lme X lmer results

Surely there is a correct denominator degrees of freedom if the design
is balanced, as Ronaldo's design seems to be. Assuming that he has
specified the design correctly to lme() and that lme() is getting the df
right, the difference is between 2 df and 878 df.  If the t-statistic
for the
second level of Xvar had been 3.0 rather than 1.1, the difference
would be between a t-statistic equal to 0.095 and 1e-6.  In a design
where there are 10 observations on each experimental unit, and all
comparisons are at the level of experimental units or above, df for
all comparisons will be inflated by a factor of at least 9.

Rather than giving df that for the comparison(s) of interest may be
highly inflated, I'd prefer to give no degrees of freedom at all, & to
encourage users to work out df for themselves if at all possible.
If they are not able to do this, then mcmcsamp() is a good alternative,
and may be the way to go in any case.  This has the further advantage
of allowing assessments in cases where the relevant distribution is
hard to get at. I'd think a warning in order that the df are upper
bounds,
and may be grossly inflated.

Incidentally, does mcmcsamp() do its calculations pretty well
independently of the lmer results?

John Maindonald.

On 29 Dec 2005, at 10:00 PM, r-help-request at stat.math.ethz.ch wrote:

From: Douglas Bates <dmbates at gmail.com>
Date: 29 December 2005 5:59:07 AM
To: "Ronaldo Reis-Jr." <chrysopa at gmail.com>
Cc: R-Help <r-help at stat.math.ethz.ch>
Subject: Re: [R] lme X lmer results


On 12/26/05, Ronaldo Reis-Jr. <chrysopa at gmail.com> wrote:

Hi,

this is not a new doubt, but is a doubt that I cant find a good
response.

Look this output:

m.lme <- lme(Yvar~Xvar,random=~1|Plot1/Plot2/Plot3)

anova(m.lme)

            numDF denDF  F-value p-value
(Intercept)     1   860 210.2457  <.0001
Xvar            1     2   1.2352  0.3821

summary(m.lme)

Linear mixed-effects model fit by REML
 Data: NULL
      AIC      BIC    logLik
  5416.59 5445.256 -2702.295

Random effects:
 Formula: ~1 | Plot1
        (Intercept)
StdDev: 0.000745924

 Formula: ~1 | Plot2 %in% Plot1
        (Intercept)
StdDev: 0.000158718

 Formula: ~1 | Plot3 %in% Plot2 %in% Plot1
        (Intercept) Residual
StdDev: 0.000196583 5.216954

Fixed effects: Yvar ~ Xvar
                   Value Std.Error  DF  t-value p-value
(Intercept)    2.3545454 0.2487091 860 9.467066  0.0000
XvarFactor2    0.3909091 0.3517278   2 1.111397  0.3821

Number of Observations: 880
Number of Groups:
                         Plot1               Plot2 %in% Plot1
                             4                              8
   Plot3 %in% Plot2 %in% Plot1
                            20

This is the correct result, de correct denDF for Xvar.

I make this using lmer.

m.lmer <- lmer(Yvar~Xvar+(1|Plot1)+(1|Plot1:Plot2)+(1|Plot3))
anova(m.lmer)

Analysis of Variance Table
           Df Sum Sq Mean Sq  Denom F value Pr(>F)
Xvar  1  33.62   33.62 878.00  1.2352 0.2667

summary(m.lmer)

Linear mixed-effects model fit by REML
Formula: Yvar ~ Xvar + (1 | Plot1) + (1 | Plot1:Plot2) + (1 | Plot3)
     AIC     BIC    logLik MLdeviance REMLdeviance
 5416.59 5445.27 -2702.295   5402.698      5404.59
Random effects:
 Groups        Name        Variance   Std.Dev.
 Plot3         (Intercept) 1.3608e-08 0.00011665
 Plot1:Plot2   (Intercept) 1.3608e-08 0.00011665
 Plot1         (Intercept) 1.3608e-08 0.00011665
 Residual                  2.7217e+01 5.21695390
# of obs: 880, groups: Plot3, 20; Plot1:Plot2, 8; Plot1, 4

Fixed effects:
                Estimate Std. Error  DF t value Pr(>|t|)
(Intercept)      2.35455    0.24871 878  9.4671   <2e-16 ***
XvarFactor2      0.39091    0.35173 878  1.1114   0.2667
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Look the wrong P value, I know that it is wrong because the DF
used. But, In
this case, the result is not correct. Dont have any difference of
the result
using random effects with lmer and using a simple analyses with lm.

You are assuming that there is a correct value of the denominator
degrees of freedom.  I don't believe there is.  The statistic that is
quoted there doesn't have exactly an F distribution so there is no
correct degrees of freedom.

One thing you can do with lmer is to form a Markov Chain Monte Carlo
sample from the posterior distribution of the parameters so you can
check to see whether the value of zero is in the middle of the
distribution of XvarFactor2 or not.

It would be possible for me to recreate in lmer the rules used in lme
for calculating denominator degrees of freedom associated with terms
of the random effects.  However, the class of models fit by lmer is
larger than the class of models fit by lme (at least as far as the
structure of the random-effects terms goes).  In particular lmer
allows for random effects associated with crossed or partially crossed
grouping factors and the rules for denominator degrees of freedom in
lme only apply cleanly to nested grouping factors.  I would prefer to
have a set of rules that would apply to the general case.

Right now I would prefer to devote my time to other aspects of lmer -
in particular I am still working on code for generalized linear mixed
models using a supernodal Cholesky factorization.  I am willing to put
this aside and code up the rules for denominator degrees of freedom
with nested grouping factors BUT first I want someone to show me an
example demonstrating that there really is a problem.  The example
must show that the p-value calculated in the anova table or the
parameter estimates table for lmer is seriously wrong compared to an
empirical p-value - obtained from simulation under the null
distribution or through MCMC sampling or something like that.  Saying
that "Software XYZ says there are n denominator d.f. and lmer says
there are m" does NOT count as an example.  I will readily concede
that the denominator degrees of freedom reported by lmer are wrong but
so are the degrees of freedom reported by Software XYZ because there
is no right answer (in general - in a few simple balanced designs
there may be a right answer).

m.lm <- lm(Yvar~Xvar)

anova(m.lm)

Analysis of Variance Table

Response: Nadultos
            Df  Sum Sq Mean Sq F value Pr(>F)
Xvar         1    33.6    33.6  1.2352 0.2667
Residuals  878 23896.2    27.2

summary(m.lm)

Call:
lm(formula = Yvar ~ Xvar)

Residuals:
    Min      1Q  Median      3Q     Max
-2.7455 -2.3545 -1.7455  0.2545 69.6455

Coefficients:
               Estimate Std. Error t value Pr(>|t|)
(Intercept)      2.3545     0.2487   9.467   <2e-16 ***
XvarFactor2      0.3909     0.3517   1.111    0.267
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.217 on 878 degrees of freedom
Multiple R-Squared: 0.001405,   Adjusted R-squared: 0.0002675
F-statistic: 1.235 on 1 and 878 DF,  p-value: 0.2667

I read the rnews about this use of the full DF in lmer, but I dont
undestand
this use with a gaussian error, I undestand this with glm data.

I need more explanations, please.

Thanks
Ronaldo