I am running a logistic regression model and get very low predicted probabilities I am running a logistic model for catastrophic health expenditure (CHE) in Argentina. The sample size is 22500. I followed Xu et al. methodology to define CHE and adjusted for 8 socioeconomic variables. The are all significant.
Then I assessed the goodness of fit by looking at the H-L statistic (p-value>0.05) and the test indicates that the model fits the data well.
Then, to assess the discrimination of the fitted model I estimated the area under the ROC curve, which is 78.1%.
So far the model looks good. 
Unfortunately when looking at predicted probabilities and residuals I realized that the predicted probabilities are very low. They range between .0001645  and .3187172. Therefore I have very large residuals and leverage. Of course this also translates into a large number of influential observations.
Any suggestions why this could be happening?
I have some ideas but I am not sure they are the right explanation:
Is it possible that the proportion of households with CHE==1 is very low (3%) in the sample and this could be affecting the results?
Can I still use this model as a descriptive model of CHE in Argentina?
Thanks you!
Mercedes  
 A: An unbalanced data set (like you suggest when you say the proportion of households with CHE = 1 make up 3% of your data) will influence the results of a logistic regression.  See the answers to this question for instance. 
In short, the class imbalance in your data set affects only the intercept term.  If you are using your model to say something about you4 8 socioeconomic variables, you should be fine.  If it is for prediction, then you could consider lowering your probability threshold below 0.5.
A: My sense is that your model has significant variables mostly due to the large sample size. However the fit statistics are also quite good. When you say that the predicted probabilities are low, are they on the order of the 3% incidence of CHE?
It would seem that the model could be used, but perhaps other methods could be used as well and could allow more complete use of the available data. Have you considered using a Poisson or negative binomial distribution to fit the model? The former would allow you to analyze CHE rates/household if your data support events/unit time.
Given the low event rate, you would need to employ some caution in interpreting the results, as what is significant may not be important. Ultimately, the utility of your model will depend on your question. 
