At 11 years of age, it is probably not cost effective to have allergy testing done but that is the only real way to know what type of food he can tolerate IF it is a food allergy. Since he is already on a limited ingredient dog food for dogs with allergies it may not be a FOOD allergy, but one related to the environment your dog is in.
I'm going to quickly go over the causes of hair loss and come back to allergies and suggest a few things that might help. Unfortunately, there are many conditions that can cause a dog to experience hair loss including ringworm, mange, hypothyroidism, cushing disease and allergies.
Ringworm can have a scaly reddish rash around the edges of the bare patches though there can be a rash in the bare area as well. This is relatively easy to diagnose, so I doubt the vets missed this. There are two types of mange and mange is actually caused by mites. Usually hair loss starts in specific areas of the body before spreading to other areas. This is diagnosed with a skin scraping and I really would have thought that one of the vets would have done this, but if they have not done it, it needs to be done to rule out parasites as the cause.
Allergies are a frequent cause of scratching and hair loss and an be caused by many different things as I've touched on. Below are websites that will go into more detail on these conditions.
Here is a website on Ringworm:
Here is two on Mange.
Hypothyroidism and cushing on these pages:
You can read up on allergies here:
Your dog is already on medications appropriate for skin dermatitis such as staph dermatitis or yeast dermatitis. Staph usually occurs on the lower regions of your pet and tends to have small pimple type bumps. Shampoo containing Chlorhexiderm and/or Oatmeal can help with this condition though it does not cure the allergy. Yeast typically shows as a greasy area that has a sweet musty odor. Sometimes the skin can become inflamed, darker and thickened due to itching. Yeast likes areas such as between toes, armpits and ears. Selsun Blue Shampoo can help with Yeast dermatitis. When shampooing, lather and leave on 15 minutes before rinsing. These shampoos are not meant to be a cure. Your shampoo might be appropriate for both types of dermatitis. However, many times owners are not told to leave the lather on for 15 minutes before relathering and rinsing. This gives the medication in the shampoo enough time to react with the bacteria or fungus on the skin. so you might want to adjust how you are bathing your dog.
There is also a condition called alopecia which is basically hair loss. You can read all about it in this article:
Some treatment methods for this are discussed on the following site. You might want to discuss this possibility with your vet.
I suspect that there is an allergy condition and possibly seasonal leading to a slight clearing of the skin before it starts over again. I suspect the itching from the allergic reaction leads to the skin dermatitis which leads to the hair loss and musty odor.
One thing that I do not see on the list of things that has been tried is atopica. It has been proved to be effective in some allergy cases where the allergen can not be identified or eliminated from the dog's environment such as pollens, grasses, etc. You can read more on this here:
So talk to your vet about adding atopica as a possible treatment and medications to clear any skin dermatitis currently present. Be sure to leave the shampoo on long enough for it to be effective before rinsing. It is worth at least trying at this point. I would normally recommend allergy testing to ensure it is allergies, but he is elderly and allergy testing can be expensive and probably wouldn't be cost efficient at this point in his life.
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