How can I know if my dog or cat is allergic to food ?

Food allergy is an abnormal immune response against one or more food ingredients.
Food intolerance is also an abnormal reaction to food, but not of immunologic nature (ie without the involvement of antibodies).
Often in Veterinary medicine we are not able to make a real distinction between allergy and intolerance and we make a generic diagnosis including both (RAC, eg. Adverse Reaction to Food).
The most common causes of food allergy in dogs and cats are the protein ingredients of the food  (eg.  beef, chicken, milk and dairy products, fish, soy etc.).
Importantly, the allergy to one food does not depend on its quality and it is a strictly individual condition (other animals can eat it without any problem). It should also be remembered that often the offending food has been eaten for months or years before inducing allergy (sensitization period).
Sometimes the animals, like humans, have more than one allergy at the same time (eg. food allergy + atopic dermatitis, or + flea allergy ).
The most common symptom in dogs and cats is itching, sometimes associated with gastrointestinal symptoms (vomiting, loose stools, frequent stools, flatulence, poor growth etc.).  In cats itching may be associated with typical skin lesions (see factsheet "Cat is not a small dog").
HOW TO MAKE A DIAGNOSIS
1st step : the elimination diet
Unfortunately allergy testings for food (skintest or serological test) do not give valid results in the dog and cat.
Thus to diagnose a food allergy you need to give an elimination diet (or deprivation or hypoallergenic diet), that is  a diet made of only two ingredients (a protein source and a carbohydrate source) that the animal has not eaten during previous six months.
It is therefore necessary that pet owner patiently fill a list of all the main ingredients of the foods that his dog or cat has eaten in the previous months (for commercial foods read the labels) and try two new ingredients.
These ingredients can be provided with a home preparation or with one of several commercial diets made for this purpose (ask your veterinarian a list).   On sale there is also another kind of diet we can use: the hydrolyzed diets, in which the ingredients molecules are reduced to a minimum size, theoretically not able to trigger the allergy.

Any type of diet is chosen, it should be administered until clinical response (ie the disappearance or noteworthy reduction of itching) or for a period of two months (some animals respond in a few weeks, others require a maximum time of 10 weeks).
It 's important that nothing, out of the diet, is administered to the animal (including treats, snacks, buffalo bones, flavored drugs etc.).  All family members should be sensitized to this purpose.
If after two months  the diet, although carried out with precision, does not cause any reduction of itching, food allergy is excluded as a diagnosis.
2nd step : the confirmation diet
If there is a positive response to elimination diet (disappearance of itching), to know which is the food the animal is allergic to, we can make confirmation (or provocation) diet.   That means re-administering 50-100 grams per meal of old ingredients , one at a time, and see which one makes itch come back: this usually occurs within a few days, so every ingredient can be reintroduced for max 7 -10 days, and then move on to the next. When itching comes back, that ingredient is suspended and will not ever administered to the animal.