[R-sig-ME] Seeming discrepancy between summary and confint; was: Confidence interval for relative contribution of random effect variance

[Martin Maechler]

>>>>>   <lorenz.gygax at agroscope.admin.ch>
>>>>>     on Fri, 12 Sep 2014 11:20:42 +0000 writes:

    > [snip ...]
    >> > A side-line: Using the confint function on one of my models and
    >> > comparing the confidence intervals with the point-estimates from the
    >> > summary of the same model, it seems that confint reports confidence
    >> > intervals for the estimated standard deviations of the random
    >> > effects as well as of the error-variability whereas summary reports
    >> > the standard deviations for the random effects but the variance for
    >> > the residuals. Is this correct? I seem to remember some such
    >> > discussion but could not find any note online that would have
    >> > verified this fact. Page 31 in "Fitting linear mixed-effects models
    >> > using lme4" discusses this part of the summary output but seems to
    >> > be using the terms standard deviation and variance somewhat
    >> > interchangeably (or, more likely, I failed to read it correctly).
    >> 
    >> Hmmm.  The output of
    >> 
    >> fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
    >> summary(fm1)
    >> 
    >> gives
    >> 
    >> 
    >> Random effects:
    >> Groups   Name        Variance Std.Dev. Corr
    >> Subject  (Intercept) 612.09   24.740
    >> Days         35.07    5.922   0.07
    >> Residual             654.94   25.592
    >> Number of obs: 180, groups:  Subject, 18
    >> 
    >> which shows both the variance and the standard deviation (i.e.
    >> *not* the uncertainty estimate, just the point estimate of the
    >> variability on both the variance and the standard deviation scales)

    > Ok. I admit that I was not very clear perhaps. Let me show an example. I am currently on lme4 version 1.1-7 in R 3.0.1 (my employer is just now updating to 3.1.1 but that always takes a while - so if that was an issue of not having the most recent version, I apologise in advance):

    > In the example which struck me odd, this was my model

    > HHbT.fin.lmer <- lmer (HHbT ~ valN +
    > (1 | ID/part/val), fNIRS.df, REML= FALSE)

    > in which the response is a transformed change in blood deoxy-hemoglobin concentration modelled by a fixed effect (three types of conditions, modelled as a linear predictor in which stimuli have been applied repeatedly) and a nested intercept random effect that accounts for the subject-to-subject variation (ID), the part-to-part variation (three different parts in the experiment) and the type of stimulus. (I am using REML= FALSE because I am conducting come model selection for the fixed effects based on information criteria.)

    > If I do the summary () this is what I get for the random effects part of the output.

    > Random effects:
    > Groups        Name        Variance Std.Dev.
    > val:(part:ID) (Intercept) 0.4599   0.6782  
    > part:ID       (Intercept) 0.1773   0.4211  
    > ID            (Intercept) 0.1278   0.3575  
    > Residual                  9.4302   3.0709  
    > Number of obs: 1833, groups:  val:(part:ID), 214; part:ID, 72; ID, 25:


    > If I do

    > confint (HHbT.fin.lmer, method= 'profile')

    > I get

    > 2.5 %     97.5 %
    > .sig01       0.41713241  0.9210729
    > .sig02       0.00000000  0.7535615
    > .sig03       0.00000000  0.6697109
    > .sigma       2.96898087  3.1786606

    > Where the above listed variances for the random effects fit nicely into the confidence intervals (.sig0x) but not the value for the residuals / .sigma where the variance from the summary seems to be approximately squared in respect to the confidence interval.

    > I guess, I am missing out on something, but on what?

Yes, the conf.ints are for the sigmas as their name suggest, and
sigmas are standard deviations aka  sqrt(<variances>).

You're welcome 
und herzlichen eidgenössischen Gruss,
Martin