 GMT

On 9/7/06, Martin Maechler <maechler@stat.math.ethz.ch> wrote:
> >>>>> "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.

Thank you for pointing that out Martin. I agree. As I mentioned a value of the denominator degrees of freedom based on the trace of the hat matrix is conditional on the estimates of the relative variances of the random effects. I think an argument could still be made for the upper bound on the dimension of the model space being rank of Z:X and hence a lower bound on the dimension of the space in which the residuals lie as being n - rank[Z:X]. One possible approach would be to use the squared length of the projection of the data vector into the orthogonal complement of Z:X as the "sum of squares" and n - rank(Z:X) as the degrees of freedom and base tests on that. Under the assumptions on the model I think an F ratio calculated using that actually would have an F distribution.

>
> Martin
>
> >> When I use these criteria and compare my "ANOVA" table to the results
> >> of analysis of Helmert contrasts using MCMC sample with highest
> >> posterior density intervals, I find that my conclusions (e.g. factor
> >> A, with three levels, has a "significant effect" on the response
> >> variable) are qualitatively the same.

>
>
> DB> I would be happy to re-institute p-values for fixed effects in the
> DB> summary and anova methods for lmer objects using a denominator degrees
> DB> of freedom based on the trace of the hat matrix or the rank of Z:X if
> DB> others will volunteer to respond to the "these answers are obviously
> DB> wrong because they don't agree with <whatever> and the idiot who wrote
> DB> this software should be thrashed to within an inch of his life"
> DB> messages. I don't have the patience.
>
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