[R] glmmADMB: Generalized Linear Mixed Models using AD Model Builder

[Roel de Jong]

One thing I forgot to mention: the data was generated and fitted with 
the binomial probit (not logit) link

Roel de Jong

Roel de Jong wrote:
> Dear professor Bates,
> 
> thank you for your reaction. To make sure that no errors occur in the 
> data generation process I used the elegant function you so neatly 
> provided to generate a couple of datasets under the model specification 
> specified earlier. Running lmer with a Laplace approximation to the 
> high-dimensional integral in the likelihood gives me a warning an then 
> this show-stopper:
> 
> Warning: IRLS iterations for PQL did not converge
> Error in objective(.par, ...) : Unable to invert singular factor of 
> downdated X'X
> 
> Fitting the dataset with glmmADMB gives no apparent problems and 
> reasonable estimates. I attached the particular dataset to the email.
> 
> The difference in computation time can be attributed to the fact that 
> glmmadmb uses a generic technique called automatic differentiation with 
> the Laplace approximation. The same technique can be employed to fit 
> much more complex nonlinear models, but I'm sure Hans & Dave can tell 
> more about it.
> 
> Best regards,
>     Roel de Jong
> 
> 
> Douglas Bates wrote:
> 
>> 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
>>>>
>>>> _____________________________
>>>> Hans Julius Skaug
>>>>
>>>> Department of Mathematics
>>>> University of Bergen
>>>> Johannes Brunsgate 12
>>>> 5008 Bergen
>>>> Norway
>>>> ph. (+47) 55 58 48 61
>>>>
>>>> ______________________________________________
>>>> R-help at stat.math.ethz.ch mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> PLEASE do read the posting guide! 
>>>> http://www.R-project.org/posting-guide.html
>>>>
>>>
>>> ______________________________________________
>>> R-help at stat.math.ethz.ch mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide! 
>>> http://www.R-project.org/posting-guide.html
>>>
>>
>>
> 
> ------------------------------------------------------------------------
> 
> dep	pred2	pred3	grp
> 0	-0.832299769325581	-1.67880317415923	1
> 0	-0.308299055464121	-0.364611545217099	1
> 1	0.630598386837566	-0.830616438489193	1
> 1	3.10913858356640	-0.521767356153283	1
> 1	0.903471812881371	1.91484272352174	1
> 0	0.90426769897266	-0.513292891585524	1
> 1	1.14495484917281	0.78473139460995	1
> 0	-0.234525952851387	-0.50697276499942	1
> 1	0.631041223657181	0.0885911624846033	1
> 0	-1.28176856712309	0.381144997887118	1
> 0	-0.199575490516357	-0.816864500080217	1
> 0	1.1185927379226	-1.89658827063818	1
> 0	-0.166954429106639	-1.72768529536314	1
> 0	0.410078885348753	-2.37013110454109	1
> 0	0.37016590773711	-0.320034871756389	1
> 0	-1.29319553111599	-0.243436474810949	1
> 1	1.44188541761604	0.46171661294066	1
> 1	1.03439142621889	-0.0934696125911877	1
> 1	0.941957044041486	1.46031495794676	1
> 0	0.63243411925538	-0.376900200288554	1
> 0	0.782797578938882	-2.17296669618680	1
> 0	-0.0434466627421846	-0.897181210612278	1
> 0	-0.918950768280257	1.40593647574051	1
> 0	-0.266797680590651	1.32852283384323	1
> 0	-1.97795808769759	-0.340096532983713	1
> 0	0.338662524103419	0.317005354534584	1
> 1	-0.778330737845841	1.83714022902796	1
> 1	0.167661592741467	-0.898198193914343	1
> 0	0.108266386948271	-1.31922947083775	1
> 1	0.799587591780234	1.20654772520622	1
> 0	-1.39792357224312	-1.17382560290430	2
> 1	1.41625128975700	0.0321732274019762	2
> 1	-0.0781989965585817	1.66218225184847	2
> 0	-1.57620982406135	0.122918359084620	2
> 0	-0.169615551813902	-1.32992468631561	2
> 0	0.0744278240167874	-0.793642608076887	2
> 1	1.33756270180946	1.43265155843536	2
> 0	0.171687202226864	-0.197474664007768	2
> 0	-0.253842074055872	0.422229019624458	2
> 1	1.51037367919741	0.708493875055813	2
> 0	0.126257617278352	-0.698370297940141	2
> 0	-1.41052021563966	-0.580830471211872	2
> 0	-0.740991848275336	-0.831723737484574	2
