# [R-sig-ME] using lme for cov structures?

Thierry Onkelinx thierry.onkelinx at inbo.be
Fri Jul 28 11:45:51 CEST 2017

Dear Ben,

The syntax of the matrix is LaTeX. You can use is to render math in
RMarkdown files. E.g.
https://bookdown.org/yihui/bookdown/markdown-extensions-by-bookdown.html#equations

Best regards,

ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for Nature and
Forest
team Biometrie & Kwaliteitszorg / team Biometrics & Quality Assurance
Kliniekstraat 25
1070 Anderlecht
Belgium

To call in the statistician after the experiment is done may be no more
than asking him to perform a post-mortem examination: he may be able to say
what the experiment died of. ~ Sir Ronald Aylmer Fisher
The plural of anecdote is not data. ~ Roger Brinner
The combination of some data and an aching desire for an answer does not
ensure that a reasonable answer can be extracted from a given body of data.
~ John Tukey

2017-07-28 10:51 GMT+02:00 Ben Pelzer <b.pelzer op maw.ru.nl>:

> Dear Thierry,
>
> After all effort and thinking about the trick with the "one" variable, I
> realised (but only after your previous mail with the syntax for the
> "group" variable) that I do not need the trick at all. With a
> grouping/context variable, lme can be used and without such
> grouping/context, gls will do. Why didn't I realize this earlier???
>
> As to your last mail, the syntax you included looks very intriguing,
> which language did you use to evaluate the cov matrix?
>
> Best regards, Ben.
>
> On 27-7-2017 15:13, Thierry Onkelinx wrote:
> > Dear Ben,
> >
> > I would look at the variance-covariance matrix of the model. Below is
> > the var-covar matrix for the model lme(opp ~ 1, random = ~1|id,
> > correlation=corCompSymm(form = ~ wave), weights=varIdent(form =
> > ~1|wave)). I worked it out for 3 id groups with each 3 waves. The
> > submatrices indicate the grouping by the random effect.
> >
> > Is this the structure you are looking for? If not please provide the
> > required structure.
> >
> > Having a random effect with only one level is nonsense. IMHO it should
> > yield an error, or at least a warning. I have no idea how lme
> > estimates the reported variance.
> >
> > Best regards,
> >
> > Thierry
> >
> > $\sigma^2_i$: random effect variance
> > $\sigma^2_e$: variance of the noise
> > $\rho$: correlation of the compound symmetry
> > $w_2$ and $w_3$: relative variance of the 2nd and 3th wave compared to
> > the 1st wave
> >
> > $$> > \begin{pmatrix} > > \begin{bmatrix} > > \sigma^2_i+\sigma^2_e & \sqrt{w_2}\sigma^2_i+\rho\sigma^2_e & > > \sigma^2_i+\sqrt{w_3}\rho\sigma^2_e \\ > > \sigma^2_i+\sqrt{w_2}\rho\sigma^2_e & w_2\sigma^2_i+\sigma^2_e & > > \sqrt{w_2w_3}\sigma^2_i+\rho\sigma^2_e \\ > > \sigma^2_i+\sqrt{w_3}\rho\sigma^2_e & > > \sqrt{w_2w_3}\sigma^2_i+w_3\rho\sigma^2_e & \sigma^2_i+\sigma^2_e > > \end{bmatrix} & > > \begin{bmatrix} > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i > > \end{bmatrix} & > > \begin{bmatrix} > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i > > \end{bmatrix} > > \\ > > \begin{bmatrix} > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i > > \end{bmatrix} & > > \begin{bmatrix} > > \sigma^2_i+\sigma^2_e & \sqrt{w_2}\sigma^2_i+\rho\sigma^2_e & > > \sigma^2_i+\sqrt{w_3}\rho\sigma^2_e \\ > > \sigma^2_i+\sqrt{w_2}\rho\sigma^2_e & w_2\sigma^2_i+\sigma^2_e & > > \sqrt{w_2w_3}\sigma^2_i+\rho\sigma^2_e \\ > > \sigma^2_i+\sqrt{w_3}\rho\sigma^2_e & > > \sqrt{w_2w_3}\sigma^2_i+w_3\rho\sigma^2_e & \sigma^2_i+\sigma^2_e > > \end{bmatrix} \\ > > \begin{bmatrix} > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i > > \end{bmatrix} & > > \begin{bmatrix} > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i\\ > > \sigma^2_i & \sigma^2_i & \sigma^2_i > > \end{bmatrix} & > > \begin{bmatrix} > > \sigma^2_i+\sigma^2_e & \sqrt{w_2}\sigma^2_i+\rho\sigma^2_e & > > \sigma^2_i+\sqrt{w_3}\rho\sigma^2_e \\ > > \sigma^2_i+\sqrt{w_2}\rho\sigma^2_e & w_2\sigma^2_i+\sigma^2_e & > > \sqrt{w_2w_3}\sigma^2_i+\rho\sigma^2_e \\ > > \sigma^2_i+\sqrt{w_3}\rho\sigma^2_e & > > \sqrt{w_2w_3}\sigma^2_i+w_3\rho\sigma^2_e & \sigma^2_i+\sigma^2_e > > \end{bmatrix} > > \end{pmatrix} > >$$
