[R-sig-ME] High correlation among random effects for longitudinal model

[Fox, John]

Dear Joshua,

I'm chiming in late, so it's possible that someone already pointed this out and I didn't notice. A better way to specify a polynomial in R is to use poly() in the model formula. By default, this produces orthogonal polynomial regressors (at least in the fixed effects) but the same fit to the data. For example,

> time <- 1:5
> X <- poly(time, 2)

> X
              1          2
[1,] -0.6324555  0.5345225
[2,] -0.3162278 -0.2672612
[3,]  0.0000000 -0.5345225
[4,]  0.3162278 -0.2672612
[5,]  0.6324555  0.5345225
attr(,"coefs")
attr(,"coefs")$alpha
[1] 3 3

attr(,"coefs")$norm2
[1]  1  5 10 14

attr(,"degree")
[1] 1 2
attr(,"class")
[1] "poly"   "matrix"

> colSums(X)
           1            2 
0.000000e+00 1.110223e-16 

> crossprod(X)
              1             2
1  1.000000e+00 -1.110223e-16
2 -1.110223e-16  1.000000e+00

My guess is that this will also reduce the correlations among the random effects. If you really must have raw polynomials, then poly(time, 2, raw=TRUE) offers the advantage that model-structure-aware functions can understand that the linear and quadratic regressors are part of the same term in the model.

Whether high correlations among the random effects are really a problem, my guess is that they aren't, because lme() uses a log-Cholesky factorization of the random-effects covariance matrix anyway. In some contexts, high correlations might produce numerical instability, but, as I said, probably not here. Ben would know.

I hope this helps,
 John

-----------------------------
John Fox, Professor Emeritus
McMaster University
Hamilton, Ontario, Canada
Web: socialsciences.mcmaster.ca/jfox/





> -----Original Message-----
> From: R-sig-mixed-models [mailto:r-sig-mixed-models-bounces at r-project.org]
> On Behalf Of Joshua Rosenberg
> Sent: Monday, April 2, 2018 5:33 PM
> To: Ben Bolker <bbolker at gmail.com>
> Cc: R SIG Mixed Models <r-sig-mixed-models at r-project.org>
> Subject: Re: [R-sig-ME] High correlation among random effects for longitudinal
> model
> 
> Dear Stuart and Ben,
> 
> Thank you, this worked to significantly reduce the correlations between the
> intercept and the linear and quadratic terms (though still quite high between the
> linear and quadratic term):
> 
> Random effects:
>  Formula: ~time + I(time^2) | student_ID
>  Structure: General positive-definite, Log-Cholesky parametrization
>             StdDev    Corr
> (Intercept) 18.671959 (Intr) time
> time        11.029842 -0.262
> I(time^2)    8.359834 -0.506  0.959
> Residual    29.006598
> 
> Could I ask if that correlation between the linear (time) and quadratic
> I(time^2) terms is cause for concern - and if so, how to think about
> (potentially) addressing this?
> Josh
> 
> On Sun, Apr 1, 2018 at 12:34 PM Ben Bolker <bbolker at gmail.com> wrote:
> 
> > On Sun, Apr 1, 2018 at 12:20 PM, Stuart Luppescu <lupp at uchicago.edu>
> > wrote:
> > > On Sun, 2018-04-01 at 12:55 +0000, Joshua Rosenberg wrote:
> > >> lme(outcome ~ time + I(time^2),
> > >>     random = ~ time + I(time^2),
> > >>     correlation = corAR1(form = ~ time | individual_ID),
> > >>     data = d_grouped)
> > >>
> > >> I have a question / concerns about the random effects, as they are
> > >> highly correlated (intercept and linear term = -.95; intercept and
> > >> quadratic term = .96; linear term and quadratic term = -.995):
> > >
> > > I think this is an ordinary occurrence for the intercept and time
> > > trend to be negatively correlated. The way to avoid this is to
> > > center the time variable at a point in the middle of the series, so,
> > > instead of setting the values of time to {0, 1, 2, 3, 4} use {-2, -1, 0, 1, 2}.
> > >
> >
> >   Agreed.  This is closely related, but not identical to, the
> > phenomenon where the *fixed effects* are highly correlated.
> >
> > > --
> > > Stuart Luppescu
> > > Chief Psychometrician (ret.)
> > > UChicago Consortium on School Research
> > > http://consortium.uchicago.edu
> > >
> > > _______________________________________________
> > > R-sig-mixed-models at r-project.org mailing list
> > > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >
> > _______________________________________________
> > R-sig-mixed-models at r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >
> --
> Joshua Rosenberg, Ph.D. Candidate
> Educational Psychology ​&​ Educational Technology Michigan State University
> http://jmichaelrosenberg.com
> 
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