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An excellent systematic review of the health impact of breast / formula feeding in developed countries. This review is held very high in the academic community. They reviewed over 9000 studies.
In short they found that in developed countries breastfeeding was associated with a lower risk of acute otitis media, non-specific gastroenteritis, severe lower respiratory tract infections, atopic dermatitis, asthma (young children), obesity, type 1 and 2 diabetes, childhood leukemia, sudden infant death syndrome (SIDS), and necrotizing enterocolitis. For mothers: a reduced risk of type 2 diabetes, breast, and ovarian cancer.
In short that means that formula feeding is associated with greater risk of illness. That is not the same as causing, or a definitive. It just raises the risk. Plenty of breastfed babies will get allergies. Just more formula fed babies will. However that does not by any means mean that all formula fed babies will get allergies or that the risk is really really high.
It is a case of knowing the risks and working out what is best for your family. For many mums this will mean that formula is best when they have weighed up the situation. However it doesnt mean that formula milk is the same as breast milk or breastfeeding is not superior to formula milk - however nothing happens in isolation. A baby may well be better off if his mother is depressed, in pain, resenting him etc. What is sad that there is not better support to enable women to overcome these problems.
You cannot conduct a randomised controlled trial on breastfeeding as we know too much about the benefits to suggest women formula feed. It certainly will never be blind or double blind
as I think women would notice whether they are breastfeeding or not (feeding from the breast has advantages over being fed expressed milk.
Most good quality studies do try and control for effects - they will partial out the effect of education, age, background etc. Although it will never be perfect.