How many observations of a binary predictor should there be for multiple regression? I have a model with ~30 binary predictors. Each predictor can be present or absent (i.e., X = 1 or X = 0).
There are ~1000 observations. The data is such that multiple predictors can be present in a given observation.
But there are some predictors that are only present in a very small number of stimuli.
Is there a rule of thumb for how many times a binary predictor should be present (i.e., how many times X = 1 instead of X = 0) for it to be included in the analysis?
I have been using 10 as a cutoff, but that is entirely arbitrary.
 A: Assuming you are modeling a continuous response, the desired properties for a binary predictor in a multivariate model are no different than those of a continuous predictor. You consider the standard deviation of that variable, its correlation with the response variable and with other predictor variables in the model.
In the bivariate modeling case, a linear regression measures association between a regressor and a response using the slope coefficient, $\beta$. The OLS estimate of $\beta$ is given by $\hat{\beta} = \text{Cov(X,Y)} / \text{SD}(X)$. OLS also produces an estimate of the residual standard error, $\hat{\sigma}^2 = \sum_{i=1}^n (Y_i - \alpha - \beta X_i)^2 / (n-2)$. And the standard error of $\hat{\beta}$ is $\text{SE}(\hat{\beta}) = \sqrt{\hat{\sigma}^2 / (\sum_{i=1}^n |X-\bar{X}|^2)}$.
So assuming that $\beta$ is a fixed magnitude, the only hypothetical issue with a low prevalence of the regressor is that the term $\sum_{i=1}^n |X-\bar{X}|^2$ becomes small, which in turn inflates $\text{SE}(\hat{\beta})$ reducing power and precision. As such, if you were asked to justify a sample size, you might find yourself choosing a larger $n$ to achieve a desired power, or not. Alternately, you can stratify a sample, but this is a useless exercise if the data are already collected. The issue of variance inflation gets worse when adding other predictors to the model. As you'd expect, few variables are exactly independent by design, and covariance between predictors inflates the standard error of the estimates all the more. There are more complicated expressions for this scenario, or you can perform simulations.
I do think it's important for analysts to turn non-specific rules of thumb into more precise arguments about known statistical concepts. So while a cutoff of $n=10$ might make sense to your collaborators, it doesn't have any statistical backbone, and it's not hard to think of a few scenarios where it will fail to work.
