The dominance myth. So we meet again.
I spend a LOT of my time as a dog trainer helping my students understand that their dog's misbehavior isn't a result of their dog thinking he is dominant or the alpha. Many of my clients come to me and say "my dog is growling when I.... (touch their bone, pick their food up, wipe off their paws, etc.). Is it because they think they're the alpha?" And the answer is no, it is never, ever related to dominance. Ever.
So why are the dogs behaving aggressively in these contexts? Usually it's insecurity about whatever it is that is happening at the time the dog growls. The dog may be insecure about having his things taken away, or may have never learned to tolerate having his paws handled. A typical archaic, old-fashioned response to these behaviors would be that it's the humans job to put the dog in "his place", whether that be by scruffing the dog, alpha rolling the dog, correcting the dog with a punitive collar, or "helicoptering" a dog (yes, there are actually trainers out there who string the dog up by it's collar and swing them around as a form of training. Yes, knowing this makes me feel ill too.) Trainers who utilize alpha/pack theory believe that the dog needs to learn that the human is the true alpha, and then the dog will acquiesce to anything it's owner wants.
What really happens is the aggression may escalate, and cause the situation to become much more dangerous than it had ever been before. When you match conflict with more conflict, what do we expect to happen? And yet it's frighteningly common advice given to dog guardians.
So what to do instead? Work to change the dog's emotions using classical conditioning and desensitization (some nice articles on this can be found here, here, and here.) Teach them that they have no reason to growl, because you approaching their coveted resource, or touching their paws predicts good things happening, not bad. If we punish an aggressive behavior, we are simply trying to treat a symptom. We need to address the underlying problem that is causing the symptom, and that is insecurity and anxiety. NOT dominance.
So, does a biting dog think that he is the "alpha" or is dominant in a relationship? Absolutely not. A dog that bites is more often than not worried or overly stressed by a situation, and feels that he has to act to protect himself. Anyone who would lead you to believe otherwise is sorely mistaken.
For more information on the incorrect use of dominance in dog training you can check out this page on the late Sophia Yin's Website, this page on Victoria Stillwell's website, and this article written by Pat Miller for the Whole Dog Journal.