[R] Do Users of Nonlinear Mixed Effects Models Know Whether Their Software Really Works?
dave fournier
otter at otter-rsch.com
Thu Oct 13 22:19:24 CEST 2005
Do Users of Nonlinear Mixed Effects Models Know
Whether Their Software Really Works?
Lesaffre et. al. (Appl. Statist. (2001) 50, Part3, pp 325-335)
analyzed
some simple clinical trials data using a logistic random effects
model. Several packages and methods MIXOR, SAS NLMIXED were employed.
They reported obtaining very different parameter estimates and
P values for the log-likelihood with the different packages and
methods. We thought it would be interesting to revisit this example
using the AD Model Builder random effects module which we feel is
the most stable software available for this problem at this time.
http://otter-rsch.com/admodel.htm
You can get Table 2 from Lesaffre et al at
http://otter-rsch.com/downloads/other/lesaffre.pdf
The data and more information are available
from the publisher.
http://www.blackwellpublishers.co.uk/rss/Volumes/Cv50p3.htm
We considered three questions:
1.) What are the estimates using the Laplace approximation
for integrating out the random effects.
2.) What are the exact MLE's.
3.) How well does hypothesis testing (likelihood-ratio) using
the Laplace approximation compare with the exact MLE's.
We first fit the data using ADMB-RE's Laplace approximation
option.
Laplace approximation estimates:
# Number of parameters = 4 log-likelihood = -629.817
value std dev P value
b_1 -2.3321e+00 7.6973e-01 < 0.0024
b_2 -6.8795e-01 6.6185e-01 0.298
b_3 -4.6134e-01 4.0000e-02 < 0.001
sigma 4.5738e+00 7.0970e-01
The parameter of interest here the treatment effect b_2 which is the
parameter reported in Lesaffre et. al.
To calculate the exact MLE we fit the model using 100 point adaptive
Gaussian integration. The ADMB-RE results were:
Gaussian integration estimates:
# Number of parameters = 4 log-likelihood = -627.481
name value std dev P value
b_1 -1.4463e+00 4.2465e-01 < 0.001
b_2 -5.2225e-01 5.5716e-01 0.348
b_3 -4.5150e-01 3.6663e-02 < 0.001
sigma 4.0137e+00 3.8083e-01
Of the estimates reported in Lesaffre et al. in table 2 only the
50 point quadrature for the program MIXOR appear to be correct
for both the log-likelihood value and the parameter estimates
while the authors concluded that the SAS NLMIXED parameter estimates
they obtained were correct. So even though these authors were looking
for pathological behaviour and were presumably very careful, and their
paper was presumably peer-reviewed, they came to the wrong conclusion
using SAS NLMIXED.
How do we know that our exact MLE's are correct? To confirm our
results we used our parameter estimates as initial values in the SAS
NLMIXED procedure using 100 point adaptive quadrature. The procedure
returned our values, that is it agreed that these are the maximum
likelihood estimates. However we verified that to get these estimates
from the SAS NLMIXED procedure one must begin with fairly good
starting values. In contrast the ADMB-RE procedure is very insensitive
to the starting values used. Our conclusion is that while SAS NLMIXED
might work for this very simple problem it probably begins to break
down when the problem is a bit more difficult.
The ADMB-RE software is more stable because it calculates exact higher
oreder derivatives by automatic differentiation for use in its
optimization procedure and calculations while other packages do not.
Gauss-Hermite integration for the random effects can
be used for this model because the Hessian for the random effects is
diagonal which permits one dimensional integration over the random
effects to great accuracy. However this procedure does not scale well
to problems where the Hessian is not diagonal. Suppose that it takes
a 20 point quadrature to obtain reliable parameter estimates with a
diagonal Hessian. Then with a block diagonal Hessian where the blocks
are of size 4x4 it would take 160,000 points.
Results using R
We fit the model using what appear to be the currently available
procedures in R. The two routines lmer (lme4 package) and glmmPQL
(MASS library) were tried.
The call
>> lmer(y ~ treat + time + (1|subject),data=lesaffre,family=binomial)
resulted in a warning message from lme4() but both routines produced
the same results.
Generalized linear mixed model fit using PQL
Formula: y ~ treat + time + (1 | subject)
Data: lesaffre
Family: binomial(logit link)
AIC BIC logLik deviance
1305.859 1333.628 -647.9295 1295.859
Random effects:
Groups Name Variance Std.Dev.
subject (Intercept) 6.8059 2.6088
# of obs: 1908, groups: subject, 294
Estimated scale (compare to 1) 0.9091945
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.626214 0.264996 -2.3631 0.01812 *
treat -0.304660 0.360866 -0.8442 0.39853
time -0.346605 0.026666 -12.9979 < 2e-16 ***
sigma 2.608
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Warning message:
optim or nlminb returned message ERROR:
ABNORMAL_TERMINATION_IN_LNSRCH
in: LMEopt(x = mer, value = cv)
The R routine correctly identifies the treatment effect as not
significant. However the parameter estimates are poor.
Likelihood Ratio Testing
Accurate calculation of the log-likelihood value is desirable so that
hypothesis testing can be carried out using likelihood ratio tests.
However as noted above the use of Gaussian integration is not
practical for many nonlinear mixed models. We were interested in
seeing how well the use of the approximate log-likelihood values
produced by ADMB-RE's Laplace approximation option would perform.
We consider the alternative model with an extra interaction term
(b_4) from Lesaffre et al.
Here are the results for the laplace approximation:
# Number of parameters = 5 log-likelihood = -627.809
name value std dev P vlaue
b_1 -2.5233e+00 7.8829e-01 < 0.002
b_2 -3.0702e-01 6.8996e-01 0.655
b_3 -4.0009e-01 4.7059e-02 < 0.001
b_4 -1.3726e-01 6.9586e-02 0.044
sigma 4.5783e+00 7.2100e-01
and the exact parameter estimates by 100 point Gaussian
integration.
# Number of parameters = 5 log-likelihood = -625.398
name value std dev P value
b_1 -1.6183e+00 4.3427e-01 < 0.001
b_2 -1.6077e-01 5.8394e-01 0.783
b_3 -3.9100e-01 4.4380e-02 < 0.001
b_4 -1.3679e-01 6.8013e-02 0.044
sigma 4.0131e+00 3.8044e-01
The log-likelihood differences are 2.01 for the Laplace
approximation and 2.08 for Gaussian integration.
Since the 95% point for hypothesis testing is 1.92
use of either model results in acceptance of the interaction
term.
Conclusions
With the exception of AD Model Builder random effect module none of
the packages tested appear to function reliably for this problem.
SAS NLMIXED was beginning to exhibit symptoms of instability which
would probably render it unreliable on more difficult problems. We
can see no reason for using "quasi-likelihoods" to fit nonlinear
mixed models when ADMB-RE can fit the models by maximum likelihood
with all the advantages that ensue.
Note
We realize that there are many other packages out there. We would
welcome results for other packages. If we can find a serious
competitor to AD Model Builder then we could move on to comparing
the relative performance on more difficult models.
Cheers,
Dave Fournier
--
David A. Fournier
P.O. Box 2040,
Sidney, B.C. V8l 3s3
Canada
http://otter-rsch.com
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