how to specify the response (dependent) variable in a logistic regression model

John,

How comfortable are you with mixed models software beyond lme4? This
seems like a perfect case for a multivariate mixed model (which you can
do with e.g. brms or MCMCglmm). The basic idea is that you do create a
single mixed model that can be thought of doing two GLMMs
simultaneously. Here's the basic syntax for doing this in brms:


brm(mvbind(Resp1, Resp2) ~ preds + ..., data=your_data, family=binomial)

You can also specify this as two formulae (which really highlights the
"two models simultaneously" intuition):


var1 = bf(Resp1 ~ preds + ....) + binomial()
var2 = bf(Resp2 ~ preds + ....) + binomial()

brm(var1 + var2, data=your_data)

The advantage to doing this as a multivariate model as opposed to
separate models is that you get simultaneous estimates across both
models, including correlation/covariance between those estimates.  See
e.g. the brms documentation
(https://paul-buerkner.github.io/brms/articles/brms_multivariate.html)
for more info. In particular, pay attention to the extra syntax for
computing shared correlation in the random effects across sub-models.

The cons for this approach are that [1] most reviewers in
(psycho)linguistics will not be familiar with it (and there was recent a
Twitter storm on this very problem) and [2] the computational costs are
noticeably higher.

Another alternative is to do something like "linked mixed models" (cf.
Hohenstein, Matuschek and Kliegl, PBR 2016). There are a few variants on
this, but the basic idea is that you use one response to predict the
other. Given the temporal ordering here, this might make sense, e.g.

mod1 = glmer(Resp1 ~ preds + ....)
mod2 = glmer(Resp2 ~ preds + YYY + ....)

where YYY is one of:
[a] Resp1
[b] fitted(mod1)
[c] fitted(mod1) + resid(mod1)

You can potentially omit mod1, in which case you have something like the
Davidson and Martin (Acta Psychologia, 2016) approach to the joint
analysis of reaction times and response accuracy.

The downside to this approach is that the variability that's in Resp1
can create problems in mod2, because standard GLMMs assume that the
predictors are measured without error/variability. Variants [b] and
especially [c] mitigate this a bit though. (And if you want to get even
more complicated, there are  "errors-within-variables" models, which can
handle this and are available in e.g. brms). I think the advantage to
the linked model approach relative to the multivariate approach is that
it's somewhat more accessible for a typical (psycho)linguistic reviewer.

Note that I am nominally originally from linguistics and do know a bit
about mixed models, so I'm a good usual suspect for a reviewer on these
things.

Best,
Phillip

PS: the multinomial models suggested by the others are also pretty good,
but again multinomial models are usually something that require getting
used to and doesn't reflect the potential covariance of Resp1 and Resp2
in an obvious way.



On 14/1/21 5:05 pm, Greg Snow wrote:

John,

I agree that ordering your responses does not make sense, but the
multinomial models are for unordered categorical data.  So you can
just treat your 4 possible outcomes as unordered categories.

Another option is to convert to a Poisson regression where the
response variable is the count (number of times each of the 4
combinations is selected) and then your categories become
explanitory/predictor variables.  You can either use a single
predictor with the 4 levels (and choose appropriate indicator
variables) or you can have 2 predictors (b vs w and 1 vs 2) as well as
their interaction.  That would give a different interpretation of the
model, but may be more what you are trying to accomplish.

On Thu, Jan 14, 2021 at 8:44 AM John Kingston <jkingstn at umass.edu>

wrote:

Dear Thierry,
Thanks for your question. Here's the reason why I think the responses
aren't multinomial (or ordinal).

The listeners were presented with spoken strings of the form CVC, where

C =

consonant and V = vowel. The rate at which the acoustics changed at the
beginning of the syllable was varied orthogonally with the duration of

the

vowel. The rate of acoustic change conveyed the identity of the initial
consonant, which was expected to sound like "b" when the rate of change

was

faster and like "w" when it was slower. The duration of the vowel

conveyed

how many syllables the string consisted of, which was expected to be "1"
when the vowel was shorter and "2" when the vowel was longer. The

listeners

were instructed to respond with "b" or "w" and "1" or "2" on every

trial.

So, unlike a truly multinomial dependent variable, such as professions

or

majors, the responses here are not unordered. They also cannot be

arranged

into a single order sensibly, because even if "b1" and "w2" responses

are

first and last in the order, there's no way of deciding *a priori* the
order of "b2" and "w1" responses.

Again, thanks for your reply.
Best,
John
John Kingston
Professor
Linguistics Department
University of Massachusetts
N434 Integrative Learning Center
650 N. Pleasant Street
Amherst, MA 01003
1-413-545-6833, fax -2792
jkingstn at umass.edu
https://blogs.umass.edu/jkingstn
<https://blogs.umass.edu/jkingstn/wp-admin/>

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