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/> [[alternative HTML version deleted]]
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