Strongly typed languages require of me to do more work upfront, to satisfy their type checker. They must necessarily reject programs that would work correctly. In this process a lot of mistakes are eliminated, and this gives me more confidence that the result will work. I like that way of working. But does it produce more robust code? Is it more productive? It feels like it, but that doesn't mean it's true.
Which basically settles the question for me as a programmer, anyway. Eliminating the possibility of a class of run time failures -- how can that not be a good thing?
The question to me is not whether type checkers are useful tools, but at what point they become a hindrance. If I may rephrase your question: The programs rejected by the type checker, how can they not be bad programs?
http://macbeth.cs.ucdavis.edu/lang_study.pdf
TL.DR. Looks like FP reduces bugs, and static typed FP reduces them a bit more, but there isn't enough data for the more interesting fine-granied conclusions.
Also Perl results are interesting (unsettling?).
Anecdotes are the best we have. Far better than a pseudoscience with agenda.
The effect of type systems is probably not that big. Otherwise the proponents of type systems would have had an easy time proving it.