> >
> >
> > ir. Thierry Onkelinx
> > Instituut voor natuur- en bosonderzoek / Research Institute for Nature
> > and Forest
> > team Biometrie & Kwaliteitszorg / team Biometrics & Quality Assurance
> > Kliniekstraat 25
> > 1070 Anderlecht
> > Belgium
> >
> > To call in the statistician after the experiment is done may be no
> > more than asking him to perform a post-mortem examination: he may be
> > able to say what the experiment died of. ~ Sir Ronald Aylmer Fisher
> > The plural of anecdote is not data. ~ Roger Brinner
> > The combination of some data and an aching desire for an answer does
> > not ensure that a reasonable answer can be extracted from a given body
> > of data. ~ John Tukey
> >
> > 2017-07-27 11:16 GMT+02:00 Ben Pelzer <b.pelzer op maw.ru.nl
> > <mailto:b.pelzer op maw.ru.nl>>:
> >
> >     Dear Thierry,
> >
> >     Thanks for your help with these models. I wasn't sure how to
> formulate
> >     them. At the department of sciology where I work (Radboud University)
> >     longitudinal data are getting more common bussiness. And very often,
> >     we'd like to estimate random country or municipality influences for
> >     repeated measures on the same person. This is not a big problem as
> >     long
> >     as we use growth modelling with random intercept and random time
> >     influence, but the covariance structure implied by such models is,
> >     say,
> >     limited. Knowing how to specify the cov. structures in R is really
> >
> >     After fiddling around with all the possibilities, the picture is
> >     slowly
> >     getting clearer here. I was puzzled by the heterogeneous compound
> >     symmetry structure and how to specify this. With specification:
> >
> >     heteroCS1 <- lme(opp ~ 1,opposites,
> >          random = ~1|id,
> >          correlation=corCompSymm(form = ~ wave),
> >          weights=varIdent(form = ~1|wave), method="REML")
> >
> >     the random person effect is estimated apart from the residuals. On
> the
> >     other hand, with:
> >
> >     heteroCS2 <- lme(opp ~ 1,opposites,
> >          random = ~1|one,
> >          correlation=corCompSymm(form = ~wave | one/id),
> >          weights=varIdent(form = ~1|wave), method="REML"))
> >
> >     the random person effect is kept part of the residuals, because it is
> >     not estimated explicitly. As a result, heteroCS2 gives the same
> >     results
> >     as obtained with the more straightforward gls specification:
> >
> >     hetroCS3 <- gls(opp ~ 1, opposites,
> >          correlation = corCompSymm(form = ~ wave|id),
> >          weights = varIdent(form = ~ 1 | wave))
> >
> >     In general, I think I would prefer to have the random person effect
> as
> >     part the residual term instead of seperating it from the residual.
> >     That
> >     is, by cutting the random person effect away from the residual, you
> >     remove the very part that causes correlation between the observations
> >     over time. The only specification (I could think of) that keeps the
> >     random person effect IN the residual is heteroCS2. But maybe
> something
> >     less artificial can be found....
