Although not having any statistics to bring up, I have experience carrying a kid with me (my stepson), mostly in a rear rack home-made "chair" (between 5 and 7 years old), with two test-rides in a trailing bike and lots of km on a tandem (last three years).
I have no experience with kids smaller than 4 years old.
I think some PRINCIPLES are valid for whatever child-carrying system, and these would be, in my opinion:
- The closer the kid is to the parent, the better;
- The closer the bid is to "the center of the bike", the better (incidentally, this ends up being equivalent to the first principle);
- Kids should not be ejected from their seat;
- Kids should not remain "trapped" in their seat after a fall.
Besides that, I think there is a compulsory speed limit depending on age. For example, for under one year old, I think 10mph is the absolute maximum acceptable speed under ideal conditions.
For toddlers and very young babies, I think the main concern is protection from mechanical shock, and ability to keep it under your eyes, preferrably with eye contact. In the event of braking, the seat must provide support. In that sense, the following "concepts" seems UTTERLY INAPPROPRIATE:
Instead of this freak, a solution similar to the photo below (a plain ol' bakfiets) is much better for the adult and for the kid, not to mention that it's almost impossible to eject the kid or use it as a bumper:
For kids that can hold a steady seating posture, but are still imature or unpredictable, I think the classic "behind the stem" position, although uncomfortable for the rider, is the preferred one because the kid surely feels protected, conversation is direct, and the kid have "nowhere to fall" without the adult holding her. That is a position with a fairly low speed limit, too, and I think a front-suspension/fat-tire combination can ease things a lot.
Then, from about 4yo until 7yo, the last step before the kid can ride his own bike or a tandem would be a rear seat. This is a situation where is desireable to allow a bit more of clearance between the kid's face and the adult's back, due to braking and also visibility. Communication is a bit harder, but the overall sensation is quite safe and stable, PROVIDED the bike has the "natural talent" to carry weight on the rear (which rule out almost any sport/race oriented bike). A longer seat stay, rigid rack brazeons, and a generous trekking rack are most desireable. Long tail bikes are the ideal solution, but shouldn't be actually necessary.
Until here, I haven't come close to answering your question, but from the most common alternatives, the one I wouldn't adopt at all (although I have some biker friends who have done so) is the baby trailer, Burley style, because if anything happens to the kid (even if she calls for help), she is too far from you, and in a position that you cannot reach untill a complete stop, coming out of the bike, etc. Not to mention the WORST problem: the kid being a bumper, gometrically WAY BELOW the average height of CAR bumpers... (I would not like to see the statistics of car/baby-trailer collisions, but the very scene of it creeps me).