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I have collected physiological data with multiple observations from 35 people, across four conditions. In planning the experiment, I had been hoping to perform inferential statistics comparing between the four conditions and thought the data would most likely be suited to a GLMM analysis as a gamma distribution with a log link.

There is a factorial predictor (within-participant), a continuous predictor, and the grouping of participant.

Here is what I collected:

enter image description here

I thought at first that perhaps the earliest spike, at about 8 on the X axis, could have been a recording error or quirk unique to particular participants. But actually these types of very short responses are shared across most people. Having looked at the raw data, it also doesn't seem to be down to an obvious equipment issue, either. Here's a break down by condition - each shares the general shape:

enter image description here

Now, I am not sure how to best characterise what I've got here. I have done some gamma mixture modelling using the mixR package in R, using the complete data set:

enter image description here

So I had the thought that I could produce gamma mixture models separately for each condition, and then perhaps characterise how the components change across the different conditions, since it looks like there might be something interesting happening there, especially with the green (lowest) line.

But what I am wondering now, is how to take my continuous predictor into account; moreover, can I address the possibility that some participants responded in different ways to the factorial predictor (condition)? I would have done so using a random slope, had I been able to analyse the data using GLMM.

Finally, should I consider performing any form of significance testing (between conditions), given the bimodality observed (and still taking this random grouping of participant into account)? By interpreting the data visually, it would appear to me that there are two different underlying processes generating the observations, but it wasn't something I expected to see, and I don't have a clear idea of what it means, which would imply to me that my original hypothesis (concerning between-condition differences) is no longer applicable.

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  • $\begingroup$ It's unclear why you need a mixture model or even a Gamma model, because the distribution of your raw observations is hardly relevant. What matters is the distribution of residuals in your model. Those will be influenced not just by the condition, but also by the "continuous predictor" to which you refer. You might begin by regressing your results on that predictor and studying how the residual distributions vary with condition. $\endgroup$
    – whuber ♦
    Commented Oct 10, 2021 at 18:00
  • $\begingroup$ @whuber Thanks, this is something I probably misunderstood from my graduate stats introduction. I have been using fitDist in gamlss to try to visualise different residuals and it settled on the Box Cox expontential dist. What are your thoughts on these diagnostics? ibb.co/z4tvsjk $\endgroup$
    – stck8888
    Commented Oct 11, 2021 at 12:04
  • $\begingroup$ I should add that the continuous predictor follows a fairly simple, linear pattern and does not seem related to the bimodality seen in the DV $\endgroup$
    – stck8888
    Commented Oct 11, 2021 at 12:32
  • $\begingroup$ Fair enough--but even so, the picture can change when you analyze the residuals. If the bimodality persists, one of the first things to consider is whether the modes reflect values of some as yet unused categorical covariate; and if so, whether you can measure it or at least infer its values. That's not always possible, but when it is, including that covariate can accomplish a lot. $\endgroup$
    – whuber ♦
    Commented Oct 11, 2021 at 14:16

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