> >
> >     And the strange thing that remains is: how can lme estimate a random
> >     effect variance in case of one single group, as in:
> >
> >     strangemodel <- lme(opp ~1, random = ~1|one, opposites)
> >
> >     which produces the summary:
> >
> >     Linear mixed-effects model fit by REML
> >       Data: opposites
> >           AIC      BIC    logLik
> >        1473.5 1482.303 -733.7498
> >
> >     Random effects:
> >       Formula: ~1 | one
> >              (Intercept) Residual
> >     StdDev:    10.50729 46.62148
> >
> >     Fixed effects: opp ~ 1
> >                     Value Std.Error  DF  t-value p-value
> >     (Intercept) 204.8143  11.22179 139 18.25148       0
> >
> >
> >     intercept variance 10.50729?
> >
> >     Best regards, Ben.
> >
> >
> >     On 26-7-2017 22:05, Thierry Onkelinx wrote:
> >     > Dear Ben,
> >     >
> >     > The correlation structure always works at the most detailed level
> of
> >     > the random effects. E.g. at the id level in the example below,
> >     not at
> >     > the group level. The correlation is only effective among
> >     observations
> >     > from the same id. Two observation within the same group but
> >     different
> >     > id have uncorrelated residuals by definition. Likewise are two
> >     > residuals from different groups uncorrelated. The correlation
> matrix
> >     > of the residuals is hence always a block diagonal matrix, with
> >     blocks
> >     > defined by the most detailed level of the random effects.
> >     >
> >     > opposites <-
> >     >
> alda/data/opposites_pp.txt
> >     <https://stats.idre.ucla.edu/stat/r/examples/alda/data/
> >     > opposites$group <- opposites$id %% 6
> >     > library(nlme)
> >     > lme(opp ~ 1, random = ~1|group/id, data = opposites)
> >     > lme(opp ~ 1, random = ~1|group/id, data = opposites, correlation =
> >     > corAR1(form = ~wave))
> >     >
> >     > Best regards,
> >     >
> >     >
> >     > ir. Thierry Onkelinx
> >     > Instituut voor natuur- en bosonderzoek / Research Institute for
> >     Nature
> >     > and Forest
> >     > team Biometrie & Kwaliteitszorg / team Biometrics & Quality
> >     Assurance
> >     > Kliniekstraat 25
> >     > 1070 Anderlecht
> >     > Belgium
> >     >
> >     > To call in the statistician after the experiment is done may be no
> >     > more than asking him to perform a post-mortem examination: he may
> be
> >     > able to say what the experiment died of. ~ Sir Ronald Aylmer Fisher
> >     > The plural of anecdote is not data. ~ Roger Brinner
> >     > The combination of some data and an aching desire for an answer
> does
> >     > not ensure that a reasonable answer can be extracted from a
> >     given body
> >     > of data. ~ John Tukey
> >     >
> >     > 2017-07-26 21:45 GMT+02:00 Ben Pelzer <b.pelzer op maw.ru.nl
> >     <mailto:b.pelzer op maw.ru.nl>
> >     > <mailto:b.pelzer op maw.ru.nl <mailto:b.pelzer op maw.ru.nl>>>:
> >     >
> >     >     Dear Thierry and Wolfgang,
> >     >
> >     >     Thanks for responding so quickly. Here is the reproducible
> >     example of
> >     >     the two models that I'm interested in.
> >     >
> >     >     # Read data.
> >     >     opposites <-
> >     >
> alda/dat/opposites_pp.txt
> >     <https://stats.idre.ucla.edu/stat/r/examples/alda/dat/
> opposites_pp.txt>
> >     >
> >      <https://stats.idre.ucla.edu/stat/r/examples/alda/dat/
> opposites_pp.txt
> >     <https://stats.idre.ucla.edu/stat/r/examples/alda/dat/
> >     >
> >     >     # Open library.
> >     >     library(nlme)
> >     >
> >     >     # Define group "one" with value 1 for all cases.
> >     >     one <- rep(1, 140)
> >     >
> >     >     #----- Model estimated with gls.
> >     >     comsym.gls <- gls(opp~1,opposites,
> >     >                        correlation=corCompSymm(form = ~ 1 |id),
> >     >     method="REML")
> >     >     summary(comsym.gls)
> >     >
> >     >
> >     >     #----- Same model estimated with lme.
