glmmADMB: Generalized Linear Mixed Models using AD Model Builder

On 12/15/05, Roel de Jong <dejongroel at gmail.com> wrote:

Dear R-users,

because lme(r) & glmmpql, which are based on Penalized Quasi Likelihood,
are not very robust with Bernoulli responses,

The current version of lmer takes method =  "PQL" (the default) or
"Laplace" or "AGQ" although AGQ is not available for vector-valued
random effects in that version so one must be content with "PQL" or
"Laplace"

I wanted to test glmmADMB.  I run the following simulation study:

500 samples are drawn with the model specification:
y = (intercept*f1+pred2*f2+pred3*f3)+(intercept*ri+pred2*rs)
    where pred2 and pred3 are predictors distributed N(0,1)
    f1..f3 are fixed effects, f1=-1, f2=1.5, f3=0.5
    ri is random intercept with associated variance var_ri=0.2
    rs is random slope with associated variance var_rs=0.4
    the covariance between ri and rs "covr"=0.1

1500 units/dataset, class size=30

Could you make the datasets, or the code that generates them,
available?  My code for such a simulation would be

genGLMM <- function(nobs, gsiz, fxd, Sigma, linkinv = binomial()$linkinv)
{
    ngrp <- nobs/gsiz
    ranef <- matrix(rnorm(ngrp * ncol(Sigma)), nr = ngrp) %*% chol(Sigma)
    pred2 <- rnorm(nobs)
    pred3 <- rnorm(nobs)
    mm <- model.matrix(~pred2 + pred3)
    rmm <- model.matrix(~pred2)
    grp <- gl(n = 1500/30, k = 30, len = 1500)
                                        # linear predictor
    lp <- as.vector(mm %*% fxd + rowSums(rmm * ranef[grp,]))
    resp <- as.integer(runif(nobs) < linkinv(lp))
    data.frame(resp = resp, pred2 = pred2, pred3 = pred3, grp = grp)
}

Running this function gives

nobs <- 1500
gsiz <- 30
fxd <- c(-1, 1.5, 0.5)
Sigma <- matrix(c(0.2, 0.1, 0.1, 0.4), nc = 2)
set.seed(123454321)
sim1 <- genGLMM(nobs, gsiz, fxd, Sigma)
(fm1 <- lmer(resp ~ pred2 + pred3 + (pred2|grp), sim1, binomial))

Generalized linear mixed model fit using PQL
Formula: resp ~ pred2 + pred3 + (pred2 | grp)
   Data: sim1
 Family: binomial(logit link)
      AIC      BIC    logLik deviance
 1403.522 1440.714 -694.7609 1389.522
Random effects:
 Groups Name        Variance Std.Dev. Corr
 grp    (Intercept) 0.44672  0.66837
        pred2       0.55629  0.74585  0.070
# of obs: 1500, groups: grp, 50

Estimated scale (compare to 1)  0.9032712

Fixed effects:
             Estimate Std. Error z value  Pr(>|z|)
(Intercept) -1.081710   0.121640 -8.8927 < 2.2e-16
pred2        1.607273   0.141697 11.3430 < 2.2e-16
pred3        0.531071   0.072643  7.3107 2.657e-13

system.time(fm1 <- lmer(resp ~ pred2 + pred3 + (pred2|grp), sim1, binomial))

[1] 0.33 0.00 0.33 0.00 0.00

(fm2 <- lmer(resp ~ pred2 + pred3 + (pred2|grp), sim1, binomial, method = "Laplace"))

Generalized linear mixed model fit using Laplace
Formula: resp ~ pred2 + pred3 + (pred2 | grp)
   Data: sim1
 Family: binomial(logit link)
      AIC      BIC    logLik deviance
 1401.396 1438.588 -693.6979 1387.396
Random effects:
 Groups Name        Variance Std.Dev. Corr
 grp    (Intercept) 0.35248  0.59370
        pred2       0.46641  0.68294  0.077
# of obs: 1500, groups: grp, 50

