What comes first: outlier detection or model selection? I'm fitting a GLMM (mixed logistic regression) in R. I have five covariates. For model selection, I'm using glmmLasso() (in R) to determine which of the five covariates and their interactions should be put in the model. 
Should outlier detection be done before or after model selection? What techniques would you recommend for outlier detection?
 A: With logistic regression if your responses are 0/1 (rather than grouped and presented as fractions) what constitutes an outlier? Getting a 1 when the probability of 0 is high? 
Responding more generally --

Should outlier detection be done before or after model selection?

You can't separate the two:


*

*outliers are only outliers with respect to some model. An outlier for model A may not be remotely unusual for model B

*model selection is impacted by points that would be outliers for some of the possible models
In a Bayesian context you could do both simultaneously, for example via MCMC. In a frequentist setting that's usually hard, but some kind of iterative process might work.

What techniques would you recommend for outlier detection?

I think that's much too general a question, but in any case my first thought would be to avoid detection (unless it's the outliers that are of central interest) and instead focus more on whether you can do something that's not especially sensitive to outliers. (Of the top of my head I doubt I can say anything sensible in that regard for GLMM)
