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? Thanks for any help. Bill Shipley North American Editor, Annals of Botany Editor, "Population and Community Biology" series, Springer Publishing D?partement de biologie, Universit? de Sherbrooke, Sherbrooke (Qu?bec) J1K 2R1 CANADA Bill.Shipley at USherbrooke.ca http://pages.usherbrooke.ca/jshipley/recherche/
On 7/26/06, Bill Shipley <bill.shipley at 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.
On 7/26/06, Bill Shipley <bill.shipley at 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 can give another perspective on the issue of degrees of freedom for a linear mixed model although it probably doesn't address the question that you want to address. The linear predictor in a mixed model has the form X\beta + Zb where \beta is the fixed-effects vector and b is the random-effects vector. The fitted values, y-hat, are the fitted values from a penalized least squares fit of the response vector, y, to this linear predictor subject to a penalty on b defined by the variance components. When the penalty is large, the fitted values approach those from the ordinary least squares fit of y on X\beta only. When the penalty is small, the fitted values approach those from an unpenalized least squares fit of y on the linear predictor. (In this case estimates of the coefficients are not well defined because the combined matrix [X:Z] is generally rank deficient but the fitted values are well defined.) If the rank of X is p and the rank of [X:Z] is r then the effective number of parameters in the linear predictor for the penalized least squares fit is somewhere between p and r. One way of defining the effective number of parameters is as the trace of the hat matrix for the penalized least squares problem. This number will change as the variance components change and is usually evaluated at the estimates of the variance components. This is exactly what Spiegelhalter, Best, Carlin and van der Linde (JRSSB, 64(4), 583-639, 2002) define to be their p_D for this model. The next release of the lme4/Matrix packages will include an extractor function to evaluate the trace of the hat matrix for a fitted lmer model, using an algorithm due to Jialiang Li. This effective number of parameters in the linear predictor is like the degrees of freedom for regression. In the limiting cases it is exactly the degrees of freedom for regression. One might then argue that the degrees of freedom for residuals would be n - hat trace and use this for the denominator degrees of freedom in the F ratios. However, this number does not vary with the numerator term and many people will claim that it must. (I admit to being a bit perplexed as to why the denominator degrees of freedom should change according to the numerator term when the denominator of the F ratio itself doesn't change, but many people insist that this is the way things must be.) So it is possible to calculate a number that can reasonably be considered to be the degrees of freedom for the denominator that is actually used in the F ratios but this will not correspond to what many people will insist is the "obviously correct" number of degrees of freedom.