<OOPS, RECAF has a 15% odds of missing that positive test and absolutely NOT the small percentage you quoted. Now, do you finally understand?>
I do, very well... I understand that you have no idea of what you are talking about but you still want to make a point.
Your mistake is first to confuse odds with specificity. That is wrong for too many reasons to explain, but I'll give you just 3:
1) As far as I can tell, the company never provided odds values and for good reason, they are irrelevant in the evaluation of a test. All they provided was number of cases, specificity and sensitivity values; it is you who by a sleight of hand turned specificity into odds.
2) Odds do not relate to the testing method: If the odds to get tails by tossing a coin is 50%, that does not change if it is dark and you misread the outcome 15% of the times when it is tails but not that much when it is heads. You cannot say that the odds of getting a tail is 35% because you misread the other 15%; they are still 50%.
3) Odds (the way you used the term, which is different that what it means in statistics) depend on the prevalence (number of undetected cancer patients) in a given population. Say the prevalence of cancer in children was 0.1% and in elders it was 2% (I am making these numbers up). Test 100,000 elders and 100,000 children. In children, the test will detect 85% of the 100 children who have cancer (that is 85 children) plus 5% of the remaining 99,900 children who do not have cancer (roughly 5,000 children). The 'odds' of being right in that example are 85:100,000 = 1:1,176. Among the elder, in 100,000 people you will have 2,000 cancers and therefore the 'odds' would be 2,000:100,000 or 1:50. The 'odds' in children and elders are very different and yet, they were found with exactly the same test. The difference in prevalence does not need to be a natural phenomenon as shown with elders and children; it can be caused by completely artificial circumstances. For example a doctor with an office on a 3rd floor with no elevator will see very few patients with arthritis. The prevalence of arthritic patients among his patients will be lower than average.
That is why you cannot use odds to evaluate a test and instead you use sensitivity and specificity.