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How to assess significance of random effect in lme4

Spencer Graves 
I don't have Pinheiro and Bates handy, but as I recall, it's somthing 
like chapter 3.  See "simulate.lme" in package nlme;  with luck, it will 
also be in the index to Pinheiro and Bates.  If not, it's still not hard 
to find.  Their chapter on this includes some marvelous figures 
presenting simulation results presumably produced with a version of 
"simulate.lme".

	  The paper that seemed most insightful to me is Crainiceanu, Ruppert 
and Vogelsang (2003) Some Properties of Likelihood Ratio Tests in Linear 
Mixed Models" Cornell U. 
(http://www.orie.cornell.edu/~davidr/papers/zeroprob_rev01.pdf). 
However, I suggest you also examine Ota et al. (2000) "Approximate 
Likelihood Ratio Tests and Marginal Distributions for Evolutionary Tree 
Models with Constraints on Parameters", Molecular Biology and Evolution 
17:798-803.

	  Just now I found other references by Googling for "likelihood ratio 
with parameter at a boundary".  My literature search on this issue is 
far from exhaustive, and I would be pleased to hear from others who have 
other articles to recommend.

	  spencer graves

#####################
Andrew Robinson wrote:
> Hi Spencer,
 >
 > thanks for this note to the r-users list.  I wonder if I could ask ...
 >
 >
 >>	  4.  With parameters at a boundary as with variance components, the
 >>best procedure seems to double the p-value from a nested anova (unless
 >>the reported p-value is already large).  This is because the
 >>2*log(likelihood ratio) in such cases is roughly a 50-50 mixture of 0
 >>and chi-square(1) [if testing only 1 variance component parameter].
 >>This is supported by a substantial amount of research, including
 >>simulations discussed in a chapter in Pinheiro and Bates (2000)
 >>Mixed-Effects Models in S and S-Plus (Springer).  The may be more
 >>accurate procedures available in the literature, but none so simple as
 >>this as far as I know.
 >
 >
 > Could you provide a page reference or references in the literature?
 > I've heard similar things before, but never seen them myself.  Don't
 > worry if they're not immediately available.
 >
 > Thanks,
 >
 > Andrew
######################################
	  Is there some reason you are NOT using "anova", as in "Examples"
section of "?lmer"?

	  Permit me to summarize what I know about this, and I'll be pleased if
someone else who thinks they know different would kindly enlighten me
and others who might otherwise be misled if anything I say is
inconsistent with the best literature available at the moment:

	  1.  Doug Bates in his PhD dissertation and later in his book with Don
Watts (1988) Nonlinear Regression Analysis and Its Applications (Wiley)
split approximation errors in nonlinear least squares into "intrinsic
curvature" and "parameter effects curvature".  He quantified these two
problems in the context of roughly three dozen published examples, if my
memory is correct, and found that in not quite all cases, the parameter
effects were at least an order of magnitude greater than the intrinsic
curvature.

	  2.  In nonnormal situations, maximum likelihood is subject to more
approximation error -- intrinsic curvature -- than "simple" nonlinear
least squares.  However, I would expect this comparison to still be
fairly accurate, even if the differences may not be quite as stark.

	  3.  The traditional use of "standard errors" to judge statistical
significance is subject to both intrinsic and parameter effects errors,
while likelihood ratio procedures such as anova are subject only to the
intrinsic curvature (assuming there are no substantive problems with
nonconvergence).  Consequently, to judge statistical significance of an
effect, anova is usually substantially better than the so-called Wald
procedure using approximate standard errors, and is almost never worse.
  If anyone knows of a case where this is NOT true, I'd like to know.

	  4.  With parameters at a boundary as with variance components, the
best procedure seems to double the p-value from a nested anova (unless
the reported p-value is already large).  This is because the
2*log(likelihood ratio) in such cases is roughly a 50-50 mixture of 0
and chi-square(1) [if testing only 1 variance component parameter].
This is supported by a substantial amount of research, including
simulations discussed in a chapter in Pinheiro and Bates (2000)
Mixed-Effects Models in S and S-Plus (Springer).  The may be more
accurate procedures available in the literature, but none so simple as
this as far as I know.

	  Comments?
	  spencer graves
p.s.  It looks like fm at bVars is a list containing vectors of length 29
and 6 in your example.  I don't know what they are, but I don't see how
they can be standard errors in the usual sense.
Doran, Harold wrote:

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

      
      
    

        

  
    

      
      
    

  


  


  


  
  

    

  

    
      
    
    
Thread (3 messages)

  

  

    
            
        Doran, Harold
        
        How to assess significance of random effect in lme4
      
      Aug 17
    
    
      
            
        Spencer Graves
        
        How to assess significance of random effect in lme4
      
      Aug 17
    
    
      
            
        Spencer Graves
        
        How to assess significance of random effect in lme4
      
      Aug 17