> >     >     comsym.lme <- lme(opp~1,opposites,
> >     >                        random= ~1|one,
> >     >                        correlation=corCompSymm(form = ~ 1 |one/id),
> >     >     method="REML")
> >     >     summary(comsym.lme)
> >     >
> >     >
> >     >     #----- Wolfgang's gls suggestion for heterogeneous CS.
> >     >
> >     >     summary(gls(opp ~ 1, opposites, correlation =
> >     corCompSymm(form = ~ 1 |
> >     >     id), weights = varIdent(form = ~ 1 | wave)))
> >     >
> >     >     # Does not work with lme.
> >     >     summary(hetercom <- lme(opp ~ 1,opposites,
> >     >                      correlation=corCompSymm(form = ~ 1 |id),
> >     >                      weights=varIdent(form = ~1|wave),
> >     method="REML"))
> >     >
> >     >     # But does work with the "one" trick.
> >     >     summary(lme(opp ~ 1,opposites,
> >     >          random = ~1|one,
> >     >          correlation=corCompSymm(form = ~ 1 |one/id),
> >     >          weights=varIdent(form = ~1|wave), method="REML"))
> >     >
> >     >
> >     >
> >     >     The main reason of my mailing to the list is this. I was
> >     wondering
> >     >     whether with lme it is possible to also estimate models with
> the
> >     >     individuals nested in higher levels like schools or
> >     hospitals etc
> >     >     and at
> >     >     the same time letting the residuals correlate within
> >     individuals over
> >     >     time with one of the nice covar structures. However, I do
> >     NOT have a
> >     >     reproducible example of such more complex models at the
> >     moment. I only
> >     >     hoped that if the lme version of the model presented above
> >     has no
> >     >     further problems than a (slightly) different AIC, BIC and std.
> >     >     error of
> >     >     the fixed intercept, it would be meaningful to proceed in this
> >     >     way, i.e.
> >     >     using lme instead of gls, since lme provides the important
> >     possibility
> >     >     of additional random effects in the model.
> >     >
> >     >     As to Wolfgang's suggestion for heretogeneous CS using:
> >     >
> >     >     gls(opp ~ 1, correlation = corCompSymm(form = ~ 1 | id),
> >     weights =
> >     >     varIdent(form = ~ 1 | timepoint))
> >     >
> >     >     I didn't find a way to estimate such a model with lme,
> >     except for the
> >     >     method with the "trick". Using:
> >     >
> >     >     lme(opp ~ 1,opposites,
> >     >                  correlation=corCompSymm(form = ~ 1 |id),
> >     >                  weights=varIdent(form = ~1|wave), method="REML")
> >     >
> >     >     leads to error-message:
> >     >
> >     >     Error in lme.formula(opp ~ 1, opposites, correlation =
> >     >     corCompSymm(form
> >     >     = ~1 | : incompatible formulas for groups in 'random' and
> >     >     'correlation'
> >     >
> >     >
> >     >     whereas
> >     >
> >     >     lme(opp ~ 1,opposites,
> >     >                  random = ~1|one,
> >     >                  correlation=corCompSymm(form = ~ 1 |one/id),
> >     >                  weights=varIdent(form = ~1|wave), method="REML")
> >     >
> >     >
> >     >
> >     >     Best regards, Ben.
> >     >
> >     >
> >     >     On 26-7-2017 14:44, Thierry Onkelinx wrote:
> >     >     > Dear Ben,
> >     >     >
> >     >     > Please be more specific on the kind of model you want to
> >     fit. That
> >     >     > would lead to a more relevant discussion that your potential
> >     >     misuse of
> >     >     > lme. A reproducible example is always useful...
