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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

  1. Is this solution for handling the zero-counts justifiable?
  2. Is there a clear Bayesian interpretation for this, like in Theils method for regression?
  3. Is there any literature which does something similar?
  4. Is there a better way to approach this?
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