A great theorem in mathematics proves rigorously that some things can never be proved rigorously. It seems that in public health too we will never know the answer to some important questions, no matter how much they are researched.
The question of whether home births are safe has been around for twenty years or more, but still the debate rumbles on. Here is the latest attempt to answer the question; and here is an editorial shredding the attempt (though some of the criticisms seem unfounded to me).
Strictly speaking the paper compares outcomes in a group delivered at home by independent (i.e. non NHS) midwives versus a group delivered in NHS hospitals. Note that most home deliveries are by NHS midwives so this study looks at a specific detailed question - the independent midwife - rather than the general problem of home birth.
And of course the problem is that no group can be found to match the sort of woman who wants a home delivery by an independent midwife. To my eye the two groups look comparable on most measures (Table 1) but the expert editorialists think not; and the authors themselves comment on some important differences in their discussion section.
So in the end (a) we find differences but (b) we have not compared like with like, leaving us no wiser than before.