I have 6 out of 167 cases of cancer in the dependent variable. I would evaluate if three independent variables predicted the cancer. Are 6 cases enough? Is there a rule to determine this? Is there difference if the indipendent variables is nominal or numeric?
I think there is no minimal number of "cases" to perform logistic regression (well at least you need to have 2 classes.).
The only downside is that, if you have a very imbalanced data, you may model minority poorly.
Also you may get some warnings from
glm, if you have perfect separably data, so you may consider adding regularization.