Solved – Weird denominator degrees of freedom in SAS glimmix

degrees of freedommixed modelrandom-effects-modelsas

I´m trying to learn generalized mixed models in SAS and I have now bumbed into a situation that I cannot figure out by myself. So I was thinking that maybe some of you could help me forward.

I´m running a model where I have normally distributed measurements of traits like tarsus etc in birds from different populations. I have few fixed factors (sampling area, sex and year) and their interactions in the model and random term "sampling site clustered within the sampling area" (2 sampling areas with both 2 sampling sites). I´m using SATTERTH as a method for computing the denominator degrees of freedom.

My problem is that for one particular trait I get really low degrees of freedom always when the sampling site is in the model, and for that particular factor. I noticed that playing around with the methods of computing the degrees of freedom, I can get "better" results, but I cannot understand why in this model, one fixed factor eats up my degrees of freedom.

If anyone can and would like to explain to me why this happens and how I should select the method for computing the DDF´s in "dummies" way, it would make my day. I'm happy to provide you with more information, I was not certain how I should present my problem.

Thank you in advance.

Best Answer

The t distribution by Satterwaite is an approximation to the distribution of a t-like statistic when the two variances are unequal and estimated separately. This is the so-called Behren's-Fisher problem. The distribution under the null hypothesis is not a t distribution but it has been found that it can be well approximated by a t with a fractional number for the degrees of freedom parameter. There is a special formula for the degrees of freedom in the approximation. You could look this up but here is a link to Wikipedia: http://en.wikipedia.org/wiki/Behrens%E2%80%93Fisher_problem.

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Solved – Differences between PROC Mixed and lme / lmer in R – degrees of freedom

For the first question, the default method in SAS to find the df is not very smart; it looks for terms in the random effect that syntactically include the fixed effect, and uses that. In this case, since trt is not found in ind, it's not doing the right thing. I've never tried BETWITHIN and don't know the details, but either the Satterthwaite option (satterth) or using ind*trt as the random effect give correct results.

PROC MIXED data=Data;
    CLASS ind fac trt;
    MODEL y = trt /s ddfm=satterth;
    RANDOM ind /s;
run;

PROC MIXED data=Data;
    CLASS ind fac trt;
    MODEL y = trt /s;
    RANDOM ind*trt /s;
run;

As for the second question, your SAS code doesn't quite match your R code; it only has a term for fac*ind, while the R code has a term for both ind and fac*ind. (See the Variance Components output to see this.) Adding this gives the same SE for trt in all models in both Q1 and Q2 (0.1892).

As you note, this is an odd model to fit as the fac*ind term has one observation for each level, so is equivalent to the error term. This is reflected in the SAS output, where the fac*ind term has zero variance. This is also what the error message from lme4 is telling you; the reason for the error is that you most likely misspecified something as you're including the error term in the model in two different ways. Interestingly, there is one slight difference in the nlme model; it's somehow finding a variance term for the fac*ind term in addition to the error term, but you will notice that the sum of these two variances equal the error term from both SAS and nlme without the fac*ind term. However, the SE for trt remains the same (0.1892) as trt is nested in ind, so these lower variance terms don't affect it.

Finally, a general note about degrees of freedom in these models: They are computed after the model is fit, and so differences in the degrees of freedom between different programs or options of a program do not necessarily mean that the model is being fit differently. For that, one must look at the estimates of the parameters, both fixed effect parameters and covariance parameters.

Also, using the t and F approximations with a given number of degrees of freedom is fairly controversial. Not only are there several ways to approximate the df, some believe the practice of doing so is not a good idea anyway. A couple words of advice:

If everything is balanced, compare the results with the traditional least squares method, as they should agree. If it's close to balanced, compute them yourself (assuming balance) so that you can make sure the ones you're using are in the right ballpark.
If you have a large sample size, the degrees of freedom don't matter very much as the distributions get close to a normal and chi-squared.
Check out Doug Bates's methods for inference. His older method is based on MCMC simulation; his newer method is based on profiling the likelihood.

Solved – Bayesian inference and degrees of freedom

At least from a theoretical point of view, identifiably is not important from a Bayesian perspective. If the data is not informative about some parameters under the model then the posterior of those parameters will just be highly influenced by the prior.

From a practical point of view if the posterior is broad then approximate methods such as MCMC will take longer maybe much longer to run.

Another practical problem is that if you have a large parameter space and little data as it sounds like you do then the results, if you can manage to compute them, are likely to be very sensitive to prior specification.

Best Answer

Related Solutions

Solved – Differences between PROC Mixed and lme / lmer in R – degrees of freedom

Solved – Bayesian inference and degrees of freedom

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