> 0	-1.38752149645338	-0.0645813982998873	2
> 0	-1.48202337888827	-0.810824176493156	2
> 0	0.434323217159074	-0.235959885057200	2
> 0	-0.3660190454198	-0.207497034351587	2
> 0	-1.55757318106926	-0.649666741994972	2
> 0	0.200298815968865	-1.25786648414456	2
> 1	0.60068195996731	1.39546043864496	2
> 1	0.92181552285589	0.169900540315928	2
> 1	0.963639582191724	-0.96652696802166	2
> 0	-1.10894528405182	-0.826447468782818	2
> 1	-0.143705873984295	0.85696358848945	2
> 1	-0.227197073907219	0.163108840069261	2
> 0	0.260778722736436	0.702350481108482	2
> 1	0.0079839656810117	0.640974302851168	2
> 1	0.520335955317318	0.82948476190548	2
> 0	-0.470053854425421	-1.41586935932929	2
> 0	0.318269807834773	0.221705522455281	2
> 0	-0.490245326301728	0.553620570804301	3
> 0	-0.276149978720162	-0.88110452374587	3
> 0	0.220721777999604	-0.653256457257882	3
> 0	-0.703075262079484	0.329683608971329	3
> 0	-0.22696243906353	-2.61950724697562	3
> 0	-1.09431005668124	-0.0232909455712828	3
> 1	1.28711954899881	0.344320685687299	3
> 0	-0.151754090955171	-0.0744843243935522	3
> 0	0.341919519960378	-0.0964112795642981	3
> 0	-1.12551027730654	-0.84933031360325	3
> 0	-0.83446590590987	-0.158563062623380	3
> 0	-0.519454023512489	-1.24574306199025	3
> 0	0.273951811480161	-0.738838977465222	3
> 1	0.644417981459438	0.48825723724319	3
> 0	0.80733294439454	0.38447235881228	3
> 0	-1.30801233701682	1.32830707446622	3
> 0	-0.691273996772228	-1.03695064187955	3
> 0	-0.55275315135725	0.712670673357253	3
> 1	1.18912706586104	-0.797779680340233	3
> 0	0.173813919484423	-0.212804192064869	3
> 0	0.510051214308946	0.407985297642093	3
> 0	0.0460404954363198	0.0781827813302751	3
> 0	-0.812726427004377	-0.961698454122676	3
> 0	0.316803703625498	0.436795213783597	3
> 0	1.59372338969642	-0.646972799801143	3
> 0	-0.264479048468343	0.436583419258436	3
> 1	1.79785791935398	-2.19022854850479	3
> 0	-1.25542914249998	0.759841939131259	3
> 0	-0.556703265843941	-0.596140993896742	3
> 0	-0.138859132587877	0.353420559076652	3
> 0	-0.54936814739631	0.5651403571148	4
> 0	0.0865897190005032	-1.28701512066411	4
> 0	-0.59359121771434	0.137688285767041	4
> 0	1.29103265636505	-1.20664947663906	4
> 0	0.593107076808349	0.398984411079726	4
> 0	-0.226418280356368	0.252338568472061	4
> 0	-0.622343771507656	-0.150521153702500	4
> 0	-0.430960330395192	-0.61365230036041	4
> 0	-0.897917002717756	1.24600197335951	4
> 0	-0.131652665221839	-1.14902442133866	4
> 1	1.05932918053364	-0.208043968099735	4
> 1	2.21374179592236	-2.11459091350310	4
> 0	-0.0984023158102187	0.620618589465858	4
> 0	-0.301419555524483	-1.51777513767374	4
> 0	0.705447946370506	0.535510970861683	4
> 0	-0.740935396882992	-0.554365880210576	4
> 1	1.05185254417717	1.44454045163646	4
> 0	-0.976862868950618	-0.28295698647597	4
> 0	-1.81719506639831	-0.214322369708696	4
> 0	-0.249925818554222	0.183451295261106	4
> 0	-2.21228328895419	0.123877231369233	4
> 1	1.38439754243109	0.505675670636377	4
> 0	-0.837221588634301	-1.50135483614086	4
> 0	-1.17154448913943	-0.708764217205316	4
> 1	1.50155644902162	0.43999620318975	4
> 0	-0.516075651409885	1.10156000900777	4
> 0	-0.583328580107054	-0.136624407275739	4
> 0	-0.805969095827047	1.23242239143692	4
> 0	0.520037308192361	-0.0532028414333528	4
> 0	-1.07675357578869	-0.246422992512653	4
> 1	0.84674606454176	1.24706822832629	5
> 1	0.50225429700895	-0.753501502939139	5
> 0	-0.110696991251422	-1.28963013849050	5
> 1	1.45669586789922	-0.0266212621300831	5
> 0	1.19164829320594	-1.6384147982708	5
> 1	1.2497582567896	-0.366144380048512	5
> 0	-1.59200279904562	-0.605612412386137	5
> 1	0.701642320563013	-0.692590957893450	5
> 0	0.826729750207121	0.104040242709383	5
> 0	-0.251589578605641	-1.20008533845167	5
> 0	1.18981892198469	-1.03838218637413	5
> 0	-0.48226798541373	-0.103516021510253	5
> 1	0.902430923145418	-1.25114657112499	5
> 0	0.480518605880176	0.223857697801260	5
> 0	-2.25438268634199	1.21126580568773	5
> 0	-0.383234731665990	-0.766293999344111	5
