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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1691.0. "Probability and AIDS screening" by AUSSIE::GARSON () Thu Nov 05 1992 04:11

Extracted without permission
    from an article by G.A.Watterson, Monash University, Australia

In order to isolate AIDS sufferers or carriers, it has been suggested in the
U.S.A. that the Government should routinely test all immigrants, all prisoners,
and all hospital patients, for AIDS. The Secretary for Health and Human
Services also announced, in June 1987, a plan to test 45000 randomly selected
people from the general population. Similar proposals have been mentioned in
Australia.

But there is one great difficulty with such proposals. If a person was tested
and diagnosed as having AIDS, there is a good chance that the diagnosis would
be wrong. If the person was then isolated, cut off from friends and family,
probably forced to give up employment, and if this isolation was by mistake,
then not only would a terrible injustice be done but no doubt such a person
would also qualify for very substantial financial recompense, via law suits
etc.

Why is there a good chance that the diagnosis would be wrong?

Denote the probability that an individual has AIDS by p(A).

If a person were tested, there might be very high probabilities that the test
would indicate the correct diagnosis. Denote the probability that an individual
tests positive given that s/he has AIDS by p(+|A) and the probability that an
individual tests negative given that s/he does not have AIDS by p(-|~A).

The following estimates are given in an article in the Journal of the American
Medical Association (1985).

p(A)    = 0.006
p(+|A)  = 0.977
p(-|~A) = 0.926

What is the probability of a false positive i.e. p(~A|+).

Making use of Bayes' Theorem...

          p(~A ^ +)
p(~A|+) = ---------
            p(+)

p(~A ^ +) = p(~A) * p(+|~A)
          = (1 - p(A)) * (1 - p(-|~A))
          = 0.994 * 0.074
          = 0.073556

p(+) = p(~A ^ +) + p(A ^ +)
     = 0.073556 + p(A) * p(+|A)
     = 0.073556 + 0.006 * 0.977
     = 0.079418

Thus p(~A|+) = 0.073556/0.079418 = 0.926

This shows that in roughly 9 cases out of 10, when the diagnosis was positive,
it would be wrong!

[DG: Indeed further analysis shows that test 'reliability', 0.977 and 0.926
above, would have to increase to somewhere around 99.9% to get false positives
down even to around 1 in 10.

P.S. This is not intended as a discussion on the merits or otherwise of AIDS
screening - only to illustrate what probability can tell us about it.]
T.RTitleUserPersonal
Name		DateLines
1691.1 associated epidemilogical analysesSGOUTL::BELDIN_RAlls well that ends: 65 daysThu Nov 05 1992 07:2816
    This kind of analysis is standard fare in epidemiological studies.  It
    seems that popular descriptions of testing for disease and/or infection
    has never made it clear when "weak" tests are being discussed.  
    
    Another analysis that is needed when one is considering immunization is
    the probability of infection by the innoculating agent.  For that, one
    needs to compare the probability of infection given innoculation to the
    probability of infection given non-innoculation.  
    
    Twenty years ago, the US stopped smallpox vaccinations because the risk
    of infection from the vaccination process was higher than without it. 
    
    Today, we are debating starting up again because the statistics have
    reversed.
    
    Dick
1691.2Its back?SSAG::LARYLaughter & hope & a sock in the eyeFri Nov 06 1992 15:5416
>    Twenty years ago, the US stopped smallpox vaccinations because the risk
>    of infection from the vaccination process was higher than without it. 
>    
>    Today, we are debating starting up again because the statistics have
>    reversed.


Are you sure about this? I thought that about 2 years ago the U.N. declared
Total Victory over smallpox, meaning that there had been no new cases in the
*entire world* for some number of months and (since, I guess, the smallpox
virus can't survive outside human hosts) therefore smallpox was No More (except
in research labs)...

Perhaps you mean some other disease, or perhaps the U.N. was fulla s2t...

							Richie
1691.3Earlier note.CADSYS::COOPERTopher CooperFri Nov 06 1992 16:443
    For what it is worth, I discussed this effect in notes 1291.1 and .2.

				Topher
1691.4STAR::ABBASIwhat happened to family values ?Fri Nov 06 1992 23:337
    
    I think if a virus disappears from people, like smallbox apparently did,
    I wish that they dont even keep samples of it alive in labs, and
    just kill the last damm ones , one and for all. one never knows , it
    might spread again by accident or something like that.
1691.5Is this the right conference?CADSYS::COOPERTopher CooperSat Nov 07 1992 10:2417
    In fact, it did.  The last reported case of the virulent form of
    smallpox (which, I think is the only one declared "extinct") was in a
    laboratory worker in, I think, Britain.  After that enforcement of the
    containment standards was increased and the number of places where the
    virus was contained in inactive form was (as planned all along)
    decreased.  I believe that there are only one or two sites left in the
    world where it is stored.  This is believed safer than the destruction
    of all stocks, since they might be needed for experimental purposes if
    (a) the disease re-emerges from an unknown resevoir (one of the reasons
    that virulent smallpox was able to be eradicated was because it cannot
    infect non-human animals, which could then re-infect the human
    population); (b) a different but related disease emerges; or (c)
    smallpox re-mutates from its root stock (probably cow-pox).  I'm told
    that smallpox is a poor candidate for biogical weapons (too slow to
    incompacitate) so, hopefuly, we don't have that spectre to worry about.

					Topher
1691.6correctionSGOUTL::BELDIN_RFree at last in 63 daysFri Nov 13 1992 09:115
    re .1
    
    Just checked.  The argument is about measles, not smallpox.
    
    Dick