It's really difficult to do this. Partly because people are addicted to the food they're allergic to. The food they just couldn't possibly stop eating on an elimination diet, is likely the food that's causing the problem.
I tried an elimination diet in 1997 and I didn't have any reactions to food challenges. Why did it fail? It wasn't gluten-free (!) for one thing. Also, I didn't quit eating fructose on the elimination diet, because I "needed" it. It turned out after a lot of trouble, I found out I actually had an allergic addiction to the fructose, because it's made from corn and I have a severe corn reaction. I didn't believe that the fructose could cause a problem, because it only has 20 ppm of corn protein - I didn't think a trace could matter. But traces of corn turned out to be crucial.
So, you should take traces of allergens seriously on an elimination diet.
A really careful elimination diet. Some of the foods you can eat on this diet.
If this seems too difficult, a less careful version will probably work.
After a week of the elimination diet, you reintroduce foods one by one to see if you're have a reaction to them. This is called a food challenge. First you eat a small amount of the food. Then wait a day or so. If nothing bad happens, try eating a larger amount. Some people have delayed reactions, so wait for a couple of days before trying a food challenge with another food. If you have a reaction a long time after eating the food, I'd try it again to verify that it really is your food challenge causing the reaction. There are cross-reactions between foods, so if you've have a definite reaction to a food, it may not be worth the risk trying other foods that probably cross-react with it.
If you have ever had an immediate reaction to a food, food challenges should be done with medical supervision. Or if you have serious problems that went away on the elimination diet, and might come back with the food challenge.
Even if you don't have a reaction to gluten grains (wheat, rye, barley) or milk, try a gluten/dairy free diet for 3 months. Not everybody who is gluten intolerant has an obvious reaction to gluten, and it may take 3 months to see an improvement on a gluten/dairy free diet. The gluten/dairy free diet is especially used for psychological symptoms. Supplement with Ca-Mg.