I have a dataset with 3726 observations. The values of '1' in the response variable is extremely rare: I have five 1's and 3721 0's. I used Firth's logistic regression to address this rarity (using logistf() and brglm() in R). I used 11 explanatory variables (mostly categorical). The results of Wald test show many of the variables significant with 90% confidence level. However, the LR test rejects the significance of all the variables, and almost all LR-based confidence intervals include 0. Why is there a huge difference and which one should I rely on? Thanks.
In this case, rely instead on the rule of thumb for logistic regression: you need 10-20 cases with the least frequent outcome per predictor variable in your model. With only 5 such cases, you perhaps should not be modeling this way at all.
In general, these tests will give the same result in the limit of a large number of cases, but there's no rule about how many is "large." My sense is that with a small number of cases the likelihood ratio (LR) test may be more reliable than the Wald test, which assumes a normal distribution of parameter value estimates. In your case, at least, the LR test has produced what seems to be the most reliable result.