Massachusetts HIV-Testing Example

1. Massachusetts HIV-Testing Example Test Characteristics ELISA: wrong on 10% of infected samples wrong on 5% of uninfected samples Western Blot: wrong on 5% of infected samples wrong on 5% of uninfected samples Prevalence of virus: 0.2% ELISA Breakdown: Before ELISA neg ELISA pos infected 0.002 0.0002 0.0018 uninfected 0.998 0.9481 0.0499

2. Breakdown after both tests Before ELISA neg WB neg WB pos infected 0.002 0.0002 0.00009 0.00171 uninfected 0.998 0.9481 0.047405 0.002495 Out of 100,000 tested: 171 infected and test positive twice 249.5 uninfected and test positive twice 180 infected and test positive on ELISA 4990 uninfected and test positive on ELISA

3. Infected .002 .998 Uninfected

4. Infected & ELISA Pos 90% Infected Infected & ELISA Neg .002 10% .998 Uninfected

5. Infected & ELISA Pos 90% Infected Infected & ELISA Neg .002 10% Uninfected & ELISA Pos .998 5% Uninfected Uninfected & ELISA Neg 95%

6. Infected & WB Pos 95% Infected & ELISA Pos 90% 5% Infected & WB Neg Infected Infected & ELISA Neg .002 10% Uninfected & ELISA Pos .998 5% Uninfected Uninfected & ELISA Neg 95%

7. Infected & WB Pos 95% Infected & ELISA Pos Infected & WB Neg 90% 5% Infected Infected & ELISA Neg .002 10% Uninfected & WB Pos 5% Uninfected & ELISA Pos .998 5% 95% Uninfected Uninfected & WB Neg 95% Uninfected & ELISA Neg

8. Concept AIDS Example Die Example sample space “low-risk” population {1,2,3,4,5,6} event A = {infected individuals} A={2,4,6} B = {positive on ELISA} B={1,2} probability P(A) = 0.002 P(A) = 1/2 P(B) = 0.0517 P(B) = 1/3 complement not A = {not infected} not A = {1,3,5} intersection (A and B) = (A and B) = {infected and ELISA pos} {2} union (A or B) = (A or B) = {infected or ELISA pos} {1,2,4,6} mutually exclusive A1 = {test pos on both} A1 = {even} A2 = {test neg on both} A2 = {odd}

9. complement: P(not A) = 1 - P(A) Addition rule: P(A or B) = P(A) + P(B) - P(A and B) Conditional probability: P(A|B) = P(A and B)/P(B) Multiplication rule: P(A and B) = P(A)P(B|A)