Clarification on the rule of 10 for logistic regression

Question

Been brushing up on my logistic regression and I've seen a couple of things about the one in ten rule.

To illustrate my current understanding (or lack thereof) lets consider a case with only two independent binary variables. Does this mean I need a minimum of 20 total samples to justify having the two independent variables?

Perhaps my intuition is wrong but that seems low. Especially if the two variables were highly correlated. Consider the extreme case where I have twenty samples and for all samples, both x_1 and x_2 are 1. In this case I would think one would want more data where the explanatory variables are distinguishable. I'm sure the confidence intervals on the betas here would be significantly large and that is to be taken into consideration, but I'm just trying to rectify this rule (and yes I know it's a rule of thumb) with extreme cases like this

Answer 1

For logistic regression the rule of thumb isn't 10 total samples per independent variable, it's 10 cases having the lower-frequency outcome. So if you're analyzing a 20%/80% outcome ratio, that rule of thumb suggests 50 total samples per independent variable.

And this is only a rule of thumb. Some thumbs are bigger than others.

Your intuition about correlated predictors adds a further important consideration. It depends somewhat on what you're trying to accomplish with your regression analysis, for example whether you really want (or even should seek) separate estimates for each predictor, or you are willing to combine information from correlated predictors in some way. Follow the regularization and multicollinearity tags on this site to see ways to deal with correlated predictors.

What would be best, of course, would be a pilot study to get preliminary data. Those initial estimates of the relations of the predictors to each other and to outcome would then inform a power analysis for definitive experimental design.