Hello Marian,

Here's the glib rule: More is better as regards numbers of cases observed for each level of a categorical variable. Your resultant parameter estimates will be better as a result. There are various guidelines offered, but these are generally guidelines and not commandments. Here are some general observations:

1. You want at least some cases in each level of a categorical variable. Otherwise, you have insufficient information to make inferences about one or more of the categories/levels. If some levels are genuinely rare in the population, and your sample reflects this, then you may find alternative models (e.g., Poisson) to be preferable.

2. Whether it makes sense to use an unordered, k-level categorical variable (when k > 2) in your analysis without re-expressing as (k - 1) dummy variates hinges on the type of analysis you plan to conduct. In a chi-square test, it could work; in a logistic or ordinal regression model, it will not.

3. It can make sense to over-sample rare categories in order to help with precision of parameter estimates.

4. In the context of chi-square contingency tables, the usual advice is to have sufficient frequencies such that expected cell frequencies are at least 5.

5. One option would be to run bootstrap estimates of standard errors for each estimated coefficient, and if that SE exceeded some threshold, then judge that variable's contribution as indeterminate (e.g., more data required).

Good luck with your work.

Frank Harrell's Author Checklist suggests a 20:1 rule of thumb. Search for 20:1 on this page to find it:

• https://discourse.datamethods.org/t/author-checklist/3407

Here is the key excerpt, with the bit relevant to binary logistic regression highlighted.

The 20:1 rule is as follows. Let m denote the effective sample size (the number of subjects if the response variable is a fully-observed continuous one; the number of events if doing a survival analysis; the lower of the number of events and number of non-events if the response is dichotomous) and p denote the number of candidate predictor terms that were examined in any way with respect to the response variable. p includes nonlinear terms, product terms, different transformations attempted, the total number of cutoffs attempted to be applied to continuous predictors, and the number of variables dropped from the final model in a way that was unblinded to the response. If the ratio of m to p exceeds 20, the model is likely to be reliable and there is less need for the model to be validated.

And notice that the next paragraph recommends the bootstrap (which David Morse also mentioned) if model validation is required.

HTH.

While I respect Frank Harrell very much If you really want to know I would recommend doing a simulation or going with the bootstrap approach. Best wishes David Booth