> 0	0.581319800323777	0.817394753080253	5
> 0	-3.3165427929598	-0.121062111785958	5
> 0	-0.1322580996873	-0.150578017302552	5
> 1	1.53193807897104	0.733939342102887	5
> 1	1.11908937603262	0.196818601090142	5
> 0	0.437875954273944	-0.0197136588508132	5
> 0	-0.0331688915453021	0.33400847926927	5
> 1	1.34543708556248	0.00907648260071153	5
> 0	0.213075347139783	-0.646249613641689	5
> 0	-0.69524684216634	-2.33922277678765	5
> 0	-1.57279905654483	-1.87538951834327	5
> 1	1.07936855487838	1.18446965022099	5
> 1	0.306985931587578	0.724405953653822	5
> 1	1.23181672693378	-0.0701925304204026	5
> 0	-0.919888911082209	0.477276404654286	6
> 0	-0.663571170286572	-1.46630908083576	6
> 0	-0.724039231476495	-0.061019673195811	6
> 0	0.0634460402565889	-1.08461362641975	6
> 1	-0.269323637008873	0.731446475758064	6
> 1	-0.146752286719258	1.51369083876776	6
> 0	-0.35095531255735	-0.234620941343001	6
> 0	0.273206507398862	-0.393903382508312	6
> 1	-0.182666577034182	1.83249929313715	6
> 0	0.681588488289945	-0.375859982611083	6
> 0	0.242434133831772	-1.64320325331244	6
> 1	0.188017184214761	-0.228076562308703	6
> 1	0.166649765922797	0.511793542958201	6
> 1	2.16402504068409	1.66375788446841	6
> 1	0.586097780802391	-0.289750400216476	6
> 0	-0.757716234538208	-0.356355379036605	6
> 0	-1.11703624150270	-1.02103411104842	6
> 1	1.32065912451240	-0.529344207548823	6
> 0	-0.563323776359257	-0.575116970325914	6
> 0	0.0761081554417085	0.466755576888024	6
> 0	-0.0393307073575922	-2.09949197028072	6
> 0	-1.45391429237201	-1.78607291521398	6
> 0	0.180700026079273	0.785140819957987	6
> 1	0.842204310528479	-0.078203655202921	6
> 0	0.957483010083191	0.0933486661821926	6
> 0	-1.34841123138060	-1.25281641699431	6
> 0	0.460944146864481	-0.516097804035054	6
> 1	1.67680834988551	-0.7539974457697	6
> 1	0.204256257528345	-0.389312192486669	6
> 1	0.546750712218541	3.06217530791803	6
> 0	0.0172078969431791	-0.101707833219385	7
> 0	-1.37010127340567	-1.02691608953111	7
> 0	2.56322137123985	-0.773240110395935	7
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> 1	0.491610220497774	-0.201909505174161	49
> 1	0.0479089033848729	0.834929949270871	49
> 1	0.0408937347976734	0.256311982993567	49
> 0	0.226321411678572	0.754127840849748	49
> 0	0.167859720959811	-0.856705902608151	49
> 0	-0.21438260903557	0.852938961109979	49
> 1	-0.137091138248145	1.37023387524365	49
> 0	-0.40768003690041	-0.605890465095008	49
> 0	-1.12149092127315	-1.78702862251756	49
> 1	0.0618578989992678	0.805416211854734	49
> 1	0.646167720744804	1.53728535762444	49
> 0	1.09985118437066	-0.51891947747395	49
> 0	-0.900129775037201	0.255426863387068	49
> 0	-1.36824937842165	-1.39605795509068	50
> 1	0.337292185184476	1.65813845617190	50
> 0	-2.59569397772874	0.838885380679198	50
> 0	0.648340806855909	0.194452874450751	50
> 0	-0.619036384971611	1.59599180284695	50
> 1	0.725811048112912	-0.219993359723846	50
> 0	-1.03964840980762	-0.875973638276369	50
> 1	1.10500418445586	0.439320468455236	50
> 1	0.906275897357221	-0.222053185810672	50
> 0	0.542431214138462	-0.503057777920217	50
> 0	-0.913489057715955	-1.54510601308067	50
> 0	-1.24147714001748	-0.626838392274742	50
> 0	0.251694589241327	-0.709916460174467	50
> 1	2.00986460944795	0.838024595423274	50
> 0	-0.305422111449992	-1.12353061011042	50
> 1	4.01604386474926	-0.69908849473065	50
> 1	0.0158673469078738	0.414729691505543	50
> 1	-0.175788159371682	0.715509498217549	50
> 0	-0.886437935370422	-0.683179013117859	50
> 1	0.762854619547605	0.629460507983583	50
> 1	1.21325869987561	-0.769799862061684	50
> 0	0.797912391730806	-0.417065865809694	50
> 0	-0.783638662157011	0.241590305559789	50
> 1	0.430123196585032	-0.299966877273855	50
> 0	-1.19015205010331	-0.161934492300737	50
> 0	-1.83358379721416	-0.633639841598871	50
> 1	1.83920861166662	1.25233060723409	50
> 1	0.859305709603053	0.0893834866951533	50
> 1	2.14654325838707	0.911332807026268	50
> 0	-1.43706014676958	-0.338272875805398	50