# [R-sig-ME] results lme unstructured covariance matrix, again

ben pelzer benpe|zer @end|ng |rom gm@||@com
Tue Jul 19 14:50:47 CEST 2022

Dear John, Wolfgang, James and Ben,

Thanks to you all for your explanations and examples! As I understand it
now, cancelling the estimation of the sigma, by fixing it to a small
positive number close to zero, produces random effects estimates of the
intercept and t2 (or of t1 and t2) which make sense. However, the
loglikelihood doesn't make sense. I guess that with sigma fixed to nearly
zero, the predictions will be meaningful too, even when including the
random effects in the predictions, though I did not verify that yet.

All in all, I think that for a model with an unstructured covariance
matrix, "gls" is more straightforward. For the example of John:

da$time <- da$t2 + 1
m4 <- gls(y ~ 1 + t2,
correlation=corSymm(form= ~time|person),
weights=varIdent(form= ~1|time),
data=da)

and the results are equal to e.g.

m2 <- lme(y ~ 1 + t2, random = ~ 1 + t2 | person, data=da)

as far as the loglikelihood and the estimated (co)variances (getVarCov) are
concerned.

My question about "lme" arose when helping someone with such an
unstructured covariances model, for data with three fixed occasions. With
"many" occasions, a random linear effect of time is often employed, or
maybe a random quadratic time effect or cubic... Anyway, time is used as a
random (linear etc) effect in such models. But with only three occasions,
and with the time effect not being linear but quadratic, the model turns
into the kind of model I was questioning about, but with three occasions
instead of two. So, instead of

modelA <- lme(y ~ 1+time+timesquared, random= ~ 1+time+timesquared |
person, data=da)

one could then use

modelB <- lme(y ~ 1+t2+t3, random= ~ 1+t2+t3 | person, data=da)

But the random effect variances of t2 and t3 are not meaningful, as I know
now. And with the often used "lmer" the model is not estimated (except when
avoiding the "counting check" as Ben Bolker explained). This means that the
"random time effect" models, which I explain to students in a course, is
not applicable to such data. I thought/hoped that "lme" could be used, and
in a way it can, with fixing the sigma, but apparently this also leads to
some "unexpected" results. On the other hand, "gls" is a new method for
students used to the "random time effect" models. So I have to choose among
these two possibilities.

Thanks again everyone, time to switch off the computer and enjoy the sun.
It's almost 40 degrees Celsius or 104 Farenheit in The Netherlands now, so
maybe enjoying is not the right word. Best to you all,

Ben Pelzer.

On Tue, 19 Jul 2022 at 04:15, John Fox <jfox using mcmaster.ca> wrote:

