[R-sig-ME] location-scale models in nlme

[Simon Harmel]

Thanks so much, Ben. I conclude that the users of lme() don't need to
convert the output for "varIdent()" or "varPower()" back due to a link
function to get the estimates of the relevant Level-1 residual variances.
This is because the former gives out the proportions between residual
variances with respect to a reference level in a categorical variable, and
the latter gives out "t", which the user can insert
in: (sigma(MODEL)^2)*abs(data$X2_numeric)^(2*t) to obtain the relationship
between  X2_numeric and the residual variance.

Ben, as both a mathematical and applied expert, which location-scale
approach, do you think, is more preferable? The one implemented in nlme()
or the one that allows modeling the scale using:  log(scale_i) = a_0 +
b_1*x_i1+ ... + b_n*x_ip  (for p predictors of scale) ??

Thank you so very much for sharing your perspective,
Simon





On Sun, Jan 14, 2024 at 2:00 PM Ben Bolker <bbolker using gmail.com> wrote:

>     For varIdent (from ?nlme::varIdent),
>
>   For identifiability reasons, the
>       coefficients of the variance function represent the ratios between
>       the variances and a reference variance (corresponding to a
>       reference group level).
>
>    I assume that this is internally parameterized via something like (1)
> a model matrix constructed with ~ <grouping factor> and (2) a log link,
> to ensure that the ratios are all positive
>
>   For varPower,
>
> s2(v) = |v|^(2*t)
>
>   -- notice it uses the absolute value of the covariate. So that term
> will also be positive.
>
> varComb uses a product; the product of two positive values will also be
> positive ...
>
> On 2024-01-14 11:40 a.m., Simon Harmel wrote:
> > Dear Ben and List Members,
> >
> > I'm following up on this
> > (https://stat.ethz.ch/pipermail/r-sig-mixed-models/2023q4/030552.html
> > <https://stat.ethz.ch/pipermail/r-sig-mixed-models/2023q4/030552.html>)
> > thread. There, Ben noted that my MODEL (below) qualifies as a
> > "location-scale" model.
> >
> > Q: Usually for the scale part of a location-scale model, the linear
> > model uses a log link to guarantee that the estimate of scale is
> positive:
> >
> > log(scale_i) = a_0 + b_1*x_i1+ ... + b_n*x_ip  (for p predictors of
> scale)
> >
> > But in the MODEL that I sketched below, how such a guarantee is made?
> >
> > Thanks, Simon
> > MODEL <- nlme::lme(y ~ X1_categorical + X2_numeric ...,
> >           random = ~1| subject,
> >           data = data,
> >           correlation = corSymm(~1|subject),
> >           weights = varComb(varIdent(form = ~ 1 |  X1_categorical ),
> >                                            varPower(form = ~  X2_numeric
> )))
> >
>
> --
> Dr. Benjamin Bolker
> Professor, Mathematics & Statistics and Biology, McMaster University
> Director, School of Computational Science and Engineering
> (Acting) Graduate chair, Mathematics & Statistics
>  > E-mail is sent at my convenience; I don't expect replies outside of
> working hours.
>

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