> >     >     >
> >     >     > Best regards,
> >     >     >
> >     >     >
> >     >     > ir. Thierry Onkelinx
> >     >     > Instituut voor natuur- en bosonderzoek / Research
> >     Institute for
> >     >     Nature
> >     >     > and Forest
> >     >     > team Biometrie & Kwaliteitszorg / team Biometrics & Quality
> >     >     Assurance
> >     >     > Kliniekstraat 25
> >     >     > 1070 Anderlecht
> >     >     > Belgium
> >     >     >
> >     >     > To call in the statistician after the experiment is done
> >     may be no
> >     >     > more than asking him to perform a post-mortem examination:
> >     he may be
> >     >     > able to say what the experiment died of. ~ Sir Ronald
> >     Aylmer Fisher
> >     >     > The plural of anecdote is not data. ~ Roger Brinner
> >     >     > The combination of some data and an aching desire for an
> >     >     > not ensure that a reasonable answer can be extracted from a
> >     >     given body
> >     >     > of data. ~ John Tukey
> >     >     >
> >     >     > 2017-07-26 13:36 GMT+02:00 Ben Pelzer <b.pelzer op maw.ru.nl
> >     <mailto:b.pelzer op maw.ru.nl>
> >     >     <mailto:b.pelzer op maw.ru.nl <mailto:b.pelzer op maw.ru.nl>>
> >     >     > <mailto:b.pelzer op maw.ru.nl <mailto:b.pelzer op maw.ru.nl>
> >     <mailto:b.pelzer op maw.ru.nl <mailto:b.pelzer op maw.ru.nl>>>>:
> >     >     >
> >     >     >     Dear list,
> >     >     >
> >     >     >     With longitudinal data, the package nlme offers the
> >     >     possibility to
> >     >     >     specify particular covariance structures for the
> >     residuals.
> >     >     In the
> >     >     >     examples below I used data from 35 individuals
> >     measured at 4
> >     >     time
> >     >     >     points, so we have 4 observations nested in each of 35
> >     >     individuals.
> >     >     >     These data are discussed in Singer and Willett, Applied
> >     >     Longitudinal
> >     >     >     Data Analysis, chapter 7.
> >     >     >
> >     >     >     In several sources I found that e.g. compound symmetry
> can
> >     >     easily be
> >     >     >     obtained with gls from package nlme, by using the
> >     correlation
> >     >     >     structure
> >     >     >     corCompSymm, as in:
> >     >     >
> >     >     >          comsym <- gls(opp~1,opposites,
> >     >     > correlation=corCompSymm(form = ~ 1 |id),
> >     >     >     method="REML")
> >     >     >
> >     >     >     where id is subject-identifier for the individual.
> >     With gls
> >     >     >     however one
> >     >     >     cannot specify random effects, as opposed to lme. To have
> >     >     lme estimate
> >     >     >     compound symmetry is simple, one would not even need
> >     have to
> >     >     specify a
> >     >     >     particular correlation structure, it would suffice to say
> >     >     >     "random=~1|id". However, for more complex covariance
> >     >     structures, e.g.
> >     >     >     heterogeneous compound symmetry, I was only able to
> >     find syntax
> >     >     >     for gls,
> >     >     >     but not for lme.
> >     >     >
> >     >     >     Then I thought that tricking lme might be an option by
> >     having a
> >     >     >     kind of
> >     >     >     "fake" random effect. That is, I constructed a variable
> >     >     "one" which
> >     >     >     takes value 1 for all cases, and let the intercept be
> >     random
> >     >     >     across "all
> >     >     >     one groups". This led to the following lme model:
> >     >     >
> >     >     >
> >     >     >          comsym.lme <- lme(opp~1,opposites, random= ~1|one,
> >     >     > correlation=corCompSymm(form = ~ 1
> >     >     |one/id),
> >     >     >     method="REML")
> >     >     >
> >     >     >     And actually to my surprise, the results of this lme
> >     are highly
> >     >     >     similar
> >     >     >     to those of the gls above.
> >     >     >     The output of both is attached below. The loglik's are
> >     >     identical, the
> >     >     >     AIC and BIC are not, which I can understand, as the
> >     lme has
> >     >     an extra
> >     >     >     variance to be estimated, compared to the gls. Also,
> >     the fixed
> >     >     >     intercept's standard error is different, the one of
> >     the lme
> >     >     being
> >     >     >     twice as large.
> >     >     >
> >     >     >     I added some independent variables (not shown below)
> >     but the
> >     >     >     similarities between gls and lme remain, with only the
> AIC
> >     >     and BIC and
> >     >     >     the standard error of the fixed intercept being different
> >     >     for the two
> >     >     >     models.
> >     >     >
> >     >     >     My question is threefold.
> >     >     >     1) Suppose the individuals (say pupils) would be nested
> in
> >     >     >     schools, then
> >     >     >     I assume that with lme I could add school as a random
> >     >     effect, and
> >     >     >     run a
> >     >     >     "usual" model, with pupils nested in schools and
> >     observations in
> >     >     >     pupils,
> >     >     >     and then use any of the possible residual covariance
> >     structures
> >     >     >     for the
> >     >     >     observations in pupils. Would you agree with this?
