Model selection with low N? I've got a study which is kinda messed up by the design...Turns out I ended up with about 50 patients of which 80% have the outcome and 20% don't (binary outcome).
I've been turning in my bed for the last month trying to figure out what to do with this. The only real answer is "don't" but I have to deliver something in the next days as it's part of a master's thesis.
I have about 10 predictors of interest for the outcome (various variable types) and the research question is whether any of these predictors can predict the outcome. Some already have an established correlation in the literature while some are original hypothesis. Several are significant with a univariate logistic regression, but it doesn't look good in any other way than the actual p-value.
Now obviously I'm not going to be able to answer this research question sufficiently but if you were in my shoes, what sort of statistical analysis would you perform to relay to your supervisors?
EDIT: Link to my (anonymized) data (CSV): https://gofile.io/?c=vwH9PS
 A: For biomedical studies, a general rule of thumb to avoid overfitting in an unpenalized logistic regression model is to have on the order of 10-20 minority-class cases per evaluated predictor. You have about 10 cases in the minority class, so without penalization you should only be evaluating 1 predictor. That predictor would need to be pre-selected based on your knowledge of the subject matter, as using the data to identify the predictor invalidates the assumptions needed to calculate p-values and confidence intervals.
If you did multiple association tests of outcome against each predictor as you propose you would at least have to correct for multiple comparisons and you would not be able to control for the values of the other predictors.
LASSO tends to return a number of predictors similar to the number that would be allowed under the rule of thumb in the first paragraph: so maybe only 1 or 2 in this case.
Logistic ridge regression (L2 penalty) might be the best way to start working with this small data set. All of your predictors would be included in the model, but their coefficients would be heavily penalized to avoid overfitting.
