The short answer is that it is impractical to appeal the loss of a dog bite case. This answer assumes that the case involved a human who was bitten by a dog, there was a claim for monetary compensation, there was no settlement, the case went to trial, and the verdict was something that favored the defendant who allegedly was responsible for the damages to the victim. In other cases, the general analysis provided in the following paragraphs might not apply. For example, if the case was a so-called "dangerous dog hearing," then it was heard by a "dog court" judge, and the appeal process is entirely different.
To appeal a case that ended in a judgment favoring the defendant, there have to be grounds. The grounds are found in the court records. An attorney must be retained to review the court records, the transcript of the trial, and the content of the evidence admitted and excluded from the trial. The attorneys who do this type of work charge by the hour. This work is not done on a contingency fee basis. It is, to put it bluntly, expensive. Just the review of the record is usually thousands of dollars.
After that, the attorney who handles the appeal also will charge by the hour. A complicated appeal can take weeks of work. Again, this part is expensive, meaning in the thousands. By time the appeal is concluded, it probably would cost tens of thousands of dollars.
Appeals usually are lost. The vast majority of the time, the lower court did not make a mistake and therefore the appeal is fruitless. But this too can cost you, because you might be ordered to pay the opposing party's costs.
Even if an appeal is won, the higher court might be simply ordering a new trial. Whatever the former trial cost, the new trial could cost.
Additionally, the new trial will require a trial attorney. The former attorney fully performed his or her obligation and therefore is entitled to a fee. The new attorney likewise will want a fee. Even if it is the same attorney, the work is being done twice and therefore a higher fee of some sort is required.
All of this takes years. It is expensive. It probably will result in losing. For that reason, appealing a dog bite case is not a practical option.