[R-sig-ME] generalized linear mixed models: large differences when using glmmPQL or lmer with laplace approximation
Douglas Bates
bates at stat.wisc.edu
Tue Oct 7 22:59:38 CEST 2008
Due to some travel and the need to attend to other projects, I haven't
been keeping up as closely with this list as I normally do. Regarding
the comparison between the PQL and Laplace methods for fitting
generalized linear mixed models, I believe that the estimates produced
by the Laplace method are more reliable than those from the PQL
method. The objective function optimized by the Laplace method is a
direct approxmation, and generally a very good approximation, to the
log-likelihood for the model being fit. The PQL method is indirect
(the "QL" part of the name stands for "quasi-likelihood") and, because
it involves alternating conditional optimization, can alternate
back-and-forth between two potential solutions, neither of which is
optimal. (To be fair, such alternating occurs more frequently in the
analogous method for nonlinear mixed-models, in which I was one of the
co-conspirators, than in the PQL method for GLMMs.)
It may be that the problem you are encountering has more to do with
the use of the quasipoisson family than with the Laplace
approximation. I am not sure that the derivation of the standard
errors in lmer when using the quasipoisson family is correct, in part
because I don't really understand the quasipoisson and quasibinomial
families. As far as I know, they don't correspond to probability
distributions so the theory is a bit iffy.
Do you need to use the quasipoisson family or could you use the
poisson family? Generally the motivation for the quasipoisson familiy
is to accomodate overdispersion. Often in a generalized linear mixed
model the problem is underdispersion rather than overdispersion.
In one of Ben's replies in this thread he discusses the degrees of
freedom attributed to certain t-statistics. Regular readers of this
list are aware that degrees of freedom is one of my least favorite
topics. If one has a reasonably large number of observations and a
reasonably large number of groups then the issue is unimportant.
(Uncertainty in degrees of freedom is important only when the value of
the degrees of freedom is small. In fact, when I first started
studying statistics we used the standard normal in place of the
t-distribution whenever the degrees of freedom exceeded 30).
Considering that the quasi-Poisson doesn't correspond to a probability
distribution in the first place, (readers should feel free to correct
me if I am wrong about this) I find the issue of the number of degrees
of freedom that should be attributed to a distribution of a quantity
calculated from a non-existent distribution to be somewhat off the
point.
I think the problem is more likely that the standard errors are not
being calculated correctly. Is that what you concluded from your
simulations, Ben?
On Tue, Oct 7, 2008 at 8:21 AM, Martijn Vandegehuchte
<martijn.vandegehuchte at ugent.be> wrote:
> Dear list,
> First of all, I am a mere ecologist, trying to get the truth out of his data, and not a statistician, so forgive me my lack of statistical background and possible conceptual misunderstandings.
> I am currently comparing generalized linear mixed models in glmmPQL and lmer, with a quasipoisson family, and have found out that parameter estimates are quite different for both methods. I read some of the discussions on the R-forum and it seems that the Laplace approximation used in the current version of lmer is generally preferred to the PQL method. I am an ex-SAS user, and in proc glimmix in SAS the default is PQL, and the estimates and p-values are almost exact the same as with glmmPQL in R. But lmer gives quite different results, and now I am wondering what would be the best option for me.
> First of all, parameter estimates of a same model can be somewhat different in lmer or glmmPQL. Second of all, in lmer, I only get t-values but no associated p-values (apparently they are omitted because of the uncertainty about the df). But if I compare the t-values generated by glmmPQL with those of a same model in lmer, the differences are substantial. My dataset consists of 120 observations, so basically you could guess the order of magnitude of the p-values in lmer based on the t-value and a "large" df.
> First example:
> In lmer:
>
>> model<-lmer(schirufu~diameter+leafvit+densroot+cover+nemcm+(1|site),family=quasipoisson)
>> summary (model)
> Generalized linear mixed model fit by the Laplace approximation
> Formula: schirufu ~ diameter + leafvit + densroot + cover + nemcm + (1 | site)
> AIC BIC logLik deviance
> 2045 2068 -1015 2029
> Random effects:
> Groups Name Variance Std.Dev.
> site (Intercept) 12.700 3.5638
> Residual 15.182 3.8964
> Number of obs: 120, groups: site, 6
>
> Fixed effects:
> Estimate Std. Error t value
> (Intercept) 1.31017 1.47249 0.890
> diameter -0.24799 0.29180 -0.850
> leafvit 1.29007 0.21041 6.131
> densroot 0.31024 0.04939 6.281
> cover -0.24544 0.22179 -1.107
> nemcm 0.24817 0.12028 2.063
>
> Correlation of Fixed Effects:
> (Intr) diamtr leafvt densrt cover
> diameter 0.031
> leafvit -0.083 0.321
> densroot 0.011 -0.017 -0.202
> cover 0.021 -0.448 0.016 0.214
> nemcm -0.014 0.114 0.114 0.310 -0.017
>>
>
> Although no p-values are given, it suggests that fixed effects leafvit, densroot and nemcm would be significant.
