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My question is about running a longitudinal trial with mixed effects and binomial response.

I have a repeated-measure trial data to analyze: Patients are subjected to 4 therapies (placebo and 3 devices) - each patient is randomized to a certain therapy, no switches - and examined three times. The outcome of the examination is binomial: TRUE/FALSE.

The three time points are: week 3, week 16, week 24. I'm thinking now whether it's better to treat is as factor or continuous.

I want to analyze the relationship between therapy arm and time by incorporating a mixed-effects model.

My model would be like this: exam <- time * arm + (1 + time + arm | patient)

I expect the intra-subject autocorrelation. I would like to experiment with: Toeplitz, AR(1) and unstructured residual covariance matrix.

The random-effect matrix, as the formula shows, involves correlation between all the random effects: both slopes and intercept. I expect patients with low initial response to have steeper response over time and across treatments. I expect that the slope will vary between treatments and placebo. I expect at least non-decreasing trend proportion of successes over time, that is, more and more patients will be positively examined over time (the examination is taken 3 times on each subject).

The distribution is binomial with logit link function.

I would also like to validate somehow the output with SAS, so optimization method should be comparable to the one used by SAS. It does not matter which SAS procedure will be used.

I can consider ML, REML, Laplace, penalized quasi-likelihood. Optimization can also involve the Nelder-Mead method.

I have heard, that PQL has the poorest quality of calculation and Laplace is the best one.

I thought about:

  • nlme - it supports residual covariance structure, but doesn't handle non-gaussian distribution

    • lme4 - handles binomial distribution, has lots of additional tools, works with single-model anova function nicely, giving me the marginal effects (in case of no strong interaction), but doesn't handle residual covariance structure other than diagonal, which is wrong in longitudinal model, as the examinations may be autocorrelated

I am afraid of using this for a trial longitudinal by nature.

  • glmmPQL handles both features, but has poor quality, no ML/REML by definition

  • glmmTMB - handles both, but is relatively new and I don't know how well it is tested and if I can trust it? Also raises issues about using AR1 for the intercept - is this because the intercept "stands" only for a single point (at X="0") so doesn't have the autocorrelation over time?

Is this kind of analysis, mixed-effect logistic longitudinal model even possible in R? I assume people managed somehow to analyze non-gaussian mixed models with R covariance in the past.

binomial
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  • I would not use `arm` as random slope, because you typically don't assume that your intervention varies accross patients (it's the outcome that varies, but you already captured that with `time` as random slope). So, your model would be `m <- glmer(exam ~ time * arm + (1 + time | patient))` or `m <- glmmTMB(exam ~ time * arm + (1 + time | patient))`. _glmmTMB_ is reliable, however, due to different default settings, _glmmTBM_ and _lme4_ differ when they give warnings etc. With `lme4::glmer()`, you can't do REML-fits for binomial outcomes, but I think you can do so with `glmmTMB()`. – Daniel Aug 13 '18 at 13:03

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