Remember, correlation is not causation -- so if a study finds a connection between two things, it doesn't mean that one causes the other
There is really so little known about the overall situation here that any correlation, rather than being significant in itself, simply further argues the point that there are problems. Why very young children are being prescribed antibiotics would be the proper place to start.
My kids came out of the generation where doctors weren't aware of the dangers in over medicating and antibiotics were given often. As babies, both my children took a ridiculous amount of abx for things that were likely viral and not bacterial. Neither child has a weight issue. Never did. Not even remotely.
Maybe supersized portions, inactivity, HFCS and junk foods are more causative of weight issues.
"The happiest people don't have the best of everything.....they know how to make the best of everything!"
I have 2 sons and they both had a lot of ABX for ear infections and strep as kids. One was always very thin and the other was heavy by the time he was 6. Both are As. Both very tall, 6'3" and 6'6". Fed the same except the younger one was always very hungry. I remember having to start him on rice cereal very young even though I was nursing. Of course I knew nothing of the BTD back in the 1970s when they were born and they were fed the standard American fare. Wish I knew then what I know now....