Re: [R] lme X lmer results

From: Douglas Bates <dmbates_at_gmail.com>
Date: Sat 31 Dec 2005 - 05:51:59 EST

On 12/29/05, John Maindonald <john.maindonald@anu.edu.au> wrote:
> 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.

I don't want to be obtuse and argumentative but I still am not convinced that there is a correct denominator degrees of freedom for _this_ F statistic. I may be wrong about this but I think you are referring to an F statistic based on a denominator from a different error stratum, which is not what is being quoted. (Those are not given because they don't generalize to unbalanced designs.)

This is why I would like to see someone undertake a simulation study to compare various approaches to inference for the fixed effects terms in a mixed model, using realistic (i.e. unbalanced) examples.

It seems peculiar to me that the F statistics are being created from the ratios of mean squares for different terms to the _same_ mean square (actually a penalized sum of squares divided by the degrees of freedom) and the adjustment suggested to take into account the presence of the random effects is to change the denominator degrees of freedom. I think the rationale for this is an attempt to generalized another approach (the use of error strata) even though it is not being used here.

> 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.

As I said, I am willing to change this if it is shown to be grossly inaccurate but please show me.

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

mcmcsamp starts from the parameter estimates when creating the chain but that is the extent to which it depends on the lmer results.

> John Maindonald.
>
> On 29 Dec 2005, at 10:00 PM, r-help-request@stat.math.ethz.ch wrote:
>
> > From: Douglas Bates <dmbates@gmail.com>
> > Date: 29 December 2005 5:59:07 AM
> > To: "Ronaldo Reis-Jr." <chrysopa@gmail.com>
> > Cc: R-Help <r-help@stat.math.ethz.ch>
> > Subject: Re: [R] lme X lmer results
> >
> >
> > On 12/26/05, Ronaldo Reis-Jr. <chrysopa@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
> >>
> >> 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
> >>
> >> undestand
> >> this use with a gaussian error, I undestand this with glm data.
> >>
> >> I need more explanations, please.
> >>
> >> Thanks
> >> Ronaldo
>
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