Problems with convergence
It would be nice to have this work properly, as I need it for certain
things and it seems that other people are having similar problems.
Getting it to work by increasing the quadrature points is a bit of an
aberration, it is not what typically happens, and I've put an example at
the end. At least in this one the profiling works which means the maximum
must be fairly close to that obtained from the optimisation.
My feeling on this, is that possibly the problem is not with the optimiser,
seeing that it fails with so many optimisers, but rather with the
calculation of the marginal likelihood. These optimisers don't tend to stop
with 0.001 gradients. When I have time I will find in the code how the node
locations are calculated and see what is happening.
Anyway, here is one that fails irrespective of nAGQ value.
thedata <- structure(list(nEvents = c(10L, 53L, 17L, 18L, 22L, 6L, 16L,
14L, 13L, 18L, 15L, 19L, 52L, 19L, 8L, 16L, 50L, 8L, 9L, 4L,
26L, 45L, 18L, 20L, 5L, 16L, 18L, 7L, 3L, 19L, 30L, 26L, 66L,
23L, 29L, 18L, 72L, 25L, 9L, 2L), total = c(200, 200, 200, 200,
200, 200, 200, 200, 200, 200, 200, 200, 200, 200, 200, 200, 200,
200, 200, 200, 200, 200, 200, 200, 200, 200, 200, 200, 200, 200,
200, 200, 200, 200, 200, 200, 200, 200, 200, 200), trt = c(0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), id = structure(c(1L,
2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L,
16L, 17L, 18L, 19L, 20L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L,
10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L), .Label = c("1",
"2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13",
"14", "15", "16", "17", "18", "19", "20"), class = "factor"),
trt12 = c(-0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5,
-0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5,
-0.5, -0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5)), .Names =
c("nEvents",
"total", "trt", "id", "trt12"), row.names = c(NA, 40L), class =
"data.frame")
glmer1a <- glmer(cbind(nEvents,total-nEvents) ~ -1 + trt + factor(id) +
(0+trt12|id), data=thedata, family=binomial, nAGQ=7)
glmer1b <- glmer(cbind(nEvents,total-nEvents) ~ -1 + trt + factor(id) +
(0+trt12|id), data=thedata, family=binomial, nAGQ=21)
On 1 April 2015 at 02:25, Viechtbauer Wolfgang (STAT) <
wolfgang.viechtbauer at maastrichtuniversity.nl> wrote:
This discussion piqued my interest. The model that Ken was fitting is in
essence one of the models that is fitted by the rma.glmm() function in the
metafor package. This is sometimes called the unconditional model with
fixed study effects. To illustrate:
### original data
thedata <- structure(list(nEvents=c(10L,53L,17L,18L,22L,6L,16L,
14L,13L,18L,15L,19L,52L,19L,8L,16L,50L,8L,9L,4L,
26L,45L,18L,20L,5L,16L,18L,7L,3L,19L,30L,26L,66L,
23L,29L,18L,72L,25L,9L,2L),total=c(200,200,200,200,
200,200,200,200,200,200,200,200,200,200,200,200,200,
200,200,200,200,200,200,200,200,200,200,200,200,200,
200,200,200,200,200,200,200,200,200,200),trt=c(0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1),id=structure(c(1L,
2L,3L,4L,5L,6L,7L,8L,9L,10L,11L,12L,13L,14L,15L,
16L,17L,18L,19L,20L,1L,2L,3L,4L,5L,6L,7L,8L,9L,
10L,11L,12L,13L,14L,15L,16L,17L,18L,19L,20L),.Label=c("1",
"2","3","4","5","6","7","8","9","10","11","12","13",
"14","15","16","17","18","19","20"),class="factor")),.Names=c("nEvents",
"total","trt","id"),row.names=c(NA,40L),class="data.frame")
### restructure data as needed for input into rma.glmm()
dat <- cbind(thedata[1:20,], thedata[21:40,])
dat$id <- dat$id <- dat$trt <- dat$trt <- NULL
colnames(dat) <- c("ci", "n2i", "ai", "n1i")
library(metafor)
library(lme4)
### model fitted by Ken
res1 <- glmer(cbind(nEvents,total-nEvents) ~ trt + factor(id) +
(0+trt|id), data=thedata, family=binomial)
### fit unconditional model with fixed study effects via rma.glmm()
res2 <- rma.glmm(measure="OR", ai=ai, n1i=n1i, ci=ci, n2i=n2i, data=dat,
nAGQ=1)
### to get exact equivalence, use +-1/2 coding for the random effects
thedata$trt12 <- thedata$trt - 1/2
res3 <- glmer(cbind(nEvents,total-nEvents) ~ -1 + trt + factor(id) +
(0+trt12|id), data=thedata, family=binomial)
summary(res1)
summary(res2)
summary(res3)
### end example
A few notes:
1) rma.glmm() uses nAGQ=7 by default, so I switched that to 1 for the
comparison.
