Excargodog is 100% correct about Bayes theorem. For the uninitiated: if you were to administer a test on a population previously screened with certainty to be free of some defect, then all positive results for the defect are necessarily false and the test is worthless. The better the pre-screening is, the more worthless the test is.
But at first glance, reference to Bayes could actually hurt his argument. If the test is shown to approximate random elimination then this test cannot possibly discriminate based on real or perceived disabilities, even if that is the employer's intent.
Here's the thing though, pilots are pre-screened primarily only on select traits that pilots require. The Hogan may still effectively discriminate among other traits where pilots have no pre-screening. These traits are the ones that could prove troublesome for the company, since it would be harder to prove a business necessity for discrimination in these areas.
So Bayes reasoning does two things here:
1. Shows that the Hogan cannot effectively discriminate among traits for which pilots are typically screened. United isn't a pilot's first job, so pre-screening defeats the test.
2. Leaves open the possibility that the Hogan does effectively discriminate among other traits. These are the traits that no one else has screened. They are not a business necessity.
Folks, this is a phycological test. The employer will use the result to assess phycological competence. Anything below the desired competence is a perceived disability. Bayes can show that these disabilities occur only in areas that are not a business necessity. If I were on United's legal team I'd be worried about this. Excargodog's argument is strong even without reference to Bayes.
The only way out of this is for the company is to insist that the Hogan is not a psychological test. If the Hogan were part of a Bayesian filter and correct answers were found and weighted only through discipline statistics and previous tests, then it could be considered a statistical risk assessment and I believe that would be legal. As a phycological test, however, with answers chosen and weighted by smart people, this test is currently illegal.