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

[Simon Harmel]

When I say "more preferable", I mean for instance, in terms of flexibility,
and generality (e.g., approach b subsuming approach a or vice versa).

On Sun, Jan 14, 2024 at 3:44 PM Simon Harmel <sim.harmel using gmail.com> wrote:

> 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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