> >     (with gls one
> >     >     >     cannot
> >     >     >     use an additional random effect, like e.g. school)
> >     >     >     2) Are the lme results indeed to be trusted when using
> >     this
> >     >     "fake"
> >     >     >     random effect, apart from the differences with gls
> >     mentioned?
> >     >     >     Could you
> >     >     >     imagine a situation where lme with this trick would
> >     produce
> >     >     wrong
> >     >     >     results?
> >     >     >     3) I don't understand the variance of the intercept in
> the
> >     >     lme output.
> >     >     >     Even when I specify a very simple model: lme(opp ~ 1,
> >     random
> >     >     = ~1|one,
> >     >     >     opposites, method="REML")
> >     >     >     lme is able to estimate the intercept variance...but
> >     what is
> >     >     this
> >     >     >     variance estimate expressing?
> >     >     >
> >     >     >     Thanks a lot for any help!!!
> >     >     >
> >     >     >     Ben.
> >     >     >
> >     >     >
> >     >     >
> >     >     >
> >     >     >
> >     >     >     *----------------- gls
> >     >     >     ---------------------------------------------------
> >     >     >
> >     >     >      > comsym <- gls(opp~1,opposites,
> >     >     > correlation=corCompSymm(form = ~ 1 |id),
> >     >     >     method="REML")
> >     >     >      > summary(comsym)
> >     >     >
> >     >     >     Generalized least squares fit by REML
> >     >     >     Model: opp ~ 1
> >     >     >     Data: opposites
> >     >     >           AIC      BIC    logLik
> >     >     >     1460.954 1469.757 -727.4769
> >     >     >
> >     >     >     Correlation Structure: Compound symmetry
> >     >     >     Formula: ~1 | id
> >     >     >     Parameter estimate(s):
> >     >     >        Rho
> >     >     >     0.2757052
> >     >     >
> >     >     >     Coefficients:
> >     >     >                     Value Std.Error t-value p-value
> >     >     >     (Intercept) 204.8143  5.341965 38.34063       0
> >     >     >
> >     >     >     Standardized residuals:
> >     >     >        Min          Q1         Med   Q3  Max
> >     >     >     -2.75474868 -0.71244027  0.00397158 0.56533908 2.24944156
> >     >     >
> >     >     >     Residual standard error: 46.76081
> >     >     >     Degrees of freedom: 140 total; 139 residual
> >     >     >
> >     >     >
> >     >     >     #------------------ lme
> >     >     >     ---------------------------------------------------
> >     >     >
> >     >     >      > comsym.lme <- lme(opp~1,opposites, random= ~1|one,
> >     >     > correlation=corCompSymm(form = ~ 1 |one/id),
> >     >     >     method="REML")
> >     >     >      > summary(comsym.lme)
> >     >     >
> >     >     >     Linear mixed-effects model fit by REML
> >     >     >     Data: opposites
> >     >     >           AIC      BIC    logLik
> >     >     >     1462.954 1474.692 -727.4769
> >     >     >
> >     >     >     Random effects:
> >     >     >        Formula: ~1 | one
> >     >     >              (Intercept) Residual
> >     >     >     StdDev:    10.53875 46.76081
> >     >     >
> >     >     >     Correlation Structure: Compound symmetry
> >     >     >     Formula: ~1 | one/id
> >     >     >     Parameter estimate(s):
> >     >     >        Rho
> >     >     >     0.2757052
> >     >     >     Fixed effects: opp ~ 1
> >     >     >                     Value Std.Error  DF t-value p-value
> >     >     >     (Intercept) 204.8143  11.81532 139 17.33463       0
> >     >     >
> >     >     >     Standardized Within-Group Residuals:
> >     >     >        Min          Q1         Med   Q3  Max
> >     >     >     -2.75474868 -0.71244027  0.00397158 0.56533908 2.24944156
> >     >     >
> >     >     >     Number of Observations: 140
> >     >     >     Number of Groups: 1
> >     >     >
> >     >     >
> >     >     >             [[alternative HTML version deleted]]
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