I meant to respond to this earlier, but now finally got around to it. The nice thing is, there is (for once) an easy answer. For testing covariance parameters, the value under the null is obviously not on the boundary of the parameter space and the concerns with testing parameters on the boundary do not apply. You can actually test this out with some data and the simulate.lme() function (see also chapter 2 in Pinheiro & Bates, 2000). For example:
library(nlme)
res1 <- lme(distance ~ age, random = ~ age | Subject, data = Orthodont)
res2 <- lme(distance ~ age, random = list(Subject = pdDiag(~ age)), data = Orthodont)
res3 <- lme(distance ~ age, random = ~ 1 | Subject, data = Orthodont)
anova(res1, res2, res3)
res.sim13 <- simulate.lme(res3, res1, nsim=1000, seed=18271) ### parameter tested on the boundary
res.sim12 <- simulate.lme(res2, res1, nsim=1000, seed=18271) ### parameter tested not on boundary
plot(res.sim13, df = c(1,2), panel = function(...) {panel.xyplot(...); panel.abline(a=0,b=1, lty="dotted")})
plot(res.sim12, df = c(1,2), panel = function(...) {panel.xyplot(...); panel.abline(a=0,b=1, lty="dotted")})
Note that the 50:50 mixture of chi^2_1 and chi^2_2 seems adequate in the first case. However, in the second case, chi^2_1 is closer to giving you a nominal test (actually it's a bit conservative, but better than the 50:50 mixture).
Best,
Wolfgang
--
Wolfgang Viechtbauer, Ph.D., Statistician
Department of Psychiatry and Psychology
School for Mental Health and Neuroscience
Faculty of Health, Medicine, and Life Sciences
Maastricht University, P.O. Box 616 (VIJV1)
6200 MD Maastricht, The Netherlands
+31 (43) 388-4170 | http://www.wvbauer.com
-----Original Message-----
From: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-
bounces at r-project.org] On Behalf Of Jake Westfall
Sent: Thursday, September 06, 2012 17:57
To: r-sig-mixed-models at r-project.org
Subject: Re: [R-sig-ME] Characterizing correlation between binomial
effects
This brings up a related question for me. Some of the issues that arise in
testing the significance of random variances are due to the null value
being on the boundary of the parameter space (i.e., testing for variance =
0). However, since COvariances can be positive or negative, this seems to
imply that these issues would not apply to the case of testing null
hypotheses about random covariances, such as covariance = 0, as Mike is
doing in his and Reinhold's solution. Is this correct?
Jake