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

[Ben Bolker]

    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
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