[R-sig-ME] What it means for rho to be 0 in lme() when using compound symmetry

[Phillip Alday]

The 'default' structure is generally the unstructured positive definite
matrix. (In lme4, this constraint is loosened to positive
semi-definite). Starting from compound symmetric is already placing a
constraint and so forcing all off-diagonal elements to zero may be the
best solution *under that constraint*. (And a multiple of the identity
matrix is a stricter constraint than a diagonal matrix, even though both
have all off diagonal elements set to zero.)

A more modern take would be using something like rePCA on the full
unstructured matrix (https://arxiv.org/abs/1506.04967 or
https://doi.org/10.33016/nextjournal.100002) to see what the effective
dimensionality is. Of course, you can fit a model with a diagonal RE
covariance matrix even if the data have some correlation, at the cost of
changing how shrinkage works
(https://doingbayesiandataanalysis.blogspot.com/2019/07/shrinkage-in-hierarchical-models-random.html).
That may be an acceptable (variance-bias) tradeoff -- less efficient
shrinkage but also less overparameterization.

All of these comments without looking at your data.

On 4/12/20 4:23 pm, Simon Harmel wrote:
> Thanks, Phillip. Given the estimated rho of 0 obtained from the default
> correlation structure in lme(), can we say that for this dataset there
> is no dependence left after fitting the 2-level model shown in my
> original post?
> 
> In other words, once getting a rho of 0 from the default correlation
> structure for this model, then one doesn't need to think of alternative
> correlation structures, because even the default correlation structure
> has shown that there is no dependence to model.
> 
> Is this a reasonable conclusion?
> 
> On Fri, Dec 4, 2020, 9:11 AM Phillip Alday <me using phillipalday.com
> <mailto:me using phillipalday.com>> wrote:
> 
>     From
>     https://stat.ethz.ch/R-manual/R-devel/library/nlme/html/pdCompSymm.html
>     :
> 
>     "This function is a constructor for the pdCompSymm class, representing a
>     positive-definite matrix with compound symmetry structure (constant
>     diagonal and constant off-diagonal elements)."
> 
>     Any multiple of the identity matrix is technically compound symmetric,
>     because all the off-diagonal elements are the same (0).
> 
>     Phillip
> 
>     On 28/11/20 2:30 am, Simon Harmel wrote:
>     > Hello All,
>     >
>     > Below, I'm using corCompSymm() (compound symmetry) for my simple
>     model.
>     >
>     > The rho is estimated to be 0. I was wondering what it means for
>     rho in the
>     > var-covariance matrix to be "0"? Is my var-covariance matrix below
>     valid?
>     > -- Thank you all, Simon
>     > #----------------------------------------------------------------
>     > library(nlme)
>     > data <-
>     read.csv('https://raw.githubusercontent.com/hkil/m/master/R.csv')
>     >
>     > m <- lme(Achieve ~ time, random = ~1|subid, data = data, correlation =
>     > corCompSymm())
>     >
>     >   aa <- corMatrix(m$modelStruct$corStruct)[[1]]
>     >   aa * sigma(m)^2
>     >
>     >          [,1]     [,2]     [,3]     [,4]
>     > [1,] 112.5003   0.0000   0.0000   0.0000
>     > [2,]   0.0000 112.5003   0.0000   0.0000
>     > [3,]   0.0000   0.0000 112.5003   0.0000
>     > [4,]   0.0000   0.0000   0.0000 112.5003
>     >
>     >       [[alternative HTML version deleted]]
>     >
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