I am using Mixed effects models (nlme package in R) to choose the model with the best random and fixed effects. I am following the procedure of Zurr et al. (2009) and read about "testing on the boundary" and the effects that it has on p values. From what I have read, it looks as though this is only something that is a problem when using REML, but I am not sure.

My questions are:

  1. When performing a LRT between 2 nested models which only differ in fixed effects (thus using ML instead of REML), do the p values have to be adjusted for testing on the boundary?

  2. If this is a problem only for REML estimation, why? Does it have to do with the REML estimation itself, or the fact that it is usually used with the LRT when comparing models that differ in random effects?

  • $\begingroup$ It's hard to say w/o more information. I suspect you are confusing testing FEs vs REs. Can you quote more of the text here? $\endgroup$ – gung Mar 27 '15 at 22:07
  • $\begingroup$ The text warning about boundary effects is in a section about choosing the best RE structure in your model by comparing models that differ in their RE structure (ie. same FE). When mentioning using the Likelyhood ratio test to test between two models with different RE structures, Zuur warns about boundary effects and that p-values have to be adjusted. He also suggests that REML be used instead of Max Likelyhood to find the optimal random structure for a model, so I was curious if boundary effects were a property of choosing REML estimation, or because we are testing between random effects. $\endgroup$ – MANOVAboard Mar 31 '15 at 21:27

Regarding your first question, if the two models only differ in the fixed effects, no parameters are on the boundary in any of the models (since all coefficients can take values on the complete real line, not just on the positive numbers, like variances must).

For example, in the smaller model, one parameter (a regression coefficient) is set to zero, but in the larger models it can be both greater or less than zero, so there is no problem.

Regarding your second question, parameters on the boundary are a problem for likelihood ratio test in general (not just for mixed-effects models). The asymptotics break down when the parameter(s) in one of the models are on the boundary of the parameter space. So yes, this is a problem for ML estimation (too).

In mixed-effects models the parameter is often a variance. In the smaller model it is set to zero, but in the larger model it can only be greater than zero. So in the smaller model, it is on the boundary of the parameter space (which goes from zero to positive infinity), and the asymptotics break down.

  • $\begingroup$ Thank you Karl. So, to clarify: 1)boundary effects are a problem whenever the parameter is a variance as variances cannot be negative. In RE the parameters are always a variance so a correction is always needed when performing a LRT to test between models that differ in their RE's. 2) The parameters of FE's can also be variances, meaning when using the LRT to test between models that differ in FE's only, you may still have to correct p-values. 3)Boundary effect occur due to the properties of the parameters, not the estimation method (REML or ML). Is that interpretation correct? $\endgroup$ – MANOVAboard Mar 31 '15 at 21:36
  • $\begingroup$ @MANOVAboard 1) Almost. The parameters can also be covariances (between two random effects), which are not constrained to be positive. 2) No, the parameters of fixed effects are coefficients in a linear model, and are not variances. 3) Basically, yes. $\endgroup$ – Karl Ove Hufthammer Apr 1 '15 at 8:09
  • $\begingroup$ thank you. I was confused by the portion of your answer in which you said parameters are often variances, but it sounds like you were referring specifically to random effects. To re-iterate 1):boundary effects are a problem whenever the parameter is a variance. In single random effects (such as a random slope or intercept ONLY model), these random effects are generally variances. In models with a random slope+intercept (or other random effects that can interact), boundary effects may not apply. Thank you for the attention and clarification. $\endgroup$ – MANOVAboard Apr 1 '15 at 13:05

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