# Re: [R] residual df in lmer and simulation results

From: Douglas Bates <bates_at_stat.wisc.edu>
Date: Fri 28 Jul 2006 - 23:48:33 EST

On 7/26/06, Bill Shipley <bill.shipley@usherbrooke.ca> wrote:

> Hello. Douglas Bates has explained in a previous posting to R why he does
> not output residual degrees of freedom, F values and probabilities in the
> mixed model (lmer) function: because the usual degrees of freedom (obs -
> fixed df -1) are not exact and are really only upper bounds. I am
> interpreting what he said but I am not a professional statistician, so I
> might be getting this wrong...

> Does anyone know of any more recent results, perhaps from simulations, that
> quantify the degree of bias that using such upper bounds for the demoninator
> degrees of freedom produces? Is it possible to calculate a lower bounds for
> such degrees of freedom?

I have not seen any responses to your request yet Bill. I was hoping that others might offer their opinions and provide some new perspectives on this issue. However, it looks as if you will be stuck with my responses for the time being.

You have phrased your question in terms of the denominator degrees of freedom associated with terms in the fixed-effects specification and, indeed, this is the way the problem is usually addressed. However, that is jumping ahead two or three steps from the iniital problem which is how to perform an hypothesis test comparing two nested models - a null model without the term in question and the alternative model including this term.

If we assume that the F statistic is a reasonable way of evaluating this hypothesis test and that the test statistic will have an F distribution with a known numerator degrees of freedom and an unknown denominator degrees of freedom then we can reduce the problem of testing the hypothesis to one of approximating the denominator degrees of freedom. However, there is a lot of assumption going on in that argument. These assumptions may be warranted or they may not.

As far as I can see, the usual argument made for those assumptions is by analogy. If we had a balanced design and if we used error strata to get expected and observed mean squares and if we equated expected and observed mean squares to obtain estimates of variance components then the test for a given term in the fixed effects specification would have a certain form. Even though we are not doing any of these things when estimating variance components by maximum likelihood or by REML, the argument is that the test for a fixed effects term should end up with the same form. I find that argument to be a bit of a stretch.

Because the results from software such as SAS PROC MIXED are based on this type of argument many people assume that it is a well-established result that the test should be conducted in this way. Current versions of PROC MIXED allow for several different ways of calculating denominator degrees of freedom, including at least one, the Kenward-Roger method, that uses two tuning parameters - denominator degrees of freedom and a scale factor.

Some simulation studies have been performed comparing the methods in SAS PROC MIXED and other simulation studies are planned but for me this is all putting the cart before the horse. There is no answer to the question "what is the _correct_ denominator degrees of freedom for this test statistic" if the test statistic doesn't have a F distribution with a known numerator degrees of freedom and an unknown denominator degrees of freedom.

I don't think there is a perfect answer to this question. I like the approach using Markov chain Monte Carlo samples from the posterior distribution of the parameters because it allows me to assess the distribution of the parameters and it takes into account the full range of the variation in the parameters (the F-test approach is conditional on estimates of the variance components). However, it does not produce a nice cryptic p-value for publication.

I understand the desire for a definitive answer that can be used in a publication. However, I am not satisfied with any of the "definitive answers" that are out there and I would rather not produce an answer than produce an answer that I don't believe in.

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