> In glmmPQL:
>
>> model<-glmmPQL(schirufu~diameter+leafvit+densroot+cover+nemcm,random=~1|site,family=quasipoisson)
> iteration 1
> iteration 2
> iteration 3
> iteration 4
> iteration 5
>> summary(model)
> Linear mixed-effects model fit by maximum likelihood
> Data: NULL
> AIC BIC logLik
> NA NA NA
>
> Random effects:
> Formula: ~1 | site
> (Intercept) Residual
> StdDev: 0.7864989 4.63591
>
> Variance function:
> Structure: fixed weights
> Formula: ~invwt
> Fixed effects: schirufu ~ diameter + leafvit + densroot + cover + nemcm
> Value Std.Error DF t-value p-value
> (Intercept) 1.4486735 0.4174843 109 3.470007 0.0007
> diameter -0.2600504 0.3477017 109 -0.747913 0.4561
> leafvit 1.2236406 0.2489291 109 4.915619 0.0000
> densroot 0.3236446 0.0596342 109 5.427164 0.0000
> cover -0.2523163 0.2698555 109 -0.935005 0.3519
> nemcm 0.2336305 0.1451751 109 1.609301 0.1104
> Correlation:
> (Intr) diamtr leafvt densrt cover
> diameter 0.130
> leafvit -0.335 0.313
> densroot 0.027 -0.022 -0.203
> cover 0.090 -0.463 0.015 0.214
> nemcm -0.056 0.097 0.107 0.301 -0.014
>
> Standardized Within-Group Residuals:
> Min Q1 Med Q3 Max
> -2.4956188 -0.4154369 -0.1333850 0.1724601 4.7355928
>
> Number of Observations: 120
> Number of Groups: 6
>>
>
> Note the difference in parameter estimates. Also, the fixed effect nemcm now is not significant any more.
>
> Second example,now with an offset:
> In lmer:
>
>> model<-lmer(nemcm~diameter+leafvit+densroot+rootvit+cover+schirufu+(1|site), offset= loglength, family=quasipoisson)
>> summary (model)
> Generalized linear mixed model fit by the Laplace approximation
> Formula: nemcm ~ diameter + leafvit + densroot + rootvit + cover + schirufu + (1 | site)
> AIC BIC logLik deviance
> 1593 1618 -787.4 1575
> Random effects:
> Groups Name Variance Std.Dev.
> site (Intercept) 21.522 4.6392
> Residual 173.888 13.1867
> Number of obs: 120, groups: site, 6
>
> Fixed effects:
> Estimate Std. Error t value
> (Intercept) 0.06733 1.92761 0.0349
> diameter 0.14665 0.60693 0.2416
> leafvit -0.19902 0.48802 -0.4078
> densroot -0.49178 0.64221 -0.7658
> rootvit 0.37699 0.46810 0.8054
> cover -0.23545 0.57896 -0.4067
> schirufu 0.23226 0.46866 0.4956
>
> Correlation of Fixed Effects:
> (Intr) diamtr leafvt densrt rootvt cover
> diameter -0.016
> leafvit 0.015 0.396
> densroot 0.055 -0.233 -0.291
> rootvit -0.038 -0.251 -0.629 0.277
> cover 0.024 -0.796 -0.133 0.253 0.117
> schirufu -0.032 0.137 -0.029 -0.505 -0.078 -0.121
>>
>
> This suggests no significant effects at all.
> In glmmPQL:
>
>> model<-glmmPQL(nemcm~diameter+leafvit+densroot+rootvit+cover+schirufu+offset(loglength),random=~1|site, family=quasipoisson)
> iteration 1
> iteration 2
> iteration 3
>> summary (model)
> Linear mixed-effects model fit by maximum likelihood
> Data: NULL
> AIC BIC logLik
> NA NA NA
>
> Random effects:
> Formula: ~1 | site
> (Intercept) Residual
> StdDev: 0.2684477 4.507758
>
> Variance function:
> Structure: fixed weights
> Formula: ~invwt
> Fixed effects: nemcm ~ diameter + leafvit + densroot + rootvit + cover + schirufu + offset(loglength)
> Value Std.Error DF t-value p-value
> (Intercept) 0.1131898 0.1656949 108 0.6831220 0.4960
> diameter 0.1225231 0.1976568 108 0.6198779 0.5366
> leafvit -0.2191361 0.1697784 108 -1.2907181 0.1996
> densroot -0.4733839 0.2221562 108 -2.1308604 0.0354
> rootvit 0.3858120 0.1615706 108 2.3878846 0.0187
> cover -0.2075038 0.1922054 108 -1.0795940 0.2827
> schirufu 0.2028444 0.1633954 108 1.2414323 0.2171
> Correlation:
> (Intr) diamtr leafvt densrt rootvt cover
> diameter -0.050
> leafvit 0.077 0.360
> densroot 0.217 -0.168 -0.262
> rootvit -0.163 -0.202 -0.632 0.257
> cover 0.084 -0.772 -0.098 0.200 0.073
> schirufu -0.103 0.099 -0.050 -0.483 -0.068 -0.075
>
> Standardized Within-Group Residuals:
> Min Q1 Med Q3 Max
> -1.1146287 -0.5208003 -0.1927005 0.2462878 7.9755368
>
> Number of Observations: 120
> Number of Groups: 6
>>
>
> Again some differences in parameter estimates, but now the two fixed effects densroot and rootvit turn out to be significant.
> So my questions are:
> - what would you recommend me to use? lmer or glmmPQL (laplace approximation or penalized quasi-likelihood)?
> - if lmer is the better option, is there a way to get a reliable p-value for the fixed effects?
> I have experienced that deleting a term and comparing models using anova() always overestimates the significance of that term, probably because the quasipoisson correction for overdispersion is not taken into account.
>
> Thank you very much beforehand,
>
> Martijn.
>
> --
> Martijn Vandegehuchte
> Ghent University
> Department Biology
> Terrestrial Ecology Unit
> K.L.Ledeganckstraat 35
> B-9000 Ghent
> telephone: +32 (0)9/264 50 84
> e-mail: martijn.vandegehuchte at ugent.be
>
> website TEREC: www.ecology.ugent.be/terec
>
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>
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