Under-reporting of Lyme actually shouldn't impact a finding of no link as as far as I can tell. Despite the under-reporting, we still know where Lyme is more or less prevalent in the US, and I believe there will be very big differences due to climate and where ticks who carry Lyme tend to live. If prevalence of Alzheimers doesn't correlate with those big differences, that seems pretty clear.

(edit) Thinking about it more, I think the anecdote you mentioned is a bit different from the parent posts. The anecdote is about a misdiagnosis (he only had Lyme, not Alzheimers), while the parent posts are about people actually having both diseases.

The worry for me is the way this thread plays out is pretty much the textbook case for how pharmaceuticals manufacture demand for a drug.

1) Develop a drug (or find one to remarket). Find a disease that it treats or invent one.

2) Pump money into advocacy groups devoted to the disease.

3) Find research that supports the use of your drug.

4) The advocacy group spreads awareness of the disease, even if it's made up, and people start asking their doctors to test for it. Some may even receive a diagnosis.

5) Public pressure reaches the FDA, AMA, et al to consider the use of the drug for treatment.

3a) If research doesn't exist, find some impecunious person with a PhD and pay them to create some.