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Mar 07, 2024 •157 likes •283 views

Conditional probabilities From Data to Insight Dr. etinkaya-Rundel July 18, 2016 DTap vaccine 2 DTaP vaccine Jackson et al. (2013) wanted to know whether it is better to give the diphtheria, tetanus and pertussis (DTaP) vaccine in

  1. Conditional probabilities From Data to Insight Dr. Çetinkaya-Rundel July 18, 2016

  2. DTap vaccine 2

  3. DTaP vaccine ‣ Jackson et al. (2013) wanted to know whether it is better to give the diphtheria, tetanus and pertussis (DTaP) vaccine in either the thigh or the arm, so they collected data on severe reactions to this vaccine in children aged 3 to 6 years old. ‣ Is the probability of a severe reaction higher for vaccines in the thigh or the arm? No severe Severe Total reaction reaction Thigh 4758 30 4788 Arm 8840 76 8916 Total 13598 106 13704 3

  4. No severe Severe Total reaction reaction Thigh 4758 30 4788 Arm 8840 76 8916 Total 13598 106 13704 P(severe reaction | thigh) = 30 / 4788 ≈ 0.0063 P(severe reaction arm) = 76 / 8916 ≈ 0.0085 4

  5. Bayes’ rule 5

  6. HIV testing with ELISA 6

  7. HIV testing with ELISA ‣ ELISA P(+ | HIV) = 0.93 ‣ Sensitivity (true positive): 93% P(- | no HIV) = 0.99 ‣ Specificity (true negative): 99% ‣ Western blot ‣ Sensitivity: 99.9% ‣ Specificity: 99.1% P(HIV) = 0.00148 ‣ Prevalance: 1.48 / 1000 ‣ P(has HIV | ELISA +) = ? 7

  8. Prior to any testing, what probability should be assigned to a recruit having HIV? P(HIV) = 0.00148 8

  9. When a recruit goes through HIV screening there are two competing claims: recruit has HIV and recruit doesn't have HIV. If the ELISA yields a positive result, what is the probability this recruit has HIV? + P(+ | HIV) P(HIV and +) P(HIV) 0.93 0.00148 x 0.93 0.00148 = 0.0013764 HIV 0.07 - P(- | HIV) P(+ | no HIV) + P(no HIV and +) no HIV 0.01 0.99852 0.99852 x 0.01 = 0.0099852 P(no HIV) - 0.99 P(- | no HIV) 9

  10. When a recruit goes through HIV screening there are two competing claims: recruit has HIV and recruit doesn't have HIV. If the ELISA yields a positive result, what is the probability this recruit has HIV? P(HIV and +) P(HIV | +) = P(+) 0.0013764 0.0013764 + 0.0099852 ≈ 0.12 = 10

  11. Drug testing [time permitting] 11

  12. Drug testing [time permitting] ‣ Most companies drug test their employees before they start employment, and sometimes regularly during their employment as well. ‣ Suppose that a drug test for an illegal drugs is 97% accurate in the case of a user of that drug, and 92% accurate in the case of a non- user for that drug. ‣ Suppose also that 5% of the entire population uses that drug. ‣ You are the hiring manager at a company that drug tests their employees. You have recently decided to hire a new employee. The prospective employee gets drug tested, and the test comes out to be positive. What is the probability that they are actually a user for the drug? 12

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