[R-sig-ME] different aic and LL in glmer(lme4) and glimmix(SAS)?

[Douglas Bates]

On Thu, Jul 1, 2010 at 11:03 AM, Jeffrey Evans
<Jeffrey.Evans at dartmouth.edu> wrote:
> Hello All,

> I have read several posts related to this previously, but haven't found any
> resolution yet. When running the same GLMM in glmer and in SAS PROC GLIMMIX,
> both programs return comparable parameter estimates, but wildly different
> likelihoods and AIC values.

> In SAS I specify use of the Laplace approximation. In R, I believe this is
> the default (no?).

> What's the difference, and [how] can I reproduce the SAS -2ll in glmer?

The difference is probably due to the way that the deviance is defined
for the binomial family in R.  A glm family object is a list of
functions and expressions.  One of the functions, called "dev.resids"
has arguments y, mu and weights.  You can specify the response for a
binomial family as the 0/1 responses or as a matrix with two columns
as you did here.  When you use the two column specification the
response y is transformed to the fraction of successes and the number
of cases is incorporated in the weights.  It turns out that this is
all the information necessary for obtaining the mle's of the
parameters but it does not give the same deviance as you would get by
listing the 0/1 responses.

I'll write an example using the cbpp data from the lme4 package.
> Thanks,
> Jeff
>
> \/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/
>> R_GLMM = glmer(cbind(SdlFinal, SdlMax-SdlFinal) ~ lnsdlmaxd*lnadultssdld +
>  (1|ID),data=sdlPCAdat,family="binomial")
>> R_GLMM
> Generalized linear mixed model fit by the Laplace approximation
> Formula: cbind(SdlFinal, SdlMax - SdlFinal) ~ lnsdlmaxd * lnadultssdld +
> (1 | ID)
>   Data: sdlPCAdat
>  AIC  BIC logLik deviance
>  1150 1165   -570     1140        <------------------ this line!!
> Random effects:
>  Groups Name        Variance Std.Dev.
>  ID     (Intercept) 1.2491   1.1176
> Number of obs: 144, groups: ID, 48
>
> Fixed effects:
>                       Estimate Std. Error z value Pr(>|z|)
> (Intercept)             4.56964    0.43148  10.591  < 2e-16 ***
> lnsdlmaxd              -0.65936    0.05686 -11.595  < 2e-16 ***
> lnadultssdld           -0.64534    0.15861  -4.069 4.73e-05 ***
> lnsdlmaxd:lnadultssdld  0.07393    0.02166   3.414  0.00064 ***
> ---
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
> Correlation of Fixed Effects:
>            (Intr) lnsdlm lndlts
> lnsdlmaxd   -0.923
> lnadltssdld -0.461  0.479
> lnsdlmxd:ln  0.482 -0.508 -0.994
>
>
>  \/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/
> title 'SAS GLMM';
> proc glimmix data=sdlPCAdat ic=pq noitprint method=laplace;
> class site id;
> model sdlfinal/sdlmax = lnsdlmaxd|lnadultssdld/ solution dist=binomial;
> random ID /;
> covtest glm/wald;
> run;
>
> //////////////////////////////////////////////////////////////////////
>
>                      SAS GLMM      19:36 Wednesday, June 30, 2010 88
>
>                   The GLIMMIX Procedure
>
>            Data Set           WORK.SDLPCADAT
>            Response Variable (Events)  SdlFinal
>            Response Variable (Trials)  SdlMax
>            Response Distribution     Binomial
>            Link Function         Logit
>            Variance Function       Default
>            Variance Matrix        Not blocked
>            Estimation Technique     Maximum Likelihood
>            Likelihood Approximation   Laplace
>            Degrees of Freedom Method   Containment
>
>
>
>                  Optimization Information
>
>             Optimization Technique    Dual Quasi-Newton
>             Parameters in Optimization  5
>             Lower Boundaries       1
>             Upper Boundaries       0
>             Fixed Effects         Not Profiled
>             Starting From         GLM estimates
>
>             Convergence criterion (GCONV=1E-8) satisfied.
>
>                     Fit Statistics
>
>               -2 Log Likelihood        1653.90  <------------------ this
> line!!
>               AIC (smaller is better)  1663.90
>               AICC (smaller is better) 1664.33
>               BIC (smaller is better)  1673.25
>               CAIC (smaller is better) 1678.25
>               HQIC (smaller is better) 1667.43
>
>
>              Fit Statistics for Conditional Distribution
>
>              -2 log L(SdlFinal | r. effects)   1436.44
>              Pearson Chi-Square          908.07
>              Pearson Chi-Square / DF        6.31
>
>
>                 Covariance Parameter Estimates
>
>            Cov         Standard     Z
>            Parm  Estimate    Error   Value   Pr > Z
>
>            ID    1.2491   0.2746   4.55   <.0001
>
>
>                 Solutions for Fixed Effects
>
>                              Standard
>     Effect         Estimate    Error    DF  t Value  Pr > |t|
>
>     Intercept           4.5696   0.4333    47    10.55  <.0001
>     lnsdlmaxd          -0.6594   0.05717   93   -11.53  <.0001
>     lnadultssdld       -0.6453   0.1593    93   -4.05   0.0001
>     lnsdlmaxd*lnadultssd0.07394  0.02174   93    3.40   0.0010
>
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