Consider a binary response $Y$ and two binary predictors $X_1$ and $X_2$. Here is some synthetic data to illustrate the problem.
| | Y=1 | Y=0 | | X1=1, X2=0 | 50 | 0 | | X1=0, X2=1 | 3 | 9 | | X1=1, X2=1 | 16 | 0 |
I would like to fit a logistic regression model of the form $$E(Y) = \beta_0 + \beta_1 X_1 + \beta_2 X_2,$$ but the zeros in the table above make this an ill-posed problem. One simple option is to add 1 dummy observation to each cell in the above table, leading to
| | Y=1 | Y=0 | | X1=1, X2=0 | 51 | 1 | | X1=0, X2=1 | 4 | 10 | | X1=1, X2=1 | 17 | 1 |
This makes it possible to fit the logistic regression model. This approach seems similar to Theils' Mixed Estimation method in econometrics, and clearly has a Bayesian flavor (and interpretation).
Some general questions are
- Is this solution for handling the zero-counts justifiable?
- Is there a clear Bayesian interpretation for this, like in Theils method for regression?
- Is there any literature which does something similar?
- Is there a better way to approach this?