[R] How to test for significance of random effects?

[Dan Bebber]

I may be out of my statistical depth here, but isn't it the case that if one 
has an experimental design with random effects, one has to include the 
random effects, even if they appear to be non-significant?
AFAIK there are two reasons: one is the possibility of 'restriction errors' 
that arise by unintentional differences in treatments among groups, so 
making analysis of among-group variance problematic; the other is that 
allocations of fixed effects to samples is no longer random and therefore 
the assumption of random errors is broken.
Real statisticians may disagree with this, however.

Dan Bebber

Department of Plant Sciences
University of Oxford

Message: 12
Date: Sun, 07 May 2006 14:25:44 -0700
From: Spencer Graves <spencer.graves at pdf.com>
Subject: Re: [R] How to test for significance of random effects?
To: Jon Olav Vik <jonovik at start.no>
Cc: r-help at stat.math.ethz.ch
Message-ID: <445E65D8.5010504 at pdf.com>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed

  1.  Ignoring the "complication of logistic regression", the
"anova(lme1,lm1)" provides the answer you seek.  See sect. 2.4 in
Pinheiro and Bates for more detail on the approximations involved and
how that answer can be refined using monte carlo.

  2.  With logistic regression, you want to do essentially the same
thing using glm and lmer (in package 'lme4'), except that many of the
required functions are not yet part of 'lme4'.  Consider the following
example:

library(lme4)

library(mlmRev)
(mlmR <- vignette("MlmSoftRev"))
#edit(mlmR) # with Rgui
#Stangle(mlmR$file) # with ESS
#   -> then open file MlmSoftRev.R

fitBin <- lmer(use ~ urban+age+livch+(1|district),
                data=Contraception, family=binomial)
fitBin0 <- glm(use ~ urban+age+livch,
                data=Contraception, family=binomial)

2*pchisq(2*as.numeric(logLik(fitBin)-
logLik(fitBin0)), 2, lower.tail=FALSE)

  Note however that this p-value computation is known to be only an
approximation;  see RSiteSearch("lmer p-values") for other perspectives.
  More accurate p-values can be obtained using Markov Chain Monte Carlo,
via "mcmcsamp".

  hope this helps,
  Spencer Graves

Jon Olav Vik wrote:
> Dear list members,
>
> I'm interested in showing that within-group statistical dependence is
> negligible, so I can use ordinary linear models without including random
> effects. However, I can find no mention of testing a model with vs.
> without random effects in either Venable & Ripley (2002) or Pinheiro and
> Bates (2000). Our in-house statisticians are not familiar with this,
> either, so I would greatly appreciate the help of this list.
>
> Pinheiro & Bates (2000:83) state that random-effect terms can be tested
> based on their likelihood ratio, if both models have the same
> fixed-effects structure and both are estimated with REML (I must admit I
> do not know exactly what REML is, although I do understand the concept of
> ML).
>
> The examples in Pinheiro & Bates 2000 deal with simple vs. complicated
> random-effects structures, both fitted with lme and method="REML".
> However, to fit a model without random effects I must use lm() or glm().
> Is there a way to tell these functions to use REML? I see that lme() can
> use ML, but Pinheiro&Bates (2000) advised against this for some reason.
>
> lme() does provide a confidence interval for the between-group variance,
> but this is constructed so as to never include zero (I guess the interval
> is as narrow as possible on log scale, or something). I would be grateful
> if anyone could tell me how to test for zero variance between groups.
>
> If lm1 and lme1 are fitted with lm() and lme() respectively, then
> anova(lm1,lme1) gives an error, whereas anova(lme1,lm1) gives an answer
> which looks reasonable enough.
>
> The command logLik() can retrieve either restricted or ordinary
> log-likelihoods from a fitted model object, but the likelihoods are then
> evaluated at the fitted parameter estimates. I guess these estimates
> differ from if the model were estimated using REML?
>
> My actual application is a logistic regression with two continuous and one
> binary predictor, in which I would like to avoid the complications of
> using generalized linear mixed models. Here is a simpler example, which is
> rather trivial but illustrates the general question:
>
> Example (run in R 2.2.1):
>
> library(nlme)
> summary(lm1 <- lm(travel~1,data=Rail)) # no random effect
> summary(lme1 <- lme(fixed=travel~1,random=~1|Rail,data=Rail)) # random
> effect
> intervals(lme1) # confidence for random effect
> anova(lm1,lme1)
> ## Outputs warning message:
> # models with response "NULL" removed because
> # response differs from model 1 in: anova.lmlist(object, ...)
> anova(lme1,lm1)
> ## Output: Can I trust this?
> #      Model df      AIC      BIC    logLik   Test  L.Ratio p-value
> # lme1     1  3 128.1770 130.6766 -61.08850
> # lm1      2  2 162.6815 164.3479 -79.34075 1 vs 2 36.50451  <.0001
> ## Various log likelihoods:
> logLik(lm1,REML=FALSE)
> logLik(lm1,REML=TRUE)
> logLik(lme1,REML=FALSE)
> logLik(lme1,REML=TRUE)
>
> Any help is highly appreciated.
>
> Best regards,
> Jon Olav Vik
>
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