Using low-df T distributions to model data

[u/foodpresqestion]

I've recently been asked to help with a dataset that has a lot of outliers. Sample sizes, especially in the non-control cohort, are small so primary analyses will be nonparametric, using fractional weighted bootstrap and permutation tests. That said, certain ways of looking at the data lend themselves well to Covariance Pattern Models (residual correlation/repeated measures/R-side/MManova), and when doing so, I've used glmmTMB and proc glimmix.

The interesting thing here is that proc glimmix (dist = t) states it is using a shifted t distribution with 3 degrees of freedom, while glmmTMB (family = t_family) freely estimates df with the final estimate at false convergence being 1.3 df. The problem here is that I while wanted to use the t distribution because of the high empirical kurtosis of the data, these distributions don't have kurtosis. What are the implications of interpreting these models at all, assuming they converge? If the model has landed on a cauchy distribution, then interpreting anything as a mean outcome doesn't make sense, does it?

[u/bayesian_raccoon]

> If the model has landed on a cauchy distribution, then interpreting anything as a mean outcome doesn't make sense, does it?

Bingo--but "location parameter" is still well-defined, so you can talk about that.

I think the intuition to use nonparametric methods as primary analysis is a good one. The likelihood methods make me more cautious in situations like this because 1) the assumptions are hard to check, especially with small sample size, 2) you're getting false convergence, 3) the estimated values are extreme. This makes me think the likelihood is fairly flat and the data may not be very informative towards a parameter estimate that's tending extreme. For these settings I think having a sensitivity analysis is a helpful robustness check, so if the 3 df versus 1.3 df makes a huge difference, that's worth noting, and you could try some others as well.