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I have a repeated measure study with continuous and two categorical predictors (time for repeated and group for control/experiment group. I checked the assumptions, and realized that there may be some problems to go with linear mixed-effects and decided to switch to generalized linear mixed effects models.

However, I cannot decide which family I should choose for the models. I have two models for two different continuous predictors. By the way, I checked the assumptions overall, not based on groups which I hope it is a better practice.

Model 1 = https://ibb.co/VLwTNq7 Figure

Model 2 = https://ibb.co/ZNFT8Hs

Just dependent: https://ibb.co/sWz44FW and its histogram: https://ibb.co/5n9mDf2 The outcome variable is Depression (measured repeatedly before and after neutral and negative stimuli exposure (between subjects). It was measured via a Likert-type questionnaire. We are looking for how adaptive/maladaptive effects pre and post depression after exposure to neutral and negative stimuli.

Which family should I choose?

Thanks in advance!

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    $\begingroup$ Please edit this question to provide more details about your study, to make it more helpful to future visitors. Describe the nature of your outcome variable and how it is measured, as that might be more helpful than qq plots and such. Describe the specific hypothesis you are testing and the structure of the model you are considering (as you did in another question). Also, please place illustrations directly into this page. It's probably best to show residuals for the whole model rather than breaking up into adaptive/maladaptive. $\endgroup$
    – EdM
    Jan 16 at 17:05
  • $\begingroup$ Hello! Actually, I was going to ask you directly. The outcome variable is Depression (measured repeatedly before and after neutral and negative stimuli exposure (between subjects). It was measured via a Likert-type questionnaire. Rather than hypotheses, we are looking for how adaptive/maladaptive effects pre and post depression after exposure to neutral and negative stimuli. $\endgroup$ Jan 16 at 17:50
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    $\begingroup$ Please edit the question to put that information, and the rest I asked for, into it directly. Comments are easy to overlook and can be deleted. Although helping with individual problems is an important part of the work on this site, it's also important to make individual questions and answers useful to later visitors. The more self-contained a question is, the better. $\endgroup$
    – EdM
    Jan 16 at 17:53
  • $\begingroup$ Thank you for making some initial amendments. What families have been considered (and why)? Why is the underlying random-effects structure used (and why)? $\endgroup$
    – usεr11852
    Jan 16 at 20:58
  • $\begingroup$ Hi. Actually, I don't know what family to consider. The question was about it. So, it is a right skewed dependent variable. We used random-effects because the participants in one of the groups and it is not important. $\endgroup$ Jan 17 at 12:38

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You might be able to do something simple with your Likert-scale outcome values that doesn't require a generalized linear model. A prior square-root or logarithmic transformation of those values might bring the residuals into a form that's good enough for a mixed-model linear regression. Those sometimes help when there is substantial skew toward large values in residuals, as your diagnostic plots suggest.

There can be problems with interpreting coefficients after such transformations. With a binary predictor and a square-root transformation of outcomes, for example, the regression coefficient is in units of "square root of Likert scale values." As Likert scales are already somewhat arbitrary, that might not be a problem for you.

The next simplest approach would be a generalized linear model with a log link. That models the log of the mean values of the outcomes, versus modeling the mean of the logs of the outcomes as you do in linear regression with log-transformed outcome values.

Ordinal regression, another class of generalized linear models, seems more involved at first but might end up being the best choice. You don't have to make any assumption about the form of spacing between consecutive outcome levels provided that they are in order. Ordinal regression can make a lot of sense for outcomes that take a number of discrete ordered values, as yours do. It even can have advantages with continuous outcomes, as it makes no assumptions about the distribution of residuals.

There are two main types of ordinal logistic regressions, proportional-odds (PO) and continuation-ratio (CR) models. With PO you model the cumulative probability of an outcome exceeding some value; with CR you model the conditional probability of an outcome taking a particular value given that the case already has at least that high a level. Choices of link function other than the logit used for logistic regressions are also possible.

Frank Harrell devotes several chapters of Regression Modeling Strategies and his course notes to ordinal regression. The R ordinal package implements ordinal regression for mixed models.

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