Estimated scale (compare to 1)  0.9854841

Fixed effects:
             Estimate Std. Error z value  Pr(>|z|)
(Intercept) -1.119008   0.121640 -9.1993 < 2.2e-16
pred2        1.680916   0.141697 11.8627 < 2.2e-16
pred3        0.543548   0.072643  7.4825 7.293e-14

system.time(fm2 <- lmer(resp ~ pred2 + pred3 + (pred2|grp), sim1, binomial, method = "Laplace"))

[1] 4.62 0.01 4.65 0.00 0.00

Fitting that model using glmmADMB gives

(fm3 <- glmm.admb(resp ~ pred2 + pred3, ~ pred2, "grp", sim1, "binomial", "logit", "full"))

...
iteration output omitted
...

GLMM's in R powered by AD Model Builder:

  Family: binomial

Fixed effects:
  Log-likelihood: -602.035
  Formula: resp ~ pred2 + pred3
(Intercept)       pred2       pred3
   -1.11990     1.69030     0.54619

Random effects:
  Grouping factor: grp
  Formula: ~pred2
Structure: General positive-definite
               StdDev      Corr
(Intercept) 0.5890755
pred2       0.6712377 0.1023698

Number of Observations: 1500
Number of Groups: 50

The "Laplace" method in lmer and the default method in glmm.admb,
which according to the documentation is the Laplace approximation,
produce essentially the same model fit.  One difference is the
reported value of the log-likelihood, which we should cross-check, and
another difference is in the execution time

system.time(fm3 <- glmm.admb(resp ~ pred2 + pred3, ~ pred2, "grp", sim1, "binomial", "logit", "full"))

...
Iteration output omitted
...
[1]  0.23  0.02 21.44 19.45  0.24

Fitting this model takes about 4.7 seconds with the Laplace
approximation in lmer (and only 0.33 seconds for PQL, which is not
that far off) and about 20 seconds in glmm.admb

convergence:
~~~~~~~~~~~~
No crashes.
5/500 Datasets had on exit a gradient of the log-likelihood > 0.001
though. Removing the datasets with questionable convergence doesn't seem
to effect the simulation analysis.

bias:
~~~~~~
f1=-1.00531376
f2= 1.49891060
f3= 0.50211520
ri= 0.20075947
covr=0.09886267
rs= 0.38948382

Only the random slope "rs" is somewhat low, but i don't think it is of
significance

coverage alpha=.95: (using asymmetric confidence intervals)
~~~~~~~~~~~~~~~~~~~~~~~~
f1=0.950
f2=0.950
f3=0.966
ri=0.974
covr=0.970
rs=0.970

While some coverages are somewhat high, confidence intervals based on
asymptotic theory will not have exactly the nominal coverage level, but
with simulations (parametric bootstrap) that can be corrected for.

I can highly recommend this excellent package to anyone fitting these
kinds of models, and want to thank Hans Skaug & Dave Fournier for their
hard work!

I agree.  I am particularly pleased that Otter Research allows access
to a Linux executable of their code (although I would, naturally,
prefer the code to be Open Source).

Roel de Jong.


Hans Julius Skaug wrote:

Dear R-users,

Half a year ago we put out the R package "glmmADMB" for fitting
overdispersed count data.

http://otter-rsch.com/admbre/examples/glmmadmb/glmmADMB.html

Several people who used this package have requested
additional features. We now have a new version ready.
The major new feature is that glmmADMB allows Bernoulli responses
with logistic and probit links. In addition there is
a "ranef.glmm.admb()" function for getting the random effects.

The download site is still:

http://otter-rsch.com/admbre/examples/glmmadmb/glmmADMB.html

The package is based on the software ADMB-RE, but the full
unrestricted R-package is made freely available by Otter Research Ltd
and does not require ADMB-RE to run. Versions for Linux and Windows
are available.

We are still happy to get feedback for users, and to get suggestions
for improvement.

We have set up a forum at http://www.otter-rsch.ca/phpbb/ for discussions
about the software.

Regards,

Hans