analysis for sparse contingency table

I am analyzing a 2x2x3x2 contingency table. The covariates are treatment (2 different kinds, TR1 and TR2, both binary), location of treatment (3 levels A, B, C), and the outcome is an event that either occurred or did not occur. The table is large compared to the sample size, and observing the event is rare, so the table is quite sparse. However, there does appear to be a strong interaction between the outcome and the covariates of interest.

Some of the suggested strategies were adding a small constant to all zero cells, and/or adjusting the degrees of freedom to account for the zeroes. I was planning on doing some kind of log-linear analysis. I am not too experienced with this kind of analysis so any help would be welcome. Thank you.
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I would also think of a log linear model.

I believe that the estimates would be OK but that the chi-squared test would maybe not be very well approximated with the chi squared distribution because of many zeros. (That could possibly be investigated with a simulation.)

Try it!