2) Some discussion of the 0/1 versus +-1/2 coding can be found in Turner
et al. (2000) and Higgins et al. (2001). I tend to prefer the +-1/2 coding,
so that is also what is currently implemented in rma.glmm(), but I may add
the 0/1 coding as an option.
3) A nice discussion of the model is provided by Senn (2000). He also
discusses a variety of other modeling options, including a model using
random study effects.
4) In fact, the unconditional model with random study effects can be
fitted with:
rma.glmm(measure="OR", ai=ai, n1i=n1i, ci=ci, n2i=n2i, data=dat, model="
UM.RS")
(which makes use of glmer() underneath). As discussed by Senn, this model
may violate what he calls the 'concurrent control principle', but his
wording is cautious ('may violate', 'may be regarded as undesirable'),
which reflects the lack of a thorough discussion in the literature
comparing the various models.
5) Yet another option is the (mixed-effects) conditional logistic model.
See, for example, Stijnen et al. (2010). This model is obtained when
conditioning on the total number of events within each study and leads to
non-central hypergeometric distributions for the data within each study.
This model can be fitted with:
rma.glmm(measure="OR", ai=ai, n1i=n1i, ci=ci, n2i=n2i, data=dat,
model="CM.EL")
Sorry, it's slow (I haven't found a clever way of speeding up the
integration over the non-central hypergeometric distributions). Much
faster, thanks to lme4, is:
rma.glmm(measure="OR", ai=ai, n1i=n1i, ci=ci, n2i=n2i, data=dat, model="
CM.AL")
which uses an approximation to the exact conditional likelihood.
6) And of course there are Bayesian implementations of such models.
7) With respect to the model fitted by Ken, it's maybe interesting to note
that NOT using the Laplace approximation, but something like 7 quadrature
points, does not cause any convergence warnings:
glmer(cbind(nEvents,total-nEvents) ~ trt + factor(id) + (0+trt|id),
data=thedata, family=binomial, nAGQ=7)
Alright, I'll shut up now.
References mentioned above:
Higgins, J. P. T., Whitehead, A., Turner, R. M., Omar, R. Z., & Thompson,
S. G. (2001). Meta-analysis of continuous outcome data from individual
patients. Statistics in Medicine, 20(15), 2219-2241.
Senn, S. (2000). The many modes of meta. Drug Information Journal, 34,
535-549.
Stijnen, T., Hamza, T. H., & Ozdemir, P. (2010). Random effects
meta-analysis of event outcome in the framework of the generalized linear
mixed model with applications in sparse data. Statistics in Medicine,
29(29), 3046-3067.
Turner, R. M., Omar, R. Z., Yang, M., Goldstein, H., & Thompson, S. G.
(2000). A multilevel model framework for meta-analysis of clinical trials
with binary outcomes. Statistics in Medicine, 19(24), 3417-3432.
Best,
Wolfgang
--
Wolfgang Viechtbauer, Ph.D., Statistician
Department of Psychiatry and Neuropsychology
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 [mailto:r-sig-mixed-models-bounces at r- project.org] On Behalf Of Ben Bolker Sent: Monday, March 30, 2015 22:21 To: r-sig-mixed-models at r-project.org Subject: Re: [R-sig-ME] Problems with convergence Ken Beath <ken.beath at ...> writes:
Yes, I was demonstrating that it fails convergence and then as a consequence fails to profile. I have my doubts about convergence for the bobyqa algorithm, I have other applications where it doesn't converge
properly.
For some of my own work I've used nlminb followed by Nelder-Mead if
there
is a convergence failure. Not optimal but it seems to work.
I'm still not sure whether you expect it to converge (I think you do), or whether you are just pointing out that the convergence warning in this case is probably justified (in the face of so many convergence warnings that turn out to be false positives, this is a useful piece of information).
While it is fairly heavily parameterised it is a real model, a frequentist implementation of Smith, T. C., Spiegelhalter, D. J., & Thomas, a.
(1995).
Bayesian approaches to random-effects meta-analysis: a comparative
study.
Statistics in Medicine, 14(24), 2685?99. The reason for having studies
as
fixed effects is probably philosophical, the overall success rates are
not
likely to be given by normally distributed random effects, and are in
many
cases specifically chosen.
I can appreciate that, but I still think it's unrealistic to expect to be able to fit 22 parameters to 40 observations except under very special circumstances. One point about switching from the Bayesian to the frequentist world is that the Bayesians (by definition) put priors on their parameters, which provides a degree of regularization that is not by default available to frequentist methods. What priors did Smith et al. use? It might be worth trying this in blme with priors on the fixed effects ...
I did find that one of the data sets that I have also failed, but
fitted
with a commercial program that is based on the EM algorithm. For this
type
of problem it is actually faster, as any type of quasi-Newton needs to calculate lots of derivatives.
I could whine about the difficulty of finding globally robust, reliable, and fast optimization algorithms, but I won't. I can certainly appreciate that there are more reliable methods for particular sub-classes of problems.
Anyway, I'm going to keep looking at the methods, and eventually the
code
for glmer and may eventually have some suggestions.
Would be happy to hear them. It's worth pointing out that lme4 is using a preliminary "nAGQ=0" step, which ignores the terms contributed by the integrals over the distributions of the conditional modes and as a result is able to fit both the fixed-effect parameters and the conditional modes in a single linear-algebra step, reducing the dimensionality of the nonlinear optimization to the length of the variance-covariance parameter vector ...
On 19 March 2015 at 14:45, Ben Bolker <bbolker <at> gmail.com> wrote:
Ken Beath <ken.beath <at> ...> writes:
The following code shows that there are convergence problem
messages
where there is a problem with convergence. The profiling shows that the maximum found is not the correct one. This is simulated data
for
a binary meta-analysis with fixed effect for study and random
effect
for treatment.
[paragraph snipped to try to make Gmane happy]
However, may I comment that this is a slightly ridiculous scenario?
The data set here has 40 observations, and the model tries to fit 22
parameters. The model that treats id as a random effect works much
better. I can believe there are scenarios where you really do
want study as a fixed effect, but did you expect it to be practical
here?
But maybe you're just trying to show that this is a "true positive"
case for the convergence warnings.
Some random code I wrote while diagnosing what was going on:
library(ggplot2); theme_set(theme_bw())
## proportion + weights is a little easier to handle
thedata <- transform(thedata,prop=nEvents/total)
ggplot(thedata,aes(trt,prop))+geom_point(aes(size=total))+
geom_line(aes(group=id),colour="gray")
glmer1 <- glmer(prop~trt+factor(id)+(0+trt|id),
weights=total,data=thedata,family=binomial)
## id as RE
glmer2 <- glmer(prop~trt+(1|id)+(0+trt|id),
weights=total,data=thedata,family=binomial)
dd <- update(glmer1,devFunOnly=TRUE)
pars <- unlist(getME(glmer1,c("theta","fixef")))
library("bbmle")
ss <- slice2D(pars,dd)
library("lattice")
plot(ss)
## too complex, but too much work to cut down significantly
library(lme4)
thedata <- structure(list(nEvents=c(10L,53L,17L,18L,22L,6L,16L,
14L,13L,18L,15L,19L,52L,19L,8L,16L,50L,8L,9L,4L,
26L,45L,18L,20L,5L,16L,18L,7L,3L,19L,30L,26L,66L,
23L,29L,18L,72L,25L,9L,2L),total=c(200,200,200,200,
200,200,200,200,200,200,200,200,200,200,200,200,200,
200,200,200,200,200,200,200,200,200,200,200,200,200,
200,200,200,200,200,200,200,200,200,200),trt=c(0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1),id=structure(c(1L,
2L,3L,4L,5L,6L,7L,8L,9L,10L,11L,12L,13L,14L,15L,
16L,17L,18L,19L,20L,1L,2L,3L,4L,5L,6L,7L,8L,9L,
10L,11L,12L,13L,14L,15L,16L,17L,18L,19L,20L),.Label=c("1",
"2","3","4","5","6","7","8","9","10","11","12","13",
"14","15","16","17","18","19","20"),class="factor")),.Names=c("nEvents",
"total","trt","id"),row.names=c(NA,40L),class="data.frame") glmer1<-glmer(cbind(nEvents,total-nEvents)~trt+factor(id)+
## (0+trt|id),data=thedata,family=binomial)
# while glmer has problems with component 9 it is 8 with a problem
profile
# I've use devtol so the discrepancy is printed prof.glmer1<-profile(glmer1,which=8,devtol=1.0e-3)
*Ken Beath* Lecturer Statistics Department MACQUARIE UNIVERSITY NSW 2109, Australia Phone: +61 (0)2 9850 8516 Building E4A, room 526 http://stat.mq.edu.au/our_staff/staff_-_alphabetical/staff/beath,_ken/ CRICOS Provider No 00002J This message is intended for the addressee named and may...{{dropped:9}}