> Dear Ben,
>
> On 2022-07-18 1:32 p.m., John Fox wrote:
> > Dear Ben,
> >
> > My apologies---I paid attention to the inessential part of your message,
> > which was the parametrization of the model rather than the
> > variance-covariance structure.
> >
> > This is a little awkward because of the LaTeX (and as far as I know, I
> > can't attach a PDF, but maybe you can paste this into a LaTeX editor):
>
> Rolf Turner pointed out to me that PDF attachments *are* acceptable, and
> so I've attached a PDF with the LaTeX part of my message.
>
> Best,
> John
>
> >
> > The model is $Y_{ij} = \alpha + \beta x_{ij} + \delta_{1i} + \delta_{2i} > > x_{ij} + \varepsilon_{ij}$ for $i = 1, \ldots, n$ and $j = 1,2$, where
> > $x_{i1} = 1$ and $x_{i2} = 0$.
> >
> > The variances of the random effects are $V(\delta_{1i}) = \psi_1^2$ and
> > $V(\delta_{2i}) = \psi_2^2$, and their covariance is $C(\delta_{1i}, > > \delta_{2i}) = \psi_{12}$. The observation-level error variance is
> > $V(\varepsilon_{ij}) = \sigma^2$.
> >
> > Then the composite error is $\zeta_{ij} = \delta_{1i} + \delta_{2i} > > x_{ij} + \varepsilon_{ij}$ with variance $V(\zeta_{ij}) = \psi_1^2 + > > x^2_{ij} \psi_2^2 + 2 x_{ij} \psi_{12} + \sigma^2$, which is
> > $V(\zeta_{i1}) = \psi_1^2 + \psi_2^2 + 2\psi_{12} + \sigma^2$ for $j = > > 1$ and  $V(\zeta_{i2}) = \psi_1^2 + \sigma^2$ for $j = 2$, and
> > covariance $C(\zeta_{i1}, \zeta_{i2}) = \psi_1^2 + \psi_2^2$.
> >
> > There are, as you say, 4 variance-covariance components, $\psi_1^2, > > \psi_2^2, \psi_{12}, \sigma^2$, and just 3 variances and covariances
> > among the composite disturbances, $V(\zeta_{i1}), V(\zeta_{i2}), > > C(\zeta_{i1}, \zeta_{i2})$, and so---as you and several others have
> > noted (but I missed)---the variance-covariance components are
> > underidentified.
> >
> > lmer(), but not lme(), appears to use a heuristic, counting the number
> > of random effects and observations, to detect underidentification. I
> > don't know whether that's bullet-proof, but it works in this case. Maybe
> > Ben Bolker, who has already responded, can comment further.
> >
> > Best,
> >   John
> >
> > John Fox, Professor Emeritus
> > McMaster University
> > Hamilton, Ontario, Canada
> > web: https://socialsciences.mcmaster.ca/jfox/
> >
> > On 2022-07-18 6:39 a.m., ben pelzer wrote:
> >> Dear John,
> >>
> >> Thank you for answering my question and your nice example with the
> >> alternative model formulations!!. But still I'm in doubt about the
> >> results. The fact that there are only three observed (co)variances
> >> which are being reproduced by four parameters estimates of lme leaves
> >> me confused. Actually, I would think that there is no unique solution
> >> without some constraint being applied. But I could not find something
> >> about such a constraint in lme documentation. The random effects of t1
> >> and t2, or of the intercept and t2 in your alternative model, should
> >> be sufficient to reproduce the observed (co)variances, so the residual
> >> variance is "unnecassary". I guess that is the reason that lmer is not
> >> able to estimate the model.
> >>
> >> library(lme4)
> >> m3 <- lmer(y ~ 1+t2+(1+t2|person), data=da)
> >>
> >> Error: number of observations (=20) <= number of random effects (=20)
> >> for term (1 + t2 | person); the random-effects parameters and the
> >> residual variance (or scale parameter) are probably unidentifiable
> >>
> >>
> >> It's interesting to notice that the two model specifications (t1+t2
> >> and 1+t2) lead to different residual variances estimated by lme. What
> >> do these residual variances mean? And of course also: what do the
> >> random effect variances estimated by both model formulations actually
> >> mean or stand for? Do you have any ideas? Kind regards,
> >>
> >> Ben.
> >>
> >> On Sat, 16 Jul 2022 at 16:50, John Fox <jfox using mcmaster.ca
> >> <mailto:jfox using mcmaster.ca>> wrote:
> >>
> >>     Dear Ben,
> >>
> >>     First, I'll make this into a reproducible example:
> >>
> >>       > set.seed(123)
> >>       > t1 <- c(rep(1, 10), rep(0, 10))
> >>       > t2 <- 1 - t1
> >>       > person <- rep(1:10, 2)
> >>       > y <- t2 + rnorm(20)
> >>       > da <- data.frame(y, t1, t2, person)
> >>
> >>       > library(nlme)
> >>
> >>     Then note that the random-effect specification 0 + t1 + t2 is
> >> simply a
> >>     reparametrization of 1 + t2 (i.e., 1 = t1 + t2), which produces the
> >>     same
> >>     fit to the data (same fixed effects, same restricted
> log-likelihood):
> >>
> >>       > m1 <- lme(y ~ 1 + t2, random = ~ 0 + t1 + t2 | person, data=da)
> >>       > m2 <- lme(y ~ 1 + t2, random = ~ 1 + t2 | person, data=da)
> >>       > m1
> >>     Linear mixed-effects model fit by REML
> >>         Data: da
> >>         Log-restricted-likelihood: -25.92726
> >>         Fixed: y ~ 1 + t2
> >>     (Intercept)          t2
> >>        0.07462564  1.13399632
> >>
> >>     Random effects:
> >>        Formula: ~0 + t1 + t2 | person
> >>        Structure: General positive-definite, Log-Cholesky
> parametrization
> >>                StdDev    Corr
> >>     t1       0.8964136 t1
> >>     t2       0.9856215 0.647
> >>     Residual 0.3258015
> >>
> >>     Number of Observations: 20
> >>     Number of Groups: 10
> >>
> >>       > m2
> >>     Linear mixed-effects model fit by REML
> >>         Data: da
> >>         Log-restricted-likelihood: -25.92726
> >>         Fixed: y ~ 1 + t2
> >>     (Intercept)          t2
> >>        0.07462564  1.13399632
> >>
> >>     Random effects:
> >>        Formula: ~1 + t2 | person
> >>        Structure: General positive-definite, Log-Cholesky
> parametrization
> >>                   StdDev    Corr
> >>     (Intercept) 0.8787887 (Intr)
> >>     t2          0.7540826 -0.302
> >>     Residual    0.3707215
> >>
> >>     Number of Observations: 20
> >>     Number of Groups: 10
> >>
> >>     Finally, it's unnecessary to supply the intercept 1 in the model
> >>     formula
> >>     since the intercept is implied if it's not explicitly excluded:
> >>
> >>       > m3 <- lme(y ~ t2, random = ~ t2 | person, data=da)
> >>       > m3
> >>     Linear mixed-effects model fit by REML
> >>         Data: da
> >>         Log-restricted-likelihood: -25.92726
> >>         Fixed: y ~ t2
> >>     (Intercept)          t2
> >>        0.07462564  1.13399632
> >>
> >>     Random effects:
> >>        Formula: ~t2 | person
> >>        Structure: General positive-definite, Log-Cholesky
> parametrization
> >>                   StdDev    Corr
> >>     (Intercept) 0.8787887 (Intr)
> >>     t2          0.7540826 -0.302
> >>     Residual    0.3707215
> >>
> >>     Number of Observations: 20
> >>     Number of Groups: 10
> >>
> >>     I hope this helps,
> >>        John
> >>
> >>     On 2022-07-16 10:05 a.m., ben pelzer wrote:
> >>      > Sorry, my previous mailed contained another question which is
> >>     irrelevant...
> >>      > I deleted that now.
> >>      >
> >>      >
> >>      > Hi all,
> >>      >
> >>      > I have a question about results from lme of package nlme.
> >>      >
> >>      > Suppose the data consists of repeated measures at two fixed time
> >>     points.
> >>      >
> >>      > I used the following equation:
> >>      >
> >>      >
> >>      >
> >>      > Model1 <- lme ( y ~ 1+t2 , random = ~ 0 + t1+t2|person, data=da)
> >>      >
> >>      >
> >>      >
> >>      > y is the dependent, t1 and t2 are binary dummy variables, valued
> >>     0 or 1,
> >>      > indicating the time point.  Model1 is estimated without any
> >>     convergence
> >>      > problems and the reproduced (co)variances found with
> >>      >
> >>      >
> >>      >
> >>      > getVarCov(Model1, type=”marginal”, indivual=”1”)
> >>      >
> >>      >
> >>      >
> >>      > are identical to the observed (co)variances.
> >>      >
> >>      >
> >>      > My question is:  how can lme estimate 4 (co)variances with only 3
> >>     known
> >>      > (co)variances?
> >>      >
> >>      >
> >>      >
> >>      > The 4 estimates concern:
> >>      >
> >>      > -          std. deviation of the random effect of dummy t1
> >>      >
> >>      > -          std. deviation of the random effect of dummy t2
> >>      >
> >>      > -          covariance of the random effects of the dummies t1 and
> >>     t2 t1
> >>      >
> >>      > -          residual std. error
> >>      >
> >>      >
> >>      >
> >>      > Related to the question above: how can the variances of the
> >>     random effects
> >>      > and the residual std. error be interpreted?
> >>      >
> >>      >
> >>      >
> >>      > Thanks for any help,
> >>      >
> >>      >
> >>      >
> >>      > Ben.
> >>      >
> >>      >       [[alternative HTML version deleted]]
> >>      >
> >>      > _______________________________________________
> >>      > R-sig-mixed-models using r-project.org
> >>     <mailto:R-sig-mixed-models using r-project.org> mailing list
> >>      > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >>     <https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models>
> >>     --     John Fox, Professor Emeritus
> >>     McMaster University
> >>     Hamilton, Ontario, Canada
> >>     web: https://socialsciences.mcmaster.ca/jfox/
> >>     <https://socialsciences.mcmaster.ca/jfox/>
> >>
> >
> > _______________________________________________
> > R-sig-mixed-models using r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> --
> John Fox, Professor Emeritus
> McMaster University
> Hamilton, Ontario, Canada
> web: https://socialsciences.mcmaster.ca/jfox/
>

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