Comparing the complication rate for  home and hospital  births by looking at past  outcomes  has no  validity whatsoever, because one  is comparing apples oranges: the two groups of women are not  comparable. Those with known or perceived risk factors will inevitably choose  hospital births. And those  who develop complications will be moved to the hospital and  give birth  there.

One can imagine any number of situations in which not being at home would be an added risk:  sudden and massive bleeding; premature separation of the placenta; a baby that doesn't breathe  and needs resuscitation, etc.  Since we know these things happen, and can happen suddenly, it is statistically impossible for  home  births  to be safe  as  hospital births.   Relatively safe,  yes; "as" safe, certainly not.