[R-sig-ME] basic question

Ben Bolker bbolker at gmail.com
Fri Aug 3 02:56:28 CEST 2012

 [cc'ing back to r-sig-mixed-models]

On 12-08-02 07:14 AM, Rachel Cohen wrote:
> Hi, thanks for your reply.  I was reading the Pinheiro and Bates book
> (pg.21) which says that for a basic model the parameters are: β, σ2b ,
> and σ2 which is why I thought the residual variance was counted as a
> parameter?  Also, on the same page: "Although the random effects, bi, i
> = 1, . . .,M may behave like parameters, formally they are just another
> level of random variation in the model so we do not “estimate” them as
> such."  That lead me to believe that the random effects are not counted
> as parameters?

   First, you need to make the distinction between the b_i value
('BLUPs', or 'conditional modes', or 'random effect estimates' -- the
estimates of what's happening in each grouping level) -- and the theta
values (the variances of the random effects).
> I am intending on using the RSE in the formula: CF = exp (RSE^2
> /2) which is supposed to correct for log-bias in estimates (in this case
> estimated tree biomass).  Each unlogged estimate is multiplied by the
> correction factor (CF).  I haven't come across any mention in the
> literature of whether it is appropriate to use the RSE from a
> mixed-effects model in this way.  I hope so?

  If you want to fit on the log scale but estimate the *mean* effects on
the unlogged scale, you should (I think) add the exp(TOTAL_variance/2);
in this case TOTAL_variance will be the sum of *all of the variance
components* in the model, including the residual variance (I'm going to
assume you have only intercept random effects in the model, otherwise
things would get a bit more complicated ...) I *think* it would be
something like sum(sapply(VarCorr(model),diag))+sigma(model)^2 ...  I
think there has been a reasonably recent post by Jarrod Hadfield
answering someone's question related to this topic ...

  If I'm right, then there's no need to go messing around with numbers
of parameters ...
  Ben Bolker

> *From:* Ben Bolker <bbolker at gmail.com>
> *To:* r-sig-mixed-models at r-project.org
> *Sent:* Thursday, 2 August 2012, 2:51
> *Subject:* Re: [R-sig-ME] basic question
> Rachel Cohen <stat.list at ...> writes:
>>  Hi, I have a very basic query (I think).  If I fit a model of the
>> specification given below how many model parameters are there?  Am I
>> correct in thinking that the model parameters are the fixed effects
>> intercept and coefficients (in this case two of them) and also a
>> parameter for within and for between group variance?  So in total 5
>> fitted parameters? or are there also covariance parameters?.  I am
>> trying to calculate the standard error of the residuals as:
>   Unfortunately, this is an easy question to state but not (in my
> opinion) an easy question to answer unambiguously.
>> RSE = ((sum(residuals^2))/(N - no.of parameters))
>> and therefore need to know how many parameters my model actually has.
>> I've gotten a little confused by the
>> literature definitions of what constitutes a parameter. 
> It's confusing!
> [snip]
> A *reasonable* definition (although not the only one) would be
> to count the number of fixed-effect parameters ('beta' in much
> of the literature) and the number of random-effect parameters
> (generally referred to as 'theta' in the lme4 documentation,
> but varying a great deal among references):
> length(fixef(model))+length(getME(model,"theta"))
>   in your case that's 3 parameters for the fixed effects
> and 6 RE parameters (you have a 3x3 variance-covariance matrix of the
> random effects, the matrix is symmetric, so counting the
> diagonal plus one triangle gives 3*(3+1)/2 = 6).  Generally one
> doesn't count the residual variance since that is estimated
> from the (penalized) residual sum of squares.
>   It really depends what you want to use the RSE for.  It may
> very well not have the properties you're expecting (i.e. the
> properties that it has in a simple (non-mixed) linear model ...)
>   There's a bit more about parameter-counting issues at
> http://glmm.wikidot.com/faq, I think ...
>> model<-lmer((log.mass)~centre.log.dbh+centre.log.height+
>> (1+centre.log.dbh+centre.log.height|species_site),
>> data=allometry_2,REML=T) 
>>  model summary output table:
>> Linear mixed model fit by REML 
>> Formula: (log.mass) ~ centre.log.dbh + centre.log.height +
>> (1 + centre.log.dbh + centre.log.height | species_site) 
> [snip]
>> Random effects:
>>  Groups       Name              Variance  Std.Dev. Corr          
>>  species_site (Intercept)       0.061636  0.24827                
>>               centre.log.dbh    0.279362  0.52855   0.575        
>>               centre.log.height 0.285753  0.53456  -0.441 -0.755 
>>  Residual                       0.086963  0.29489                
>> Number of obs: 337, groups: species_site, 18
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