One of my friends had a kids diagnosed with multiple food allergies before he was a year old. Now I want to check with her to see what they did. They may have done a scratch test.
My older kids both had blood tests for allergies by the time they were 4. Our ped was ambivalent at first, said we should just avoid what we thought they were reacting to, but that was hard to pinpoint. When we finally saw an actual allergist a few years ago, she told me that the most accurate diagnosis comes from the combination of scratch test and blood test.
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That said, as others have mentioned, there could be other things causing those symptoms. I have a couple of friends who swear that if they ate onions or spicy foods, baby had issues.