Winky, like many aspects health, it can be hard to establish direct cause and effect with bf/ff, but this does not mean (as RandA says) that no 'proper' studies can be done. She means you cannot 'randomise' a trial, ensuring that an entire cohort be ff and a matching cohort be bf, 'double blinding' it so neither the parents or the front-line researchers know which of the babies is bf and which ff until the results come in.
Obviously not possible :)
But observational studies can be done perfectly respectably and there is a whole discipline of epidemiology which looks at health in this way.
With infant feeding, there are many studies that show more ff babies need hospital treatment than bf babies.
www.ncbi.nlm.nih.gov/pubmed/17403827 is a good UK study showing this for diarrhoea and respiratory infection.
There were 15,000 plus in the study - nice big study - and it looked at babies under 8 mths old, all born healthy and at term.
Diarrhoea and respiratory infection are 'good' topics to study, as you are ruling out things like meningitis, or febrile convulsion, or conditions related to disability - you're using 2 fairly common conditions known to have a plausible link with feeding from previous research, and also things which have to be pretty bad to send a young baby into hospital, so knowing if something increases the risk is worth it.
Number-crunching led the researchers in this study to remove 'confounders' which might skew the results - so you are comparing only the feeding method, not the social background (for instance) of the babies. I think the study also removed smoking in the parents, but I would have to check.
They found that if all the babies in their study had been fully or partially breastfed, 84 per cent of hospital admissions for diarrhoea could have been avoided. 52 per cent of admissions for respiratory infection could have been avoided. This does not mean direct cause and effect on individuals and it does not mean 'no breastfed baby ever goes into hospital for diarrhoea or respiratory infection'. But it does show an increased risk if you ff.They also showed that the effect was stronger if you fully ff - partial breastfeeding exerted a protective effect, though full breastfeeding was better.
This still does not link cause and effect in individual babies. And hospitalisation for babies for these reasons is still unusual - they found 1.1 per cent for diarrhoea and 3.2 per cent for respiratory infection. Most babies, ff or bf, will not need to go into hospital for either of these reasons, in the UK, according to